cdlemk31

Description: Part of proof of Lemma K of Crawley p. 118. TODO: fix comment. Part of attempt to simplify hypotheses. (Contributed by NM, 17-Jul-2013)

	Ref	Expression
Hypotheses	cdlemk3.b	\|- B = ( Base ` K )
	cdlemk3.l	\|- .<_ = ( le ` K )
	cdlemk3.j	\|- .\/ = ( join ` K )
	cdlemk3.m	\|- ./\ = ( meet ` K )
	cdlemk3.a	\|- A = ( Atoms ` K )
	cdlemk3.h	\|- H = ( LHyp ` K )
	cdlemk3.t	\|- T = ( ( LTrn ` K ) ` W )
	cdlemk3.r	\|- R = ( ( trL ` K ) ` W )
	cdlemk3.s	\|- S = ( f e. T \|-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
	cdlemk3.u1	\|- Y = ( d e. T , e e. T \|-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) )
Assertion	cdlemk31	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ b e. T /\ N e. T ) /\ G e. T ) /\ ( ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( F =/= ( _I \|` B ) /\ b =/= ( _I \|` B ) /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( b Y G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( ( S ` b ) ` P ) .\/ ( R ` ( G o. `' b ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemk3.b	\|- B = ( Base ` K )
2		cdlemk3.l	\|- .<_ = ( le ` K )
3		cdlemk3.j	\|- .\/ = ( join ` K )
4		cdlemk3.m	\|- ./\ = ( meet ` K )
5		cdlemk3.a	\|- A = ( Atoms ` K )
6		cdlemk3.h	\|- H = ( LHyp ` K )
7		cdlemk3.t	\|- T = ( ( LTrn ` K ) ` W )
8		cdlemk3.r	\|- R = ( ( trL ` K ) ` W )
9		cdlemk3.s	\|- S = ( f e. T \|-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
10		cdlemk3.u1	\|- Y = ( d e. T , e e. T \|-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) )
11		simp2l2	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ b e. T /\ N e. T ) /\ G e. T ) /\ ( ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( F =/= ( _I \|` B ) /\ b =/= ( _I \|` B ) /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> b e. T )
12		simp2r	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ b e. T /\ N e. T ) /\ G e. T ) /\ ( ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( F =/= ( _I \|` B ) /\ b =/= ( _I \|` B ) /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> G e. T )
13		eqid	\|- ( S ` b ) = ( S ` b )
14		eqid	\|- ( e e. T \|-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` b ) ` P ) .\/ ( R ` ( e o. `' b ) ) ) ) ) ) = ( e e. T \|-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` b ) ` P ) .\/ ( R ` ( e o. `' b ) ) ) ) ) )
15	1 2 3 4 5 6 7 8 9 10 13 14	cdlemkuu	\|- ( ( b e. T /\ G e. T ) -> ( b Y G ) = ( ( e e. T \|-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` b ) ` P ) .\/ ( R ` ( e o. `' b ) ) ) ) ) ) ` G ) )
16	11 12 15	syl2anc	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ b e. T /\ N e. T ) /\ G e. T ) /\ ( ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( F =/= ( _I \|` B ) /\ b =/= ( _I \|` B ) /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( b Y G ) = ( ( e e. T \|-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` b ) ` P ) .\/ ( R ` ( e o. `' b ) ) ) ) ) ) ` G ) )
17	16	fveq1d	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ b e. T /\ N e. T ) /\ G e. T ) /\ ( ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( F =/= ( _I \|` B ) /\ b =/= ( _I \|` B ) /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( b Y G ) ` P ) = ( ( ( e e. T \|-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` b ) ` P ) .\/ ( R ` ( e o. `' b ) ) ) ) ) ) ` G ) ` P ) )
18		simp1l	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ b e. T /\ N e. T ) /\ G e. T ) /\ ( ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( F =/= ( _I \|` B ) /\ b =/= ( _I \|` B ) /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( K e. HL /\ W e. H ) )
19		simp1r	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ b e. T /\ N e. T ) /\ G e. T ) /\ ( ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( F =/= ( _I \|` B ) /\ b =/= ( _I \|` B ) /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` F ) = ( R ` N ) )
20		simp2l	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ b e. T /\ N e. T ) /\ G e. T ) /\ ( ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( F =/= ( _I \|` B ) /\ b =/= ( _I \|` B ) /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( F e. T /\ b e. T /\ N e. T ) )
21		simp31	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ b e. T /\ N e. T ) /\ G e. T ) /\ ( ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( F =/= ( _I \|` B ) /\ b =/= ( _I \|` B ) /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) )
22		simp321	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ b e. T /\ N e. T ) /\ G e. T ) /\ ( ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( F =/= ( _I \|` B ) /\ b =/= ( _I \|` B ) /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F =/= ( _I \|` B ) )
23		simp323	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ b e. T /\ N e. T ) /\ G e. T ) /\ ( ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( F =/= ( _I \|` B ) /\ b =/= ( _I \|` B ) /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> G =/= ( _I \|` B ) )
24		simp322	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ b e. T /\ N e. T ) /\ G e. T ) /\ ( ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( F =/= ( _I \|` B ) /\ b =/= ( _I \|` B ) /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> b =/= ( _I \|` B ) )
25	22 23 24	3jca	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ b e. T /\ N e. T ) /\ G e. T ) /\ ( ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( F =/= ( _I \|` B ) /\ b =/= ( _I \|` B ) /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( F =/= ( _I \|` B ) /\ G =/= ( _I \|` B ) /\ b =/= ( _I \|` B ) ) )
26		simp33	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ b e. T /\ N e. T ) /\ G e. T ) /\ ( ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( F =/= ( _I \|` B ) /\ b =/= ( _I \|` B ) /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( P e. A /\ -. P .<_ W ) )
27	1 2 3 4 5 6 7 8 9 13 14	cdlemkuv2	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ b e. T /\ N e. T ) /\ ( ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( F =/= ( _I \|` B ) /\ G =/= ( _I \|` B ) /\ b =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( ( e e. T \|-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` b ) ` P ) .\/ ( R ` ( e o. `' b ) ) ) ) ) ) ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( ( S ` b ) ` P ) .\/ ( R ` ( G o. `' b ) ) ) ) )
28	18 19 12 20 21 25 26 27	syl313anc	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ b e. T /\ N e. T ) /\ G e. T ) /\ ( ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( F =/= ( _I \|` B ) /\ b =/= ( _I \|` B ) /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( ( e e. T \|-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` b ) ` P ) .\/ ( R ` ( e o. `' b ) ) ) ) ) ) ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( ( S ` b ) ` P ) .\/ ( R ` ( G o. `' b ) ) ) ) )
29	17 28	eqtrd	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ b e. T /\ N e. T ) /\ G e. T ) /\ ( ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( F =/= ( _I \|` B ) /\ b =/= ( _I \|` B ) /\ G =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( b Y G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( ( S ` b ) ` P ) .\/ ( R ` ( G o. `' b ) ) ) ) )

Metamath Proof Explorer

Theorem cdlemk31

Proof