cdlemk42

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

	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	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 )

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		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 )
13	1 2 3 4 5 6 7 8 9 10 11	cdlemk36	\|- ( ( ( ( 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 ) ) ) ) -> ( X ` P ) = Y )
14	13	sbcth	\|- ( G e. T -> [. G / g ]. ( ( ( ( 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 ) ) ) ) -> ( X ` P ) = Y ) )
15		sbcimg	\|- ( G e. T -> ( [. G / g ]. ( ( ( ( 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 ) ) ) ) -> ( X ` P ) = Y ) <-> ( [. G / g ]. ( ( ( 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 ) = Y ) ) )
16	14 15	mpbid	\|- ( G e. T -> ( [. G / g ]. ( ( ( 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 ) = Y ) )
17		eleq1	\|- ( g = G -> ( g e. T <-> G e. T ) )
18		neeq1	\|- ( g = G -> ( g =/= ( _I \|` B ) <-> G =/= ( _I \|` B ) ) )
19	17 18	anbi12d	\|- ( g = G -> ( ( g e. T /\ g =/= ( _I \|` B ) ) <-> ( G e. T /\ G =/= ( _I \|` B ) ) ) )
20	19	3anbi3d	\|- ( g = G -> ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) ) /\ ( g e. T /\ g =/= ( _I \|` B ) ) ) <-> ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) ) ) )
21		fveq2	\|- ( g = G -> ( R ` g ) = ( R ` G ) )
22	21	neeq2d	\|- ( g = G -> ( ( R ` b ) =/= ( R ` g ) <-> ( R ` b ) =/= ( R ` G ) ) )
23	22	3anbi3d	\|- ( g = G -> ( ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) <-> ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) )
24	23	anbi2d	\|- ( g = G -> ( ( 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 ) ) ) ) )
25	20 24	3anbi13d	\|- ( g = G -> ( ( ( ( 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 ) ) ) ) <-> ( ( ( 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 ) ) ) ) ) )
26	25	sbcieg	\|- ( G e. T -> ( [. G / g ]. ( ( ( 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 ) ) ) ) <-> ( ( ( 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 ) ) ) ) ) )
27		sbceqg	\|- ( G e. T -> ( [. G / g ]. ( X ` P ) = Y <-> [_ G / g ]_ ( X ` P ) = [_ G / g ]_ Y ) )
28		csbfv12	\|- [_ G / g ]_ ( X ` P ) = ( [_ G / g ]_ X ` [_ G / g ]_ P )
29		csbconstg	\|- ( G e. T -> [_ G / g ]_ P = P )
30	29	fveq2d	\|- ( G e. T -> ( [_ G / g ]_ X ` [_ G / g ]_ P ) = ( [_ G / g ]_ X ` P ) )
31	28 30	eqtrid	\|- ( G e. T -> [_ G / g ]_ ( X ` P ) = ( [_ G / g ]_ X ` P ) )
32	31	eqeq1d	\|- ( G e. T -> ( [_ G / g ]_ ( X ` P ) = [_ G / g ]_ Y <-> ( [_ G / g ]_ X ` P ) = [_ G / g ]_ Y ) )
33	27 32	bitrd	\|- ( G e. T -> ( [. G / g ]. ( X ` P ) = Y <-> ( [_ G / g ]_ X ` P ) = [_ G / g ]_ Y ) )
34	16 26 33	3imtr3d	\|- ( G e. T -> ( ( ( ( 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 ) )
35	12 34	mpcom	\|- ( ( ( ( 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 )

Metamath Proof Explorer

Theorem cdlemk42

Proof