cdlemk7u-2N

Description: Part of proof of Lemma K of Crawley p. 118. Line 5, p. 119 for the sigma_2 case. (Contributed by NM, 5-Jul-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdlemk2.b	\|- B = ( Base ` K )
	cdlemk2.l	\|- .<_ = ( le ` K )
	cdlemk2.j	\|- .\/ = ( join ` K )
	cdlemk2.m	\|- ./\ = ( meet ` K )
	cdlemk2.a	\|- A = ( Atoms ` K )
	cdlemk2.h	\|- H = ( LHyp ` K )
	cdlemk2.t	\|- T = ( ( LTrn ` K ) ` W )
	cdlemk2.r	\|- R = ( ( trL ` K ) ` W )
	cdlemk2.s	\|- S = ( f e. T \|-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
	cdlemk2.q	\|- Q = ( S ` C )
	cdlemk2.v	\|- V = ( d e. T \|-> ( iota_ k e. T ( k ` P ) = ( ( P .\/ ( R ` d ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( d o. `' C ) ) ) ) ) )
	cdlemk2.z	\|- Z = ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' C ) ) .\/ ( R ` ( X o. `' C ) ) ) )
Assertion	cdlemk7u-2N	\|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( X e. T /\ X =/= ( _I \|` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( V ` G ) ` P ) .<_ ( ( ( V ` X ) ` P ) .\/ Z ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemk2.b	\|- B = ( Base ` K )
2		cdlemk2.l	\|- .<_ = ( le ` K )
3		cdlemk2.j	\|- .\/ = ( join ` K )
4		cdlemk2.m	\|- ./\ = ( meet ` K )
5		cdlemk2.a	\|- A = ( Atoms ` K )
6		cdlemk2.h	\|- H = ( LHyp ` K )
7		cdlemk2.t	\|- T = ( ( LTrn ` K ) ` W )
8		cdlemk2.r	\|- R = ( ( trL ` K ) ` W )
9		cdlemk2.s	\|- S = ( f e. T \|-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
10		cdlemk2.q	\|- Q = ( S ` C )
11		cdlemk2.v	\|- V = ( d e. T \|-> ( iota_ k e. T ( k ` P ) = ( ( P .\/ ( R ` d ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( d o. `' C ) ) ) ) ) )
12		cdlemk2.z	\|- Z = ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' C ) ) .\/ ( R ` ( X o. `' C ) ) ) )
13		simp11	\|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( X e. T /\ X =/= ( _I \|` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> K e. HL )
14		simp12	\|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( X e. T /\ X =/= ( _I \|` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> W e. H )
15	13 14	jca	\|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( X e. T /\ X =/= ( _I \|` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( K e. HL /\ W e. H ) )
16		simp211	\|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( X e. T /\ X =/= ( _I \|` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F e. T )
17		simp212	\|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( X e. T /\ X =/= ( _I \|` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> C e. T )
18		simp213	\|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( X e. T /\ X =/= ( _I \|` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> N e. T )
19		simp22l	\|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( X e. T /\ X =/= ( _I \|` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> G e. T )
20		simp23l	\|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( X e. T /\ X =/= ( _I \|` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> X e. T )
21	18 19 20	3jca	\|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( X e. T /\ X =/= ( _I \|` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( N e. T /\ G e. T /\ X e. T ) )
22		simp33	\|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( X e. T /\ X =/= ( _I \|` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( P e. A /\ -. P .<_ W ) )
23		simp13	\|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( X e. T /\ X =/= ( _I \|` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` F ) = ( R ` N ) )
24		simp32l	\|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( X e. T /\ X =/= ( _I \|` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F =/= ( _I \|` B ) )
25		simp32r	\|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( X e. T /\ X =/= ( _I \|` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> C =/= ( _I \|` B ) )
26		simp22r	\|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( X e. T /\ X =/= ( _I \|` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> G =/= ( _I \|` B ) )
27	24 25 26	3jca	\|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( X e. T /\ X =/= ( _I \|` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( F =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) /\ G =/= ( _I \|` B ) ) )
28		simp23r	\|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( X e. T /\ X =/= ( _I \|` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> X =/= ( _I \|` B ) )
29		simp31	\|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( X e. T /\ X =/= ( _I \|` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) )
30	1 2 3 4 5 6 7 8 9 10 11 12	cdlemk7u	\|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ C e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) /\ G =/= ( _I \|` B ) ) /\ X =/= ( _I \|` B ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) ) ) -> ( ( V ` G ) ` P ) .<_ ( ( ( V ` X ) ` P ) .\/ Z ) )
31	15 16 17 21 22 23 27 28 29 30	syl333anc	\|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) ) /\ ( X e. T /\ X =/= ( _I \|` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( F =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( V ` G ) ` P ) .<_ ( ( ( V ` X ) ` P ) .\/ Z ) )

Metamath Proof Explorer

Theorem cdlemk7u-2N

Proof