Theorem cdlemg1bOLDN

Description: This theorem can be used to shorten F = hypothesis that have the form of the conclusion. TODO: fix comment. (Contributed by NM, 16-Apr-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdlemg1.b	|- B = ( Base ` K )
	cdlemg1.l	|- .<_ = ( le ` K )
	cdlemg1.j	|- .\/ = ( join ` K )
	cdlemg1.m	|- ./\ = ( meet ` K )
	cdlemg1.a	|- A = ( Atoms ` K )
	cdlemg1.h	|- H = ( LHyp ` K )
	cdlemg1b.u	|- U = ( ( P .\/ Q ) ./\ W )
	cdlemg1b.d	|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
	cdlemg1b.e	|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
	cdlemg1b.f	|- F = ( iota_ f e. T ( f ` P ) = Q )
	cdlemg1b.t	|- T = ( ( LTrn ` K ) ` W )
Assertion	cdlemg1bOLDN	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemg1.b	|- B = ( Base ` K )
2		cdlemg1.l	|- .<_ = ( le ` K )
3		cdlemg1.j	|- .\/ = ( join ` K )
4		cdlemg1.m	|- ./\ = ( meet ` K )
5		cdlemg1.a	|- A = ( Atoms ` K )
6		cdlemg1.h	|- H = ( LHyp ` K )
7		cdlemg1b.u	|- U = ( ( P .\/ Q ) ./\ W )
8		cdlemg1b.d	|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
9		cdlemg1b.e	|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
10		cdlemg1b.f	|- F = ( iota_ f e. T ( f ` P ) = Q )
11		cdlemg1b.t	|- T = ( ( LTrn ` K ) ` W )
12		eqid	|- ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) ) = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) )
13	1 2 3 4 5 6 7 8 9 12 11 10	cdlemg1b2	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) ) )