Theorem cdlemk42yN

Description: Part of proof of Lemma K of Crawley p. 118. TODO: fix comment. (Contributed by NM, 20-Jul-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdlemk5.b	|- B = ( Base ` K )
	cdlemk5.l	|- .<_ = ( le ` K )
	cdlemk5.j	|- .\/ = ( join ` K )
	cdlemk5.m	|- ./\ = ( meet ` K )
	cdlemk5.a	|- A = ( Atoms ` K )
	cdlemk5.h	|- H = ( LHyp ` K )
	cdlemk5.t	|- T = ( ( LTrn ` K ) ` W )
	cdlemk5.r	|- R = ( ( trL ` K ) ` W )
	cdlemk5.z	|- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
	cdlemk5.y	|- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
	cdlemk5.x	|- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
Assertion	cdlemk42yN	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( [_ G / g ]_ X ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemk5.b	|- B = ( Base ` K )
2		cdlemk5.l	|- .<_ = ( le ` K )
3		cdlemk5.j	|- .\/ = ( join ` K )
4		cdlemk5.m	|- ./\ = ( meet ` K )
5		cdlemk5.a	|- A = ( Atoms ` K )
6		cdlemk5.h	|- H = ( LHyp ` K )
7		cdlemk5.t	|- T = ( ( LTrn ` K ) ` W )
8		cdlemk5.r	|- R = ( ( trL ` K ) ` W )
9		cdlemk5.z	|- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
10		cdlemk5.y	|- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
11		cdlemk5.x	|- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
12	1 2 3 4 5 6 7 8 9 10 11	cdlemk42	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( [_ G / g ]_ X ` P ) = [_ G / g ]_ Y )
13		simp13l	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> G e. T )
14	10	cdlemk41	|- ( G e. T -> [_ G / g ]_ Y = ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) ) )
15	13 14	syl	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> [_ G / g ]_ Y = ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) ) )
16	12 15	eqtrd	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( [_ G / g ]_ X ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) ) )