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Metamath Proof Explorer

Theorem cdlemg2jlemOLDN

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT. f preserves join: f(r \/ s) = f(r) \/ s, p. 115 10th line from bottom. TODO: Combine with cdlemg2jOLDN ? (Contributed by NM, 22-Apr-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemg2.b 
|- B = ( Base ` K )
cdlemg2.l 
|- .<_ = ( le ` K )
cdlemg2.j 
|- .\/ = ( join ` K )
cdlemg2.m 
|- ./\ = ( meet ` K )
cdlemg2.a 
|- A = ( Atoms ` K )
cdlemg2.h 
|- H = ( LHyp ` K )
cdlemg2.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemg2ex.u 
|- U = ( ( p .\/ q ) ./\ W )
cdlemg2ex.d 
|- D = ( ( t .\/ U ) ./\ ( q .\/ ( ( p .\/ t ) ./\ W ) ) )
cdlemg2ex.e 
|- E = ( ( p .\/ q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
cdlemg2ex.g 
|- G = ( 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 ) )
Assertion cdlemg2jlemOLDN 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F e. T ) -> ( F ` ( P .\/ Q ) ) = ( ( F ` P ) .\/ ( F ` Q ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg2.b 
 |-  B = ( Base ` K )
2 cdlemg2.l 
 |-  .<_ = ( le ` K )
3 cdlemg2.j 
 |-  .\/ = ( join ` K )
4 cdlemg2.m 
 |-  ./\ = ( meet ` K )
5 cdlemg2.a 
 |-  A = ( Atoms ` K )
6 cdlemg2.h 
 |-  H = ( LHyp ` K )
7 cdlemg2.t 
 |-  T = ( ( LTrn ` K ) ` W )
8 cdlemg2ex.u 
 |-  U = ( ( p .\/ q ) ./\ W )
9 cdlemg2ex.d 
 |-  D = ( ( t .\/ U ) ./\ ( q .\/ ( ( p .\/ t ) ./\ W ) ) )
10 cdlemg2ex.e 
 |-  E = ( ( p .\/ q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
11 cdlemg2ex.g 
 |-  G = ( 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 ) )
12 fveq1 
 |-  ( F = G -> ( F ` ( P .\/ Q ) ) = ( G ` ( P .\/ Q ) ) )
13 fveq1 
 |-  ( F = G -> ( F ` P ) = ( G ` P ) )
14 fveq1 
 |-  ( F = G -> ( F ` Q ) = ( G ` Q ) )
15 13 14 oveq12d 
 |-  ( F = G -> ( ( F ` P ) .\/ ( F ` Q ) ) = ( ( G ` P ) .\/ ( G ` Q ) ) )
16 12 15 eqeq12d 
 |-  ( F = G -> ( ( F ` ( P .\/ Q ) ) = ( ( F ` P ) .\/ ( F ` Q ) ) <-> ( G ` ( P .\/ Q ) ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) )
17 vex 
 |-  s e. _V
18 eqid 
 |-  ( ( s .\/ U ) ./\ ( q .\/ ( ( p .\/ s ) ./\ W ) ) ) = ( ( s .\/ U ) ./\ ( q .\/ ( ( p .\/ s ) ./\ W ) ) )
19 9 18 cdleme31sc 
 |-  ( s e. _V -> [_ s / t ]_ D = ( ( s .\/ U ) ./\ ( q .\/ ( ( p .\/ s ) ./\ W ) ) ) )
20 17 19 ax-mp 
 |-  [_ s / t ]_ D = ( ( s .\/ U ) ./\ ( q .\/ ( ( p .\/ s ) ./\ W ) ) )
21 eqid 
 |-  ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( p .\/ q ) ) -> y = E ) ) = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( p .\/ q ) ) -> y = E ) )
22 eqid 
 |-  if ( s .<_ ( p .\/ q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( p .\/ q ) ) -> y = E ) ) , [_ s / t ]_ D ) = if ( s .<_ ( p .\/ q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( p .\/ q ) ) -> y = E ) ) , [_ s / t ]_ D )
23 eqid 
 |-  ( 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 ) ) ) ) = ( 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 ) ) ) )
24 1 2 3 4 5 6 8 20 9 10 21 22 23 11 cdleme42mgN 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ -. p .<_ W ) /\ ( q e. A /\ -. q .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( G ` ( P .\/ Q ) ) = ( ( G ` P ) .\/ ( G ` Q ) ) )
25 1 2 3 4 5 6 7 8 9 10 11 16 24 cdlemg2ce 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( F ` ( P .\/ Q ) ) = ( ( F ` P ) .\/ ( F ` Q ) ) )
26 25 3com23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F e. T ) -> ( F ` ( P .\/ Q ) ) = ( ( F ` P ) .\/ ( F ` Q ) ) )