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Metamath Proof Explorer

Theorem rlocbas

Description: The base set of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025)

Ref Expression
Hypotheses rlocbas.b 
|- B = ( Base ` R )
rlocbas.1 
|- .0. = ( 0g ` R )
rlocbas.2 
|- .x. = ( .r ` R )
rlocbas.3 
|- .- = ( -g ` R )
rlocbas.w 
|- W = ( B X. S )
rlocbas.l 
|- L = ( R RLocal S )
rlocbas.4 
|- .~ = ( R ~RL S )
rlocbas.r 
|- ( ph -> R e. V )
rlocbas.s 
|- ( ph -> S C_ B )
Assertion rlocbas 
|- ( ph -> ( W /. .~ ) = ( Base ` L ) )

Proof

Step Hyp Ref Expression
1 rlocbas.b 
 |-  B = ( Base ` R )
2 rlocbas.1 
 |-  .0. = ( 0g ` R )
3 rlocbas.2 
 |-  .x. = ( .r ` R )
4 rlocbas.3 
 |-  .- = ( -g ` R )
5 rlocbas.w 
 |-  W = ( B X. S )
6 rlocbas.l 
 |-  L = ( R RLocal S )
7 rlocbas.4 
 |-  .~ = ( R ~RL S )
8 rlocbas.r 
 |-  ( ph -> R e. V )
9 rlocbas.s 
 |-  ( ph -> S C_ B )
10 eqid 
 |-  ( +g ` R ) = ( +g ` R )
11 eqid 
 |-  ( le ` R ) = ( le ` R )
12 eqid 
 |-  ( Scalar ` R ) = ( Scalar ` R )
13 eqid 
 |-  ( Base ` ( Scalar ` R ) ) = ( Base ` ( Scalar ` R ) )
14 eqid 
 |-  ( .s ` R ) = ( .s ` R )
15 eqid 
 |-  ( TopSet ` R ) = ( TopSet ` R )
16 eqid 
 |-  ( dist ` R ) = ( dist ` R )
17 eqid 
 |-  ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) = ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. )
18 eqid 
 |-  ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) = ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. )
19 eqid 
 |-  ( k e. ( Base ` ( Scalar ` R ) ) , a e. W |-> <. ( k ( .s ` R ) ( 1st ` a ) ) , ( 2nd ` a ) >. ) = ( k e. ( Base ` ( Scalar ` R ) ) , a e. W |-> <. ( k ( .s ` R ) ( 1st ` a ) ) , ( 2nd ` a ) >. )
20 eqid 
 |-  { <. a , b >. | ( ( a e. W /\ b e. W ) /\ ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( le ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) } = { <. a , b >. | ( ( a e. W /\ b e. W ) /\ ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( le ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) }
21 eqid 
 |-  ( a e. W , b e. W |-> ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( dist ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) ) = ( a e. W , b e. W |-> ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( dist ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) )
22 1 2 3 4 10 11 12 13 14 5 7 15 16 17 18 19 20 21 8 9 rlocval 
 |-  ( ph -> ( R RLocal S ) = ( ( ( { <. ( Base ` ndx ) , W >. , <. ( +g ` ndx ) , ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. , <. ( .r ` ndx ) , ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( k e. ( Base ` ( Scalar ` R ) ) , a e. W |-> <. ( k ( .s ` R ) ( 1st ` a ) ) , ( 2nd ` a ) >. ) >. , <. ( .i ` ndx ) , (/) >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopSet ` R ) tX ( ( TopSet ` R ) |`t S ) ) >. , <. ( le ` ndx ) , { <. a , b >. | ( ( a e. W /\ b e. W ) /\ ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( le ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) } >. , <. ( dist ` ndx ) , ( a e. W , b e. W |-> ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( dist ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) ) >. } ) /s .~ ) )
23 6 22 eqtrid 
 |-  ( ph -> L = ( ( ( { <. ( Base ` ndx ) , W >. , <. ( +g ` ndx ) , ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. , <. ( .r ` ndx ) , ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( k e. ( Base ` ( Scalar ` R ) ) , a e. W |-> <. ( k ( .s ` R ) ( 1st ` a ) ) , ( 2nd ` a ) >. ) >. , <. ( .i ` ndx ) , (/) >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopSet ` R ) tX ( ( TopSet ` R ) |`t S ) ) >. , <. ( le ` ndx ) , { <. a , b >. | ( ( a e. W /\ b e. W ) /\ ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( le ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) } >. , <. ( dist ` ndx ) , ( a e. W , b e. W |-> ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( dist ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) ) >. } ) /s .~ ) )
24 eqidd 
 |-  ( ph -> ( ( { <. ( Base ` ndx ) , W >. , <. ( +g ` ndx ) , ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. , <. ( .r ` ndx ) , ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( k e. ( Base ` ( Scalar ` R ) ) , a e. W |-> <. ( k ( .s ` R ) ( 1st ` a ) ) , ( 2nd ` a ) >. ) >. , <. ( .i ` ndx ) , (/) >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopSet ` R ) tX ( ( TopSet ` R ) |`t S ) ) >. , <. ( le ` ndx ) , { <. a , b >. | ( ( a e. W /\ b e. W ) /\ ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( le ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) } >. , <. ( dist ` ndx ) , ( a e. W , b e. W |-> ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( dist ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) ) >. } ) = ( ( { <. ( Base ` ndx ) , W >. , <. ( +g ` ndx ) , ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. , <. ( .r ` ndx ) , ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( k e. ( Base ` ( Scalar ` R ) ) , a e. W |-> <. ( k ( .s ` R ) ( 1st ` a ) ) , ( 2nd ` a ) >. ) >. , <. ( .i ` ndx ) , (/) >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopSet ` R ) tX ( ( TopSet ` R ) |`t S ) ) >. , <. ( le ` ndx ) , { <. a , b >. | ( ( a e. W /\ b e. W ) /\ ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( le ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) } >. , <. ( dist ` ndx ) , ( a e. W , b e. W |-> ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( dist ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) ) >. } ) )
25 eqid 
 |-  ( ( { <. ( Base ` ndx ) , W >. , <. ( +g ` ndx ) , ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. , <. ( .r ` ndx ) , ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( k e. ( Base ` ( Scalar ` R ) ) , a e. W |-> <. ( k ( .s ` R ) ( 1st ` a ) ) , ( 2nd ` a ) >. ) >. , <. ( .i ` ndx ) , (/) >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopSet ` R ) tX ( ( TopSet ` R ) |`t S ) ) >. , <. ( le ` ndx ) , { <. a , b >. | ( ( a e. W /\ b e. W ) /\ ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( le ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) } >. , <. ( dist ` ndx ) , ( a e. W , b e. W |-> ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( dist ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) ) >. } ) = ( ( { <. ( Base ` ndx ) , W >. , <. ( +g ` ndx ) , ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. , <. ( .r ` ndx ) , ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( k e. ( Base ` ( Scalar ` R ) ) , a e. W |-> <. ( k ( .s ` R ) ( 1st ` a ) ) , ( 2nd ` a ) >. ) >. , <. ( .i ` ndx ) , (/) >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopSet ` R ) tX ( ( TopSet ` R ) |`t S ) ) >. , <. ( le ` ndx ) , { <. a , b >. | ( ( a e. W /\ b e. W ) /\ ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( le ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) } >. , <. ( dist ` ndx ) , ( a e. W , b e. W |-> ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( dist ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) ) >. } )
26 25 imasvalstr 
 |-  ( ( { <. ( Base ` ndx ) , W >. , <. ( +g ` ndx ) , ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. , <. ( .r ` ndx ) , ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( k e. ( Base ` ( Scalar ` R ) ) , a e. W |-> <. ( k ( .s ` R ) ( 1st ` a ) ) , ( 2nd ` a ) >. ) >. , <. ( .i ` ndx ) , (/) >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopSet ` R ) tX ( ( TopSet ` R ) |`t S ) ) >. , <. ( le ` ndx ) , { <. a , b >. | ( ( a e. W /\ b e. W ) /\ ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( le ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) } >. , <. ( dist ` ndx ) , ( a e. W , b e. W |-> ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( dist ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) ) >. } ) Struct <. 1 , ; 1 2 >.
27 baseid 
 |-  Base = Slot ( Base ` ndx )
28 snsstp1 
 |-  { <. ( Base ` ndx ) , W >. } C_ { <. ( Base ` ndx ) , W >. , <. ( +g ` ndx ) , ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. , <. ( .r ` ndx ) , ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. }
29 ssun1 
 |-  { <. ( Base ` ndx ) , W >. , <. ( +g ` ndx ) , ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. , <. ( .r ` ndx ) , ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. } C_ ( { <. ( Base ` ndx ) , W >. , <. ( +g ` ndx ) , ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. , <. ( .r ` ndx ) , ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( k e. ( Base ` ( Scalar ` R ) ) , a e. W |-> <. ( k ( .s ` R ) ( 1st ` a ) ) , ( 2nd ` a ) >. ) >. , <. ( .i ` ndx ) , (/) >. } )
30 ssun1 
 |-  ( { <. ( Base ` ndx ) , W >. , <. ( +g ` ndx ) , ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. , <. ( .r ` ndx ) , ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( k e. ( Base ` ( Scalar ` R ) ) , a e. W |-> <. ( k ( .s ` R ) ( 1st ` a ) ) , ( 2nd ` a ) >. ) >. , <. ( .i ` ndx ) , (/) >. } ) C_ ( ( { <. ( Base ` ndx ) , W >. , <. ( +g ` ndx ) , ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. , <. ( .r ` ndx ) , ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( k e. ( Base ` ( Scalar ` R ) ) , a e. W |-> <. ( k ( .s ` R ) ( 1st ` a ) ) , ( 2nd ` a ) >. ) >. , <. ( .i ` ndx ) , (/) >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopSet ` R ) tX ( ( TopSet ` R ) |`t S ) ) >. , <. ( le ` ndx ) , { <. a , b >. | ( ( a e. W /\ b e. W ) /\ ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( le ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) } >. , <. ( dist ` ndx ) , ( a e. W , b e. W |-> ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( dist ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) ) >. } )
31 29 30 sstri 
 |-  { <. ( Base ` ndx ) , W >. , <. ( +g ` ndx ) , ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. , <. ( .r ` ndx ) , ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. } C_ ( ( { <. ( Base ` ndx ) , W >. , <. ( +g ` ndx ) , ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. , <. ( .r ` ndx ) , ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( k e. ( Base ` ( Scalar ` R ) ) , a e. W |-> <. ( k ( .s ` R ) ( 1st ` a ) ) , ( 2nd ` a ) >. ) >. , <. ( .i ` ndx ) , (/) >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopSet ` R ) tX ( ( TopSet ` R ) |`t S ) ) >. , <. ( le ` ndx ) , { <. a , b >. | ( ( a e. W /\ b e. W ) /\ ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( le ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) } >. , <. ( dist ` ndx ) , ( a e. W , b e. W |-> ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( dist ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) ) >. } )
32 28 31 sstri 
 |-  { <. ( Base ` ndx ) , W >. } C_ ( ( { <. ( Base ` ndx ) , W >. , <. ( +g ` ndx ) , ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. , <. ( .r ` ndx ) , ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( k e. ( Base ` ( Scalar ` R ) ) , a e. W |-> <. ( k ( .s ` R ) ( 1st ` a ) ) , ( 2nd ` a ) >. ) >. , <. ( .i ` ndx ) , (/) >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopSet ` R ) tX ( ( TopSet ` R ) |`t S ) ) >. , <. ( le ` ndx ) , { <. a , b >. | ( ( a e. W /\ b e. W ) /\ ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( le ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) } >. , <. ( dist ` ndx ) , ( a e. W , b e. W |-> ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( dist ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) ) >. } )
33 1 fvexi 
 |-  B e. _V
34 33 a1i 
 |-  ( ph -> B e. _V )
35 34 9 ssexd 
 |-  ( ph -> S e. _V )
36 34 35 xpexd 
 |-  ( ph -> ( B X. S ) e. _V )
37 5 36 eqeltrid 
 |-  ( ph -> W e. _V )
38 eqid 
 |-  ( Base ` ( ( { <. ( Base ` ndx ) , W >. , <. ( +g ` ndx ) , ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. , <. ( .r ` ndx ) , ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( k e. ( Base ` ( Scalar ` R ) ) , a e. W |-> <. ( k ( .s ` R ) ( 1st ` a ) ) , ( 2nd ` a ) >. ) >. , <. ( .i ` ndx ) , (/) >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopSet ` R ) tX ( ( TopSet ` R ) |`t S ) ) >. , <. ( le ` ndx ) , { <. a , b >. | ( ( a e. W /\ b e. W ) /\ ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( le ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) } >. , <. ( dist ` ndx ) , ( a e. W , b e. W |-> ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( dist ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) ) >. } ) ) = ( Base ` ( ( { <. ( Base ` ndx ) , W >. , <. ( +g ` ndx ) , ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. , <. ( .r ` ndx ) , ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( k e. ( Base ` ( Scalar ` R ) ) , a e. W |-> <. ( k ( .s ` R ) ( 1st ` a ) ) , ( 2nd ` a ) >. ) >. , <. ( .i ` ndx ) , (/) >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopSet ` R ) tX ( ( TopSet ` R ) |`t S ) ) >. , <. ( le ` ndx ) , { <. a , b >. | ( ( a e. W /\ b e. W ) /\ ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( le ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) } >. , <. ( dist ` ndx ) , ( a e. W , b e. W |-> ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( dist ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) ) >. } ) )
39 24 26 27 32 37 38 strfv3 
 |-  ( ph -> ( Base ` ( ( { <. ( Base ` ndx ) , W >. , <. ( +g ` ndx ) , ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. , <. ( .r ` ndx ) , ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( k e. ( Base ` ( Scalar ` R ) ) , a e. W |-> <. ( k ( .s ` R ) ( 1st ` a ) ) , ( 2nd ` a ) >. ) >. , <. ( .i ` ndx ) , (/) >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopSet ` R ) tX ( ( TopSet ` R ) |`t S ) ) >. , <. ( le ` ndx ) , { <. a , b >. | ( ( a e. W /\ b e. W ) /\ ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( le ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) } >. , <. ( dist ` ndx ) , ( a e. W , b e. W |-> ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( dist ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) ) >. } ) ) = W )
40 39 eqcomd 
 |-  ( ph -> W = ( Base ` ( ( { <. ( Base ` ndx ) , W >. , <. ( +g ` ndx ) , ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. , <. ( .r ` ndx ) , ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( k e. ( Base ` ( Scalar ` R ) ) , a e. W |-> <. ( k ( .s ` R ) ( 1st ` a ) ) , ( 2nd ` a ) >. ) >. , <. ( .i ` ndx ) , (/) >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopSet ` R ) tX ( ( TopSet ` R ) |`t S ) ) >. , <. ( le ` ndx ) , { <. a , b >. | ( ( a e. W /\ b e. W ) /\ ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( le ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) } >. , <. ( dist ` ndx ) , ( a e. W , b e. W |-> ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( dist ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) ) >. } ) ) )
41 7 ovexi 
 |-  .~ e. _V
42 41 a1i 
 |-  ( ph -> .~ e. _V )
43 tpex 
 |-  { <. ( Base ` ndx ) , W >. , <. ( +g ` ndx ) , ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. , <. ( .r ` ndx ) , ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. } e. _V
44 tpex 
 |-  { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( k e. ( Base ` ( Scalar ` R ) ) , a e. W |-> <. ( k ( .s ` R ) ( 1st ` a ) ) , ( 2nd ` a ) >. ) >. , <. ( .i ` ndx ) , (/) >. } e. _V
45 43 44 unex 
 |-  ( { <. ( Base ` ndx ) , W >. , <. ( +g ` ndx ) , ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. , <. ( .r ` ndx ) , ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( k e. ( Base ` ( Scalar ` R ) ) , a e. W |-> <. ( k ( .s ` R ) ( 1st ` a ) ) , ( 2nd ` a ) >. ) >. , <. ( .i ` ndx ) , (/) >. } ) e. _V
46 tpex 
 |-  { <. ( TopSet ` ndx ) , ( ( TopSet ` R ) tX ( ( TopSet ` R ) |`t S ) ) >. , <. ( le ` ndx ) , { <. a , b >. | ( ( a e. W /\ b e. W ) /\ ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( le ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) } >. , <. ( dist ` ndx ) , ( a e. W , b e. W |-> ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( dist ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) ) >. } e. _V
47 45 46 unex 
 |-  ( ( { <. ( Base ` ndx ) , W >. , <. ( +g ` ndx ) , ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. , <. ( .r ` ndx ) , ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( k e. ( Base ` ( Scalar ` R ) ) , a e. W |-> <. ( k ( .s ` R ) ( 1st ` a ) ) , ( 2nd ` a ) >. ) >. , <. ( .i ` ndx ) , (/) >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopSet ` R ) tX ( ( TopSet ` R ) |`t S ) ) >. , <. ( le ` ndx ) , { <. a , b >. | ( ( a e. W /\ b e. W ) /\ ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( le ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) } >. , <. ( dist ` ndx ) , ( a e. W , b e. W |-> ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( dist ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) ) >. } ) e. _V
48 47 a1i 
 |-  ( ph -> ( ( { <. ( Base ` ndx ) , W >. , <. ( +g ` ndx ) , ( a e. W , b e. W |-> <. ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( +g ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. , <. ( .r ` ndx ) , ( a e. W , b e. W |-> <. ( ( 1st ` a ) .x. ( 1st ` b ) ) , ( ( 2nd ` a ) .x. ( 2nd ` b ) ) >. ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( k e. ( Base ` ( Scalar ` R ) ) , a e. W |-> <. ( k ( .s ` R ) ( 1st ` a ) ) , ( 2nd ` a ) >. ) >. , <. ( .i ` ndx ) , (/) >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopSet ` R ) tX ( ( TopSet ` R ) |`t S ) ) >. , <. ( le ` ndx ) , { <. a , b >. | ( ( a e. W /\ b e. W ) /\ ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( le ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) } >. , <. ( dist ` ndx ) , ( a e. W , b e. W |-> ( ( ( 1st ` a ) .x. ( 2nd ` b ) ) ( dist ` R ) ( ( 1st ` b ) .x. ( 2nd ` a ) ) ) ) >. } ) e. _V )
49 23 40 42 48 qusbas 
 |-  ( ph -> ( W /. .~ ) = ( Base ` L ) )