Theorem mapdhval0

Description: Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015)

	Ref	Expression
Hypotheses	mapdh.q	|- Q = ( 0g ` C )
	mapdh.i	|- I = ( x e. _V |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) )
	mapdh0.o	|- .0. = ( 0g ` U )
	mapdh0.x	|- ( ph -> X e. A )
	mapdh0.f	|- ( ph -> F e. B )
Assertion	mapdhval0	|- ( ph -> ( I ` <. X , F , .0. >. ) = Q )

Proof

Step	Hyp	Ref	Expression
1		mapdh.q	|- Q = ( 0g ` C )
2		mapdh.i	|- I = ( x e. _V |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) )
3		mapdh0.o	|- .0. = ( 0g ` U )
4		mapdh0.x	|- ( ph -> X e. A )
5		mapdh0.f	|- ( ph -> F e. B )
6	3	fvexi	|- .0. e. _V
7	6	a1i	|- ( ph -> .0. e. _V )
8	1 2 4 5 7	mapdhval	|- ( ph -> ( I ` <. X , F , .0. >. ) = if ( .0. = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { .0. } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- .0. ) } ) ) = ( J ` { ( F R h ) } ) ) ) ) )
9		eqid	|- .0. = .0.
10	9	iftruei	|- if ( .0. = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { .0. } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- .0. ) } ) ) = ( J ` { ( F R h ) } ) ) ) ) = Q
11	8 10	eqtrdi	|- ( ph -> ( I ` <. X , F , .0. >. ) = Q )