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Metamath Proof Explorer

Theorem cdleme51finvfvN

Description: Part of proof of Lemma E in Crawley p. 113. TODO: fix comment. (Contributed by NM, 14-Apr-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemef50.b 
|- B = ( Base ` K )
cdlemef50.l 
|- .<_ = ( le ` K )
cdlemef50.j 
|- .\/ = ( join ` K )
cdlemef50.m 
|- ./\ = ( meet ` K )
cdlemef50.a 
|- A = ( Atoms ` K )
cdlemef50.h 
|- H = ( LHyp ` K )
cdlemef50.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdlemef50.d 
|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdlemefs50.e 
|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
cdlemef50.f 
|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) )
cdlemef51.v 
|- V = ( ( Q .\/ P ) ./\ W )
cdlemef51.n 
|- N = ( ( v .\/ V ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) )
cdlemefs51.o 
|- O = ( ( Q .\/ P ) ./\ ( N .\/ ( ( u .\/ v ) ./\ W ) ) )
cdlemef51.g 
|- G = ( a e. B |-> if ( ( Q =/= P /\ -. a .<_ W ) , ( iota_ c e. B A. u e. A ( ( -. u .<_ W /\ ( u .\/ ( a ./\ W ) ) = a ) -> c = ( if ( u .<_ ( Q .\/ P ) , ( iota_ b e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( Q .\/ P ) ) -> b = O ) ) , [_ u / v ]_ N ) .\/ ( a ./\ W ) ) ) ) , a ) )
Assertion cdleme51finvfvN 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ X e. B ) -> ( `' F ` X ) = ( G ` X ) )

Proof

Step Hyp Ref Expression
1 cdlemef50.b 
 |-  B = ( Base ` K )
2 cdlemef50.l 
 |-  .<_ = ( le ` K )
3 cdlemef50.j 
 |-  .\/ = ( join ` K )
4 cdlemef50.m 
 |-  ./\ = ( meet ` K )
5 cdlemef50.a 
 |-  A = ( Atoms ` K )
6 cdlemef50.h 
 |-  H = ( LHyp ` K )
7 cdlemef50.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdlemef50.d 
 |-  D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
9 cdlemefs50.e 
 |-  E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
10 cdlemef50.f 
 |-  F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) )
11 cdlemef51.v 
 |-  V = ( ( Q .\/ P ) ./\ W )
12 cdlemef51.n 
 |-  N = ( ( v .\/ V ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) )
13 cdlemefs51.o 
 |-  O = ( ( Q .\/ P ) ./\ ( N .\/ ( ( u .\/ v ) ./\ W ) ) )
14 cdlemef51.g 
 |-  G = ( a e. B |-> if ( ( Q =/= P /\ -. a .<_ W ) , ( iota_ c e. B A. u e. A ( ( -. u .<_ W /\ ( u .\/ ( a ./\ W ) ) = a ) -> c = ( if ( u .<_ ( Q .\/ P ) , ( iota_ b e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( Q .\/ P ) ) -> b = O ) ) , [_ u / v ]_ N ) .\/ ( a ./\ W ) ) ) ) , a ) )
15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 cdleme48fgv 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ X e. B ) -> ( F ` ( G ` X ) ) = X )
16 1 2 3 4 5 6 7 8 9 10 cdleme50f1o 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> F : B -1-1-onto-> B )
17 16 adantr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ X e. B ) -> F : B -1-1-onto-> B )
18 1 2 3 4 5 6 11 12 13 14 cdlemeg46fvcl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ X e. B ) -> ( G ` X ) e. B )
19 f1ocnvfv 
 |-  ( ( F : B -1-1-onto-> B /\ ( G ` X ) e. B ) -> ( ( F ` ( G ` X ) ) = X -> ( `' F ` X ) = ( G ` X ) ) )
20 17 18 19 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ X e. B ) -> ( ( F ` ( G ` X ) ) = X -> ( `' F ` X ) = ( G ` X ) ) )
21 15 20 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ X e. B ) -> ( `' F ` X ) = ( G ` X ) )