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 ) ) )