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Metamath Proof Explorer

Theorem cdlemk21N

Description: Part of proof of Lemma K of Crawley p. 118. Lines 26-27, p. 119 for i=0 and j=1. (Contributed by NM, 5-Jul-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemk1.b 
|- B = ( Base ` K )
cdlemk1.l 
|- .<_ = ( le ` K )
cdlemk1.j 
|- .\/ = ( join ` K )
cdlemk1.m 
|- ./\ = ( meet ` K )
cdlemk1.a 
|- A = ( Atoms ` K )
cdlemk1.h 
|- H = ( LHyp ` K )
cdlemk1.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemk1.r 
|- R = ( ( trL ` K ) ` W )
cdlemk1.s 
|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
cdlemk1.o 
|- O = ( S ` D )
cdlemk1.u 
|- U = ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( O ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) )
Assertion cdlemk21N 
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( ( S ` G ) ` P ) = ( ( U ` G ) ` P ) )

Proof

Step Hyp Ref Expression
1 cdlemk1.b 
 |-  B = ( Base ` K )
2 cdlemk1.l 
 |-  .<_ = ( le ` K )
3 cdlemk1.j 
 |-  .\/ = ( join ` K )
4 cdlemk1.m 
 |-  ./\ = ( meet ` K )
5 cdlemk1.a 
 |-  A = ( Atoms ` K )
6 cdlemk1.h 
 |-  H = ( LHyp ` K )
7 cdlemk1.t 
 |-  T = ( ( LTrn ` K ) ` W )
8 cdlemk1.r 
 |-  R = ( ( trL ` K ) ` W )
9 cdlemk1.s 
 |-  S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
10 cdlemk1.o 
 |-  O = ( S ` D )
11 cdlemk1.u 
 |-  U = ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( O ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) )
12 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( K e. HL /\ W e. H ) )
13 simp21r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> G e. T )
14 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
15 2 3 5 6 7 8 trljat1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( R ` G ) ) = ( P .\/ ( G ` P ) ) )
16 12 13 14 15 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( P .\/ ( R ` G ) ) = ( P .\/ ( G ` P ) ) )
17 10 fveq1i 
 |-  ( O ` P ) = ( ( S ` D ) ` P )
18 17 a1i 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( O ` P ) = ( ( S ` D ) ` P ) )
19 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> D e. T )
20 6 7 8 trlcocnv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ D e. T ) -> ( R ` ( G o. `' D ) ) = ( R ` ( D o. `' G ) ) )
21 12 13 19 20 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( R ` ( G o. `' D ) ) = ( R ` ( D o. `' G ) ) )
22 18 21 oveq12d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) = ( ( ( S ` D ) ` P ) .\/ ( R ` ( D o. `' G ) ) ) )
23 16 22 oveq12d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) = ( ( P .\/ ( G ` P ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( D o. `' G ) ) ) ) )
24 simp23 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( R ` F ) = ( R ` N ) )
25 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> F e. T )
26 simp21l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> N e. T )
27 simp3r1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( R ` D ) =/= ( R ` F ) )
28 simp3r2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( R ` G ) =/= ( R ` D ) )
29 28 necomd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( R ` D ) =/= ( R ` G ) )
30 27 29 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) )
31 simp3l1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> F =/= ( _I |` B ) )
32 simp3l3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> G =/= ( _I |` B ) )
33 simp3l2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> D =/= ( _I |` B ) )
34 31 32 33 3jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) )
35 1 2 3 4 5 6 7 8 9 10 11 cdlemkuv2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( U ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) )
36 12 24 13 25 19 26 30 34 14 35 syl333anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( ( U ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) )
37 26 19 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( N e. T /\ D e. T ) )
38 simp3r3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( R ` G ) =/= ( R ` F ) )
39 38 27 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( ( R ` G ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) )
40 1 2 3 5 6 7 8 4 9 cdlemk12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ D e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> ( ( S ` G ) ` P ) = ( ( P .\/ ( G ` P ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( D o. `' G ) ) ) ) )
41 12 25 13 37 14 24 34 39 28 40 syl333anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( ( S ` G ) ` P ) = ( ( P .\/ ( G ` P ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( D o. `' G ) ) ) ) )
42 23 36 41 3eqtr4rd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( ( S ` G ) ` P ) = ( ( U ` G ) ` P ) )