cdlemk11tc

Description: Part of proof of Lemma K of Crawley p. 118. Lemma for Eq. 5, p. 119. G , I stand for g, h. TODO: fix comment. (Contributed by NM, 21-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	cdlemk11tc	\|- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( [_ G / g ]_ X ` P ) .<_ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` ( I o. `' G ) ) ) )

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	cdlemk11tb	\|- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> [_ G / g ]_ Y .<_ ( [_ I / g ]_ Y .\/ ( R ` ( I o. `' G ) ) ) )
13		simp31	\|- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> b e. T )
14		simp32	\|- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) )
15	13 14	jca	\|- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( b e. T /\ ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) )
16	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 )
17	15 16	syld3an3	\|- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( [_ G / g ]_ X ` P ) = [_ G / g ]_ Y )
18		simp11	\|- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( K e. HL /\ W e. H ) )
19		simp12	\|- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( F e. T /\ F =/= ( _I \|` B ) ) )
20		simp331	\|- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> I e. T )
21		simp332	\|- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> I =/= ( _I \|` B ) )
22	20 21	jca	\|- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( I e. T /\ I =/= ( _I \|` B ) ) )
23		simp2	\|- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) )
24		simp321	\|- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> b =/= ( _I \|` B ) )
25		simp322	\|- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( R ` b ) =/= ( R ` F ) )
26		simp333	\|- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( R ` b ) =/= ( R ` I ) )
27	24 25 26	3jca	\|- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( b =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` I ) ) )
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 ) ) /\ ( I e. T /\ I =/= ( _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 ` I ) ) ) ) -> ( [_ I / g ]_ X ` P ) = [_ I / g ]_ Y )
29	18 19 22 23 13 27 28	syl312anc	\|- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( [_ I / g ]_ X ` P ) = [_ I / g ]_ Y )
30	29	oveq1d	\|- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( ( [_ I / g ]_ X ` P ) .\/ ( R ` ( I o. `' G ) ) ) = ( [_ I / g ]_ Y .\/ ( R ` ( I o. `' G ) ) ) )
31	12 17 30	3brtr4d	\|- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I \|` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( [_ G / g ]_ X ` P ) .<_ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` ( I o. `' G ) ) ) )

Metamath Proof Explorer

Theorem cdlemk11tc

Proof