Theorem cdlemkuel-2N

Description: Part of proof of Lemma K of Crawley p. 118. Conditions for the sigma_2 (p) function to be a translation. TODO: combine cdlemkj ? (Contributed by NM, 2-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 ) ) ) ) ) )
Assertion	cdlemkuel-2N	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( V ` G ) e. T )

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	1 2 3 4 5 6 7 8 9 10 11	cdlemkuel	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( V ` G ) e. T )