Theorem cdlemkuvN

Description: Part of proof of Lemma K of Crawley p. 118. Value of the sigma_1 (p) function U . (Contributed by NM, 2-Jul-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdlemk1.b	|- B = ( Base ` K )
	cdlemk1.l	|- .<_ = ( le ` K )
	cdlemk1.j	|- .\/ = ( join ` K )
	cdlemk1.m	|- ./\ = ( meet ` K )
	cdlemk1.a	|- A = ( Atoms ` K )
	cdlemk1.h	|- H = ( LHyp ` K )
	cdlemk1.t	|- T = ( ( LTrn ` K ) ` W )
	cdlemk1.r	|- R = ( ( trL ` K ) ` W )
	cdlemk1.s	|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
	cdlemk1.o	|- O = ( S ` D )
	cdlemk1.u	|- U = ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( O ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) )
Assertion	cdlemkuvN	|- ( G e. T -> ( U ` G ) = ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemk1.b	|- B = ( Base ` K )
2		cdlemk1.l	|- .<_ = ( le ` K )
3		cdlemk1.j	|- .\/ = ( join ` K )
4		cdlemk1.m	|- ./\ = ( meet ` K )
5		cdlemk1.a	|- A = ( Atoms ` K )
6		cdlemk1.h	|- H = ( LHyp ` K )
7		cdlemk1.t	|- T = ( ( LTrn ` K ) ` W )
8		cdlemk1.r	|- R = ( ( trL ` K ) ` W )
9		cdlemk1.s	|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
10		cdlemk1.o	|- O = ( S ` D )
11		cdlemk1.u	|- U = ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( O ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) )
12	1 2 3 5 6 7 8 4 11	cdlemksv	|- ( G e. T -> ( U ` G ) = ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) )