Metamath Proof Explorer

Theorem cdlemk33N

Description: Part of proof of Lemma K of Crawley p. 118. TODO: fix comment. Part of attempt to simplify hypotheses. TODO: not needed, is embodied in cdlemk34 . (Contributed by NM, 18-Jul-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemk3.b 
|- B = ( Base ` K )
cdlemk3.l 
|- .<_ = ( le ` K )
cdlemk3.j 
|- .\/ = ( join ` K )
cdlemk3.m 
|- ./\ = ( meet ` K )
cdlemk3.a 
|- A = ( Atoms ` K )
cdlemk3.h 
|- H = ( LHyp ` K )
cdlemk3.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemk3.r 
|- R = ( ( trL ` K ) ` W )
cdlemk3.s 
|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
cdlemk3.u1 
|- Y = ( d e. T , e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) )
cdlemk3.x 
|- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> z = ( b Y G ) ) )
Assertion cdlemk33N 
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = ( ( b Y G ) ` P ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemk3.b 
 |-  B = ( Base ` K )
2 cdlemk3.l 
 |-  .<_ = ( le ` K )
3 cdlemk3.j 
 |-  .\/ = ( join ` K )
4 cdlemk3.m 
 |-  ./\ = ( meet ` K )
5 cdlemk3.a 
 |-  A = ( Atoms ` K )
6 cdlemk3.h 
 |-  H = ( LHyp ` K )
7 cdlemk3.t 
 |-  T = ( ( LTrn ` K ) ` W )
8 cdlemk3.r 
 |-  R = ( ( trL ` K ) ` W )
9 cdlemk3.s 
 |-  S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
10 cdlemk3.u1 
 |-  Y = ( d e. T , e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) )
11 cdlemk3.x 
 |-  X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> z = ( b Y G ) ) )
12 fveq1 
 |-  ( z = ( b Y G ) -> ( z ` P ) = ( ( b Y G ) ` P ) )
13 simpl11 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) /\ ( z ` P ) = ( ( b Y G ) ` P ) ) -> K e. HL )
14 simpl12 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) /\ ( z ` P ) = ( ( b Y G ) ` P ) ) -> W e. H )
15 13 14 jca 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) /\ ( z ` P ) = ( ( b Y G ) ` P ) ) -> ( K e. HL /\ W e. H ) )
16 simpl31 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) /\ ( z ` P ) = ( ( b Y G ) ` P ) ) -> z e. T )
17 simp11 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> K e. HL )
18 simp12 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> W e. H )
19 17 18 jca 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( K e. HL /\ W e. H ) )
20 simp13 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( R ` F ) = ( R ` N ) )
21 simp22l 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> G e. T )
22 19 20 21 3jca 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) )
23 22 adantr 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) /\ ( z ` P ) = ( ( b Y G ) ` P ) ) -> ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) )
24 simp211 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> F e. T )
25 simp32 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> b e. T )
26 simp213 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> N e. T )
27 24 25 26 3jca 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( F e. T /\ b e. T /\ N e. T ) )
28 27 adantr 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) /\ ( z ` P ) = ( ( b Y G ) ` P ) ) -> ( F e. T /\ b e. T /\ N e. T ) )
29 simp332 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( R ` b ) =/= ( R ` F ) )
30 simp333 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( R ` b ) =/= ( R ` G ) )
31 29 30 jca 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) )
32 simp212 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> F =/= ( _I |` B ) )
33 simp22r 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> G =/= ( _I |` B ) )
34 simp331 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> b =/= ( _I |` B ) )
35 32 33 34 3jca 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ b =/= ( _I |` B ) ) )
36 simp23 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
37 31 35 36 3jca 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ b =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) )
38 37 adantr 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) /\ ( z ` P ) = ( ( b Y G ) ` P ) ) -> ( ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ b =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) )
39 1 2 3 4 5 6 7 8 9 10 cdlemkuel-3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ b e. T /\ N e. T ) /\ ( ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ b =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( b Y G ) e. T )
40 23 28 38 39 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) /\ ( z ` P ) = ( ( b Y G ) ` P ) ) -> ( b Y G ) e. T )
41 simpl23 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) /\ ( z ` P ) = ( ( b Y G ) ` P ) ) -> ( P e. A /\ -. P .<_ W ) )
42 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) /\ ( z ` P ) = ( ( b Y G ) ` P ) ) -> ( z ` P ) = ( ( b Y G ) ` P ) )
43 2 5 6 7 cdlemd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ z e. T /\ ( b Y G ) e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( z ` P ) = ( ( b Y G ) ` P ) ) -> z = ( b Y G ) )
44 15 16 40 41 42 43 syl311anc 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) /\ ( z ` P ) = ( ( b Y G ) ` P ) ) -> z = ( b Y G ) )
45 44 ex 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( ( z ` P ) = ( ( b Y G ) ` P ) -> z = ( b Y G ) ) )
46 12 45 impbid2 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( z = ( b Y G ) <-> ( z ` P ) = ( ( b Y G ) ` P ) ) )
47 46 3expia 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( z e. T /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) -> ( z = ( b Y G ) <-> ( z ` P ) = ( ( b Y G ) ` P ) ) ) )
48 47 3expd 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( z e. T -> ( b e. T -> ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z = ( b Y G ) <-> ( z ` P ) = ( ( b Y G ) ` P ) ) ) ) ) )
49 48 imp31 
 |-  ( ( ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) /\ z e. T ) /\ b e. T ) -> ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z = ( b Y G ) <-> ( z ` P ) = ( ( b Y G ) ` P ) ) ) )
50 49 pm5.74d 
 |-  ( ( ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) /\ z e. T ) /\ b e. T ) -> ( ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> z = ( b Y G ) ) <-> ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = ( ( b Y G ) ` P ) ) ) )
51 50 ralbidva 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) /\ z e. T ) -> ( A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> z = ( b Y G ) ) <-> A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = ( ( b Y G ) ` P ) ) ) )
52 51 riotabidva 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> z = ( b Y G ) ) ) = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = ( ( b Y G ) ` P ) ) ) )
53 11 52 syl5eq 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = ( ( b Y G ) ` P ) ) ) )