cdlemk43N

Description: Part of proof of Lemma K of Crawley p. 118. TODO: fix comment. (Contributed by NM, 31-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 ) )
	cdlemk5.u	\|- U = ( g e. T \|-> if ( F = N , g , X ) )
Assertion	cdlemk43N	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ N e. T /\ F =/= N ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( ( U ` G ) ` P ) = [_ G / g ]_ Y )

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		cdlemk5.u	\|- U = ( g e. T \|-> if ( F = N , g , X ) )
13		simp213	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ N e. T /\ F =/= N ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> F =/= N )
14		simp22l	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ N e. T /\ F =/= N ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> G e. T )
15	11 12	cdlemk40f	\|- ( ( F =/= N /\ G e. T ) -> ( U ` G ) = [_ G / g ]_ X )
16	13 14 15	syl2anc	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ N e. T /\ F =/= N ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( U ` G ) = [_ G / g ]_ X )
17	16	fveq1d	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ N e. T /\ F =/= N ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( ( U ` G ) ` P ) = ( [_ G / g ]_ X ` P ) )
18		simp1l	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ N e. T /\ F =/= N ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( K e. HL /\ W e. H ) )
19		simp211	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ N e. T /\ F =/= N ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> F e. T )
20		simp212	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ N e. T /\ F =/= N ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> N e. T )
21		simp1r	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ N e. T /\ F =/= N ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( R ` F ) = ( R ` N ) )
22	1 6 7 8	trlnid	\|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( F =/= N /\ ( R ` F ) = ( R ` N ) ) ) -> F =/= ( _I \|` B ) )
23	18 19 20 13 21 22	syl122anc	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ N e. T /\ F =/= N ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> F =/= ( _I \|` B ) )
24	19 23	jca	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ N e. T /\ F =/= N ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( F e. T /\ F =/= ( _I \|` B ) ) )
25		simp22	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ N e. T /\ F =/= N ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( G e. T /\ G =/= ( _I \|` B ) ) )
26		simp23	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ N e. T /\ F =/= N ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
27		simp3	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ N e. T /\ F =/= N ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( b e. T /\ ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) )
28	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 )
29	18 24 25 20 26 21 27 28	syl331anc	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ N e. T /\ F =/= N ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( [_ G / g ]_ X ` P ) = [_ G / g ]_ Y )
30	17 29	eqtrd	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ N e. T /\ F =/= N ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( ( U ` G ) ` P ) = [_ G / g ]_ Y )

Metamath Proof Explorer

Theorem cdlemk43N

Proof