cdleme28c

Description: Part of proof of Lemma E in Crawley p. 113. Eliminate the s =/= t antecedent in cdleme28b . TODO: FIX COMMENT. (Contributed by NM, 6-Feb-2013)

	Ref	Expression
Hypotheses	cdleme26.b	\|- B = ( Base ` K )
	cdleme26.l	\|- .<_ = ( le ` K )
	cdleme26.j	\|- .\/ = ( join ` K )
	cdleme26.m	\|- ./\ = ( meet ` K )
	cdleme26.a	\|- A = ( Atoms ` K )
	cdleme26.h	\|- H = ( LHyp ` K )
	cdleme27.u	\|- U = ( ( P .\/ Q ) ./\ W )
	cdleme27.f	\|- F = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
	cdleme27.z	\|- Z = ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) )
	cdleme27.n	\|- N = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( s .\/ z ) ./\ W ) ) )
	cdleme27.d	\|- D = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) )
	cdleme27.c	\|- C = if ( s .<_ ( P .\/ Q ) , D , F )
	cdleme27.g	\|- G = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
	cdleme27.o	\|- O = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) )
	cdleme27.e	\|- E = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = O ) )
	cdleme27.y	\|- Y = if ( t .<_ ( P .\/ Q ) , E , G )
Assertion	cdleme28c	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( C .\/ ( X ./\ W ) ) = ( Y .\/ ( X ./\ W ) ) )

Proof

Step	Hyp	Ref	Expression
1		cdleme26.b	\|- B = ( Base ` K )
2		cdleme26.l	\|- .<_ = ( le ` K )
3		cdleme26.j	\|- .\/ = ( join ` K )
4		cdleme26.m	\|- ./\ = ( meet ` K )
5		cdleme26.a	\|- A = ( Atoms ` K )
6		cdleme26.h	\|- H = ( LHyp ` K )
7		cdleme27.u	\|- U = ( ( P .\/ Q ) ./\ W )
8		cdleme27.f	\|- F = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
9		cdleme27.z	\|- Z = ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) )
10		cdleme27.n	\|- N = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( s .\/ z ) ./\ W ) ) )
11		cdleme27.d	\|- D = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) )
12		cdleme27.c	\|- C = if ( s .<_ ( P .\/ Q ) , D , F )
13		cdleme27.g	\|- G = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
14		cdleme27.o	\|- O = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) )
15		cdleme27.e	\|- E = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = O ) )
16		cdleme27.y	\|- Y = if ( t .<_ ( P .\/ Q ) , E , G )
17	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16	cdleme27b	\|- ( s = t -> C = Y )
18	17	oveq1d	\|- ( s = t -> ( C .\/ ( X ./\ W ) ) = ( Y .\/ ( X ./\ W ) ) )
19	18	adantl	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s = t ) -> ( C .\/ ( X ./\ W ) ) = ( Y .\/ ( X ./\ W ) ) )
20		simpl11	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( K e. HL /\ W e. H ) )
21		simpl12	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( P e. A /\ -. P .<_ W ) )
22		simpl13	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( Q e. A /\ -. Q .<_ W ) )
23		simpl21	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> P =/= Q )
24		simpl22	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( s e. A /\ -. s .<_ W ) )
25		simpl23	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( t e. A /\ -. t .<_ W ) )
26		simpr	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> s =/= t )
27		simpl31	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( s .\/ ( X ./\ W ) ) = X )
28		simpl32	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( t .\/ ( X ./\ W ) ) = X )
29	27 28	jca	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) )
30		simpl33	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( X e. B /\ -. X .<_ W ) )
31	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16	cdleme28b	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( C .\/ ( X ./\ W ) ) = ( Y .\/ ( X ./\ W ) ) )
32	20 21 22 23 24 25 26 29 30 31	syl333anc	\|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( C .\/ ( X ./\ W ) ) = ( Y .\/ ( X ./\ W ) ) )
33	19 32	pm2.61dane	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( C .\/ ( X ./\ W ) ) = ( Y .\/ ( X ./\ W ) ) )

Metamath Proof Explorer

Theorem cdleme28c

Proof