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Theorem mapdhval 37013
Description: Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 6-May-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 ) } ) ) ) ) )
mapdh.x |- ( ph -> X e. A )
mapdh.f |- ( ph -> F e. B )
mapdh.y |- ( ph -> Y e. E )
Assertion
Ref Expression
mapdhval |- ( ph -> ( I ` <. X , F , Y >. ) = if ( Y = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) ) )
Distinct variable groups:   x, D   x, h, F   x, J   x, M   x, N   x, .0.   x, Q   x, R   x, .-   h, X, x   h, Y, x   ph, h
Allowed substitution hints:   ph( x)   A( x, h)   B( x, h)   C( x, h)   D( h)   Q( h)   R( h)   E( x, h)   I( x, h)   J( h)   M( h)   .- ( h)   N( h)   .0. ( h)
Proof of Theorem mapdhval
StepHypRef Expression
1 otex 4933 . . 3 |- <. X , F , Y >. e. _V
2 fveq2 6191 . . . . . 6 |- ( x = <. X , F , Y >. -> ( 2nd ` x ) = ( 2nd ` <. X , F , Y >. ) )
32eqeq1d 2624 . . . . 5 |- ( x = <. X , F , Y >. -> ( ( 2nd ` x ) = .0. <-> ( 2nd ` <. X , F , Y >. ) = .0. ) )
42sneqd 4189 . . . . . . . . . 10 |- ( x = <. X , F , Y >. -> { ( 2nd ` x ) } = { ( 2nd ` <. X , F , Y >. ) } )
54fveq2d 6195 . . . . . . . . 9 |- ( x = <. X , F , Y >. -> ( N ` { ( 2nd ` x ) } ) = ( N ` { ( 2nd ` <. X , F , Y >. ) } ) )
65fveq2d 6195 . . . . . . . 8 |- ( x = <. X , F , Y >. -> ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( M ` ( N ` { ( 2nd ` <. X , F , Y >. ) } ) ) )
76eqeq1d 2624 . . . . . . 7 |- ( x = <. X , F , Y >. -> ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) <-> ( M ` ( N ` { ( 2nd ` <. X , F , Y >. ) } ) ) = ( J ` { h } ) ) )
8 fveq2 6191 . . . . . . . . . . . . 13 |- ( x = <. X , F , Y >. -> ( 1st ` x ) = ( 1st ` <. X , F , Y >. ) )
98fveq2d 6195 . . . . . . . . . . . 12 |- ( x = <. X , F , Y >. -> ( 1st ` ( 1st ` x ) ) = ( 1st ` ( 1st ` <. X , F , Y >. ) ) )
109, 2oveq12d 6668 . . . . . . . . . . 11 |- ( x = <. X , F , Y >. -> ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) = ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) )
1110sneqd 4189 . . . . . . . . . 10 |- ( x = <. X , F , Y >. -> { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } = { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } )
1211fveq2d 6195 . . . . . . . . 9 |- ( x = <. X , F , Y >. -> ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) = ( N ` { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } ) )
1312fveq2d 6195 . . . . . . . 8 |- ( x = <. X , F , Y >. -> ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( M ` ( N ` { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } ) ) )
148fveq2d 6195 . . . . . . . . . . 11 |- ( x = <. X , F , Y >. -> ( 2nd ` ( 1st ` x ) ) = ( 2nd ` ( 1st ` <. X , F , Y >. ) ) )
1514oveq1d 6665 . . . . . . . . . 10 |- ( x = <. X , F , Y >. -> ( ( 2nd ` ( 1st ` x ) ) R h ) = ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) )
1615sneqd 4189 . . . . . . . . 9 |- ( x = <. X , F , Y >. -> { ( ( 2nd ` ( 1st ` x ) ) R h ) } = { ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) } )
1716fveq2d 6195 . . . . . . . 8 |- ( x = <. X , F , Y >. -> ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) = ( J ` { ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) } ) )
1813, 17eqeq12d 2637 . . . . . . 7 |- ( x = <. X , F , Y >. -> ( ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) <-> ( M ` ( N ` { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) } ) ) )
197, 18anbi12d 747 . . . . . 6 |- ( x = <. X , F , Y >. -> ( ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) <-> ( ( M ` ( N ` { ( 2nd ` <. X , F , Y >. ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) } ) ) ) )
2019riotabidv 6613 . . . . 5 |- ( x = <. X , F , Y >. -> ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) = ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` <. X , F , Y >. ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) } ) ) ) )
213, 20ifbieq2d 4111 . . . 4 |- ( x = <. X , F , Y >. -> 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 ) } ) ) ) ) = if ( ( 2nd ` <. X , F , Y >. ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` <. X , F , Y >. ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) } ) ) ) ) )
22 mapdh.i . . . 4 |- 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 ) } ) ) ) ) )
23 mapdh.q . . . . . 6 |- Q = ( 0g ` C )
24 fvex 6201 . . . . . 6 |- ( 0g ` C ) e. _V
2523, 24eqeltri 2697 . . . . 5 |- Q e. _V
26 riotaex 6615 . . . . 5 |- ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` <. X , F , Y >. ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) } ) ) ) e. _V
2725, 26ifex 4156 . . . 4 |- if ( ( 2nd ` <. X , F , Y >. ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` <. X , F , Y >. ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) } ) ) ) ) e. _V
2821, 22, 27fvmpt 6282 . . 3 |- ( <. X , F , Y >. e. _V -> ( I ` <. X , F , Y >. ) = if ( ( 2nd ` <. X , F , Y >. ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` <. X , F , Y >. ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) } ) ) ) ) )
291, 28mp1i 13 . 2 |- ( ph -> ( I ` <. X , F , Y >. ) = if ( ( 2nd ` <. X , F , Y >. ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` <. X , F , Y >. ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) } ) ) ) ) )
30 mapdh.y . . . . 5 |- ( ph -> Y e. E )
31 ot3rdg 7184 . . . . 5 |- ( Y e. E -> ( 2nd ` <. X , F , Y >. ) = Y )
3230, 31syl 17 . . . 4 |- ( ph -> ( 2nd ` <. X , F , Y >. ) = Y )
3332eqeq1d 2624 . . 3 |- ( ph -> ( ( 2nd ` <. X , F , Y >. ) = .0. <-> Y = .0. ) )
3432sneqd 4189 . . . . . . . 8 |- ( ph -> { ( 2nd ` <. X , F , Y >. ) } = { Y } )
3534fveq2d 6195 . . . . . . 7 |- ( ph -> ( N ` { ( 2nd ` <. X , F , Y >. ) } ) = ( N ` { Y } ) )
3635fveq2d 6195 . . . . . 6 |- ( ph -> ( M ` ( N ` { ( 2nd ` <. X , F , Y >. ) } ) ) = ( M ` ( N ` { Y } ) ) )
3736eqeq1d 2624 . . . . 5 |- ( ph -> ( ( M ` ( N ` { ( 2nd ` <. X , F , Y >. ) } ) ) = ( J ` { h } ) <-> ( M ` ( N ` { Y } ) ) = ( J ` { h } ) ) )
38 mapdh.x . . . . . . . . . . 11 |- ( ph -> X e. A )
39 mapdh.f . . . . . . . . . . 11 |- ( ph -> F e. B )
40 ot1stg 7182 . . . . . . . . . . 11 |- ( ( X e. A /\ F e. B /\ Y e. E ) -> ( 1st ` ( 1st ` <. X , F , Y >. ) ) = X )
4138, 39, 30, 40syl3anc 1326 . . . . . . . . . 10 |- ( ph -> ( 1st ` ( 1st ` <. X , F , Y >. ) ) = X )
4241, 32oveq12d 6668 . . . . . . . . 9 |- ( ph -> ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) = ( X .- Y ) )
4342sneqd 4189 . . . . . . . 8 |- ( ph -> { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } = { ( X .- Y ) } )
4443fveq2d 6195 . . . . . . 7 |- ( ph -> ( N ` { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } ) = ( N ` { ( X .- Y ) } ) )
4544fveq2d 6195 . . . . . 6 |- ( ph -> ( M ` ( N ` { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } ) ) = ( M ` ( N ` { ( X .- Y ) } ) ) )
46 ot2ndg 7183 . . . . . . . . . 10 |- ( ( X e. A /\ F e. B /\ Y e. E ) -> ( 2nd ` ( 1st ` <. X , F , Y >. ) ) = F )
4738, 39, 30, 46syl3anc 1326 . . . . . . . . 9 |- ( ph -> ( 2nd ` ( 1st ` <. X , F , Y >. ) ) = F )
4847oveq1d 6665 . . . . . . . 8 |- ( ph -> ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) = ( F R h ) )
4948sneqd 4189 . . . . . . 7 |- ( ph -> { ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) } = { ( F R h ) } )
5049fveq2d 6195 . . . . . 6 |- ( ph -> ( J ` { ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) } ) = ( J ` { ( F R h ) } ) )
5145, 50eqeq12d 2637 . . . . 5 |- ( ph -> ( ( M ` ( N ` { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) } ) <-> ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) )
5237, 51anbi12d 747 . . . 4 |- ( ph -> ( ( ( M ` ( N ` { ( 2nd ` <. X , F , Y >. ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) } ) ) <-> ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) )
5352riotabidv 6613 . . 3 |- ( ph -> ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` <. X , F , Y >. ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) } ) ) ) = ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) )
5433, 53ifbieq2d 4111 . 2 |- ( ph -> if ( ( 2nd ` <. X , F , Y >. ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` <. X , F , Y >. ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) } ) ) ) ) = if ( Y = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) ) )
5529, 54eqtrd 2656 1 |- ( ph -> ( I ` <. X , F , Y >. ) = if ( Y = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) ) )
Colors of variables: wff setvar class
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:  mapdhval0  37014  mapdhval2  37015  hdmap1valc  37093
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