This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cdlemn11pre

Description: Part of proof of Lemma N of Crawley p. 121 line 37. TODO: combine cdlemn11a , cdlemn11b , cdlemn11c , cdlemn11pre into one? (Contributed by NM, 27-Feb-2014)

Ref Expression
Hypotheses cdlemn11a.b 
|- B = ( Base ` K )
cdlemn11a.l 
|- .<_ = ( le ` K )
cdlemn11a.j 
|- .\/ = ( join ` K )
cdlemn11a.a 
|- A = ( Atoms ` K )
cdlemn11a.h 
|- H = ( LHyp ` K )
cdlemn11a.p 
|- P = ( ( oc ` K ) ` W )
cdlemn11a.o 
|- O = ( h e. T |-> ( _I |` B ) )
cdlemn11a.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemn11a.r 
|- R = ( ( trL ` K ) ` W )
cdlemn11a.e 
|- E = ( ( TEndo ` K ) ` W )
cdlemn11a.i 
|- I = ( ( DIsoB ` K ) ` W )
cdlemn11a.J 
|- J = ( ( DIsoC ` K ) ` W )
cdlemn11a.u 
|- U = ( ( DVecH ` K ) ` W )
cdlemn11a.d 
|- .+ = ( +g ` U )
cdlemn11a.s 
|- .(+) = ( LSSum ` U )
cdlemn11a.f 
|- F = ( iota_ h e. T ( h ` P ) = Q )
cdlemn11a.g 
|- G = ( iota_ h e. T ( h ` P ) = N )
Assertion cdlemn11pre 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> N .<_ ( Q .\/ X ) )

Proof

Step Hyp Ref Expression
1 cdlemn11a.b 
 |-  B = ( Base ` K )
2 cdlemn11a.l 
 |-  .<_ = ( le ` K )
3 cdlemn11a.j 
 |-  .\/ = ( join ` K )
4 cdlemn11a.a 
 |-  A = ( Atoms ` K )
5 cdlemn11a.h 
 |-  H = ( LHyp ` K )
6 cdlemn11a.p 
 |-  P = ( ( oc ` K ) ` W )
7 cdlemn11a.o 
 |-  O = ( h e. T |-> ( _I |` B ) )
8 cdlemn11a.t 
 |-  T = ( ( LTrn ` K ) ` W )
9 cdlemn11a.r 
 |-  R = ( ( trL ` K ) ` W )
10 cdlemn11a.e 
 |-  E = ( ( TEndo ` K ) ` W )
11 cdlemn11a.i 
 |-  I = ( ( DIsoB ` K ) ` W )
12 cdlemn11a.J 
 |-  J = ( ( DIsoC ` K ) ` W )
13 cdlemn11a.u 
 |-  U = ( ( DVecH ` K ) ` W )
14 cdlemn11a.d 
 |-  .+ = ( +g ` U )
15 cdlemn11a.s 
 |-  .(+) = ( LSSum ` U )
16 cdlemn11a.f 
 |-  F = ( iota_ h e. T ( h ` P ) = Q )
17 cdlemn11a.g 
 |-  G = ( iota_ h e. T ( h ` P ) = N )
18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 cdlemn11c 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> E. y e. ( J ` Q ) E. z e. ( I ` X ) <. G , ( _I |` T ) >. = ( y .+ z ) )
19 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> ( K e. HL /\ W e. H ) )
20 simp21 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
21 2 4 5 6 8 10 12 16 dicelval3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( y e. ( J ` Q ) <-> E. s e. E y = <. ( s ` F ) , s >. ) )
22 19 20 21 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> ( y e. ( J ` Q ) <-> E. s e. E y = <. ( s ` F ) , s >. ) )
23 simp23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> ( X e. B /\ X .<_ W ) )
24 1 2 5 8 9 7 11 dibelval3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( z e. ( I ` X ) <-> E. g e. T ( z = <. g , O >. /\ ( R ` g ) .<_ X ) ) )
25 19 23 24 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> ( z e. ( I ` X ) <-> E. g e. T ( z = <. g , O >. /\ ( R ` g ) .<_ X ) ) )
26 22 25 anbi12d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> ( ( y e. ( J ` Q ) /\ z e. ( I ` X ) ) <-> ( E. s e. E y = <. ( s ` F ) , s >. /\ E. g e. T ( z = <. g , O >. /\ ( R ` g ) .<_ X ) ) ) )
27 reeanv 
 |-  ( E. s e. E E. g e. T ( y = <. ( s ` F ) , s >. /\ ( z = <. g , O >. /\ ( R ` g ) .<_ X ) ) <-> ( E. s e. E y = <. ( s ` F ) , s >. /\ E. g e. T ( z = <. g , O >. /\ ( R ` g ) .<_ X ) ) )
28 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) /\ ( ( s e. E /\ g e. T ) /\ ( R ` g ) .<_ X /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( K e. HL /\ W e. H ) )
29 simpl21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) /\ ( ( s e. E /\ g e. T ) /\ ( R ` g ) .<_ X /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
30 simpl22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) /\ ( ( s e. E /\ g e. T ) /\ ( R ` g ) .<_ X /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( N e. A /\ -. N .<_ W ) )
31 simpl23 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) /\ ( ( s e. E /\ g e. T ) /\ ( R ` g ) .<_ X /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( X e. B /\ X .<_ W ) )
32 simpr1r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) /\ ( ( s e. E /\ g e. T ) /\ ( R ` g ) .<_ X /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> g e. T )
33 simpr1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) /\ ( ( s e. E /\ g e. T ) /\ ( R ` g ) .<_ X /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> s e. E )
34 simpr3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) /\ ( ( s e. E /\ g e. T ) /\ ( R ` g ) .<_ X /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) )
35 1 2 4 5 6 7 8 10 13 14 16 17 cdlemn9 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( g ` Q ) = N )
36 28 29 30 33 32 34 35 syl123anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) /\ ( ( s e. E /\ g e. T ) /\ ( R ` g ) .<_ X /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( g ` Q ) = N )
37 simpr2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) /\ ( ( s e. E /\ g e. T ) /\ ( R ` g ) .<_ X /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( R ` g ) .<_ X )
38 1 2 3 4 5 8 9 cdlemn10 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( g e. T /\ ( g ` Q ) = N /\ ( R ` g ) .<_ X ) ) -> N .<_ ( Q .\/ X ) )
39 28 29 30 31 32 36 37 38 syl133anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) /\ ( ( s e. E /\ g e. T ) /\ ( R ` g ) .<_ X /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> N .<_ ( Q .\/ X ) )
40 39 3exp2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> ( ( s e. E /\ g e. T ) -> ( ( R ` g ) .<_ X -> ( <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) -> N .<_ ( Q .\/ X ) ) ) ) )
41 oveq12 
 |-  ( ( y = <. ( s ` F ) , s >. /\ z = <. g , O >. ) -> ( y .+ z ) = ( <. ( s ` F ) , s >. .+ <. g , O >. ) )
42 41 eqeq2d 
 |-  ( ( y = <. ( s ` F ) , s >. /\ z = <. g , O >. ) -> ( <. G , ( _I |` T ) >. = ( y .+ z ) <-> <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) )
43 42 imbi1d 
 |-  ( ( y = <. ( s ` F ) , s >. /\ z = <. g , O >. ) -> ( ( <. G , ( _I |` T ) >. = ( y .+ z ) -> N .<_ ( Q .\/ X ) ) <-> ( <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) -> N .<_ ( Q .\/ X ) ) ) )
44 43 imbi2d 
 |-  ( ( y = <. ( s ` F ) , s >. /\ z = <. g , O >. ) -> ( ( ( R ` g ) .<_ X -> ( <. G , ( _I |` T ) >. = ( y .+ z ) -> N .<_ ( Q .\/ X ) ) ) <-> ( ( R ` g ) .<_ X -> ( <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) -> N .<_ ( Q .\/ X ) ) ) ) )
45 44 biimprd 
 |-  ( ( y = <. ( s ` F ) , s >. /\ z = <. g , O >. ) -> ( ( ( R ` g ) .<_ X -> ( <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) -> N .<_ ( Q .\/ X ) ) ) -> ( ( R ` g ) .<_ X -> ( <. G , ( _I |` T ) >. = ( y .+ z ) -> N .<_ ( Q .\/ X ) ) ) ) )
46 45 com23 
 |-  ( ( y = <. ( s ` F ) , s >. /\ z = <. g , O >. ) -> ( ( R ` g ) .<_ X -> ( ( ( R ` g ) .<_ X -> ( <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) -> N .<_ ( Q .\/ X ) ) ) -> ( <. G , ( _I |` T ) >. = ( y .+ z ) -> N .<_ ( Q .\/ X ) ) ) ) )
47 46 impr 
 |-  ( ( y = <. ( s ` F ) , s >. /\ ( z = <. g , O >. /\ ( R ` g ) .<_ X ) ) -> ( ( ( R ` g ) .<_ X -> ( <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) -> N .<_ ( Q .\/ X ) ) ) -> ( <. G , ( _I |` T ) >. = ( y .+ z ) -> N .<_ ( Q .\/ X ) ) ) )
48 47 com12 
 |-  ( ( ( R ` g ) .<_ X -> ( <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) -> N .<_ ( Q .\/ X ) ) ) -> ( ( y = <. ( s ` F ) , s >. /\ ( z = <. g , O >. /\ ( R ` g ) .<_ X ) ) -> ( <. G , ( _I |` T ) >. = ( y .+ z ) -> N .<_ ( Q .\/ X ) ) ) )
49 40 48 syl6 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> ( ( s e. E /\ g e. T ) -> ( ( y = <. ( s ` F ) , s >. /\ ( z = <. g , O >. /\ ( R ` g ) .<_ X ) ) -> ( <. G , ( _I |` T ) >. = ( y .+ z ) -> N .<_ ( Q .\/ X ) ) ) ) )
50 49 rexlimdvv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> ( E. s e. E E. g e. T ( y = <. ( s ` F ) , s >. /\ ( z = <. g , O >. /\ ( R ` g ) .<_ X ) ) -> ( <. G , ( _I |` T ) >. = ( y .+ z ) -> N .<_ ( Q .\/ X ) ) ) )
51 27 50 biimtrrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> ( ( E. s e. E y = <. ( s ` F ) , s >. /\ E. g e. T ( z = <. g , O >. /\ ( R ` g ) .<_ X ) ) -> ( <. G , ( _I |` T ) >. = ( y .+ z ) -> N .<_ ( Q .\/ X ) ) ) )
52 26 51 sylbid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> ( ( y e. ( J ` Q ) /\ z e. ( I ` X ) ) -> ( <. G , ( _I |` T ) >. = ( y .+ z ) -> N .<_ ( Q .\/ X ) ) ) )
53 52 rexlimdvv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> ( E. y e. ( J ` Q ) E. z e. ( I ` X ) <. G , ( _I |` T ) >. = ( y .+ z ) -> N .<_ ( Q .\/ X ) ) )
54 18 53 mpd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> N .<_ ( Q .\/ X ) )