Theorem cdlemg2dN

Description: This theorem can be used to shorten G = hypothesis. TODO: Fix comment. (Contributed by NM, 21-Apr-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdlemg2.b	|- B = ( Base ` K )
	cdlemg2.l	|- .<_ = ( le ` K )
	cdlemg2.j	|- .\/ = ( join ` K )
	cdlemg2.m	|- ./\ = ( meet ` K )
	cdlemg2.a	|- A = ( Atoms ` K )
	cdlemg2.h	|- H = ( LHyp ` K )
	cdlemg2.t	|- T = ( ( LTrn ` K ) ` W )
	cdlemg2.u	|- U = ( ( P .\/ Q ) ./\ W )
	cdlemg2.d	|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
	cdlemg2.e	|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
	cdlemg2.g	|- G = ( 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 ) )
Assertion	cdlemg2dN	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ ( F ` P ) = Q ) ) -> F = G )

Proof

Step	Hyp	Ref	Expression
1		cdlemg2.b	|- B = ( Base ` K )
2		cdlemg2.l	|- .<_ = ( le ` K )
3		cdlemg2.j	|- .\/ = ( join ` K )
4		cdlemg2.m	|- ./\ = ( meet ` K )
5		cdlemg2.a	|- A = ( Atoms ` K )
6		cdlemg2.h	|- H = ( LHyp ` K )
7		cdlemg2.t	|- T = ( ( LTrn ` K ) ` W )
8		cdlemg2.u	|- U = ( ( P .\/ Q ) ./\ W )
9		cdlemg2.d	|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
10		cdlemg2.e	|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
11		cdlemg2.g	|- G = ( 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 ) )
12		simp3l	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ ( F ` P ) = Q ) ) -> F e. T )
13		simp1	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ ( F ` P ) = Q ) ) -> ( K e. HL /\ W e. H ) )
14		simp2l	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ ( F ` P ) = Q ) ) -> ( P e. A /\ -. P .<_ W ) )
15		simp2r	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ ( F ` P ) = Q ) ) -> ( Q e. A /\ -. Q .<_ W ) )
16		simp3r	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ ( F ` P ) = Q ) ) -> ( F ` P ) = Q )
17	1 2 3 4 5 6 7 8 9 10 11	cdlemg2cN	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = Q ) -> ( F e. T <-> F = G ) )
18	13 14 15 16 17	syl31anc	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ ( F ` P ) = Q ) ) -> ( F e. T <-> F = G ) )
19	12 18	mpbid	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ ( F ` P ) = Q ) ) -> F = G )