Theorem mapdhval0 37014

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

Hypotheses

Ref	Expression
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

Ref	Expression
mapdhval0	|- ( ph -> ( I ` <. X , F , .0. >. ) = Q )

Distinct variable groups:   x, D   x, h, F   x, J   x, M   x, N   x, .0.   x, Q   x, R   x, .-   h, X, x   ph, h   .0. , h

Allowed substitution hints:   ph( x)   A( x, h)   B( x, h)   C( x, h)   D( h)   Q( h)   R( h)   U( x, h)   I( x, h)   J( h)   M( h)   .- ( h)   N( h)

Proof of Theorem mapdhval0

Step	Hyp	Ref	Expression
1		mapdh.q	. . 3 |- Q = ( 0g ` C )
2		mapdh.i	. . 3 |- 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.x	. . 3 |- ( ph -> X e. A )
4		mapdh0.f	. . 3 |- ( ph -> F e. B )
5		mapdh0.o	. . . . 5 |- .0. = ( 0g ` U )
6		fvex 6201	. . . . 5 |- ( 0g ` U ) e. _V
7	5, 6	eqeltri 2697	. . . 4 |- .0. e. _V
8	7	a1i 11	. . 3 |- ( ph -> .0. e. _V )
9	1, 2, 3, 4, 8	mapdhval 37013	. 2 |- ( 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 ) } ) ) ) ) )
10		eqid 2622	. . 3 |- .0. = .0.
11	10	iftruei 4093	. 2 |- if ( .0. = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { .0. } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- .0. ) } ) ) = ( J ` { ( F R h ) } ) ) ) ) = Q
12	9, 11	syl6eq 2672	1 |- ( ph -> ( I ` <. X , F , .0. >. ) = Q )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   ifcif 4086   {csn 4177   <.cotp 4185   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   0gc0g 16100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-ot 4186  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653  df-1st 7168  df-2nd 7169

This theorem is referenced by:  mapdhcl  37016  mapdh6bN  37026  mapdh6cN  37027  mapdh6dN  37028  mapdh8  37078