Theorem cdlemk13

Description: Part of proof of Lemma K of Crawley p. 118. Line 13 on p. 119. O , D are k_1, f_1. (Contributed by NM, 1-Jul-2013)

	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 )
Assertion	cdlemk13	|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( O ` P ) = ( ( P .\/ ( R ` D ) ) ./\ ( ( N ` P ) .\/ ( R ` ( D o. `' F ) ) ) ) )

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	10	fveq1i	|- ( O ` P ) = ( ( S ` D ) ` P )
12	1 2 3 5 6 7 8 4 9	cdlemksv2	|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( ( S ` D ) ` P ) = ( ( P .\/ ( R ` D ) ) ./\ ( ( N ` P ) .\/ ( R ` ( D o. `' F ) ) ) ) )
13	11 12	syl5eq	|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( O ` P ) = ( ( P .\/ ( R ` D ) ) ./\ ( ( N ` P ) .\/ ( R ` ( D o. `' F ) ) ) ) )