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Theorem mapdhval 36521
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 4712 . . 3  |-  <. X ,  F ,  Y >.  e. 
_V
2 fveq2 5864 . . . . . 6  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( 2nd `  x )  =  ( 2nd `  <. X ,  F ,  Y >. ) )
32eqeq1d 2469 . . . . 5  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( ( 2nd `  x )  =  .0.  <->  ( 2nd ` 
<. X ,  F ,  Y >. )  =  .0.  ) )
42sneqd 4039 . . . . . . . . . 10  |-  ( x  =  <. X ,  F ,  Y >.  ->  { ( 2nd `  x ) }  =  { ( 2nd `  <. X ,  F ,  Y >. ) } )
54fveq2d 5868 . . . . . . . . 9  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( N `
 { ( 2nd `  x ) } )  =  ( N `  { ( 2nd `  <. X ,  F ,  Y >. ) } ) )
65fveq2d 5868 . . . . . . . 8  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( M `
 ( N `  { ( 2nd `  x
) } ) )  =  ( M `  ( N `  { ( 2nd `  <. X ,  F ,  Y >. ) } ) ) )
76eqeq1d 2469 . . . . . . 7  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( ( M `  ( N `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  <->  ( M `  ( N `  { ( 2nd `  <. X ,  F ,  Y >. ) } ) )  =  ( J `  {
h } ) ) )
8 fveq2 5864 . . . . . . . . . . . . 13  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( 1st `  x )  =  ( 1st `  <. X ,  F ,  Y >. ) )
98fveq2d 5868 . . . . . . . . . . . 12  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( 1st `  ( 1st `  x
) )  =  ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) ) )
109, 2oveq12d 6300 . . . . . . . . . . 11  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( ( 1st `  ( 1st `  x ) )  .-  ( 2nd `  x ) )  =  ( ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) )
1110sneqd 4039 . . . . . . . . . 10  |-  ( x  =  <. X ,  F ,  Y >.  ->  { ( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) }  =  { ( ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) } )
1211fveq2d 5868 . . . . . . . . 9  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( N `
 { ( ( 1st `  ( 1st `  x ) )  .-  ( 2nd `  x ) ) } )  =  ( N `  {
( ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) } ) )
1312fveq2d 5868 . . . . . . . 8  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( M `
 ( N `  { ( ( 1st `  ( 1st `  x
) )  .-  ( 2nd `  x ) ) } ) )  =  ( M `  ( N `  { (
( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) } ) ) )
148fveq2d 5868 . . . . . . . . . . 11  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( 2nd `  ( 1st `  x
) )  =  ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) )
1514oveq1d 6297 . . . . . . . . . 10  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( ( 2nd `  ( 1st `  x ) ) R h )  =  ( ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) R h ) )
1615sneqd 4039 . . . . . . . . 9  |-  ( x  =  <. X ,  F ,  Y >.  ->  { ( ( 2nd `  ( 1st `  x ) ) R h ) }  =  { ( ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) R h ) } )
1716fveq2d 5868 . . . . . . . 8  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } )  =  ( J `  { ( ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) R h ) } ) )
1813, 17eqeq12d 2489 . . . . . . 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 710 . . . . . 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 6245 . . . . 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 3964 . . . 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 5874 . . . . . 6  |-  ( 0g
`  C )  e. 
_V
2523, 24eqeltri 2551 . . . . 5  |-  Q  e. 
_V
26 riotaex 6247 . . . . 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 4008 . . . 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 5948 . . 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 12 . 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 6797 . . . . 5  |-  ( Y  e.  E  ->  ( 2nd `  <. X ,  F ,  Y >. )  =  Y )
3230, 31syl 16 . . . 4  |-  ( ph  ->  ( 2nd `  <. X ,  F ,  Y >. )  =  Y )
3332eqeq1d 2469 . . 3  |-  ( ph  ->  ( ( 2nd `  <. X ,  F ,  Y >. )  =  .0.  <->  Y  =  .0.  ) )
3432sneqd 4039 . . . . . . . 8  |-  ( ph  ->  { ( 2nd `  <. X ,  F ,  Y >. ) }  =  { Y } )
3534fveq2d 5868 . . . . . . 7  |-  ( ph  ->  ( N `  {
( 2nd `  <. X ,  F ,  Y >. ) } )  =  ( N `  { Y } ) )
3635fveq2d 5868 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( 2nd `  <. X ,  F ,  Y >. ) } ) )  =  ( M `
 ( N `  { Y } ) ) )
3736eqeq1d 2469 . . . . 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 6795 . . . . . . . . . . 11  |-  ( ( X  e.  A  /\  F  e.  B  /\  Y  e.  E )  ->  ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  =  X )
4138, 39, 30, 40syl3anc 1228 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  =  X )
4241, 32oveq12d 6300 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) )  =  ( X 
.-  Y ) )
4342sneqd 4039 . . . . . . . 8  |-  ( ph  ->  { ( ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) }  =  { ( X 
.-  Y ) } )
4443fveq2d 5868 . . . . . . 7  |-  ( ph  ->  ( N `  {
( ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) } )  =  ( N `  {
( X  .-  Y
) } ) )
4544fveq2d 5868 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { (
( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) } ) )  =  ( M `  ( N `  { ( X  .-  Y ) } ) ) )
46 ot2ndg 6796 . . . . . . . . . 10  |-  ( ( X  e.  A  /\  F  e.  B  /\  Y  e.  E )  ->  ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) )  =  F )
4738, 39, 30, 46syl3anc 1228 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) )  =  F )
4847oveq1d 6297 . . . . . . . 8  |-  ( ph  ->  ( ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) R h )  =  ( F R h ) )
4948sneqd 4039 . . . . . . 7  |-  ( ph  ->  { ( ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) R h ) }  =  {
( F R h ) } )
5049fveq2d 5868 . . . . . 6  |-  ( ph  ->  ( J `  {
( ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) R h ) } )  =  ( J `  { ( F R h ) } ) )
5145, 50eqeq12d 2489 . . . . 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 710 . . . 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 6245 . . 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 3964 . 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 2508 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 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   ifcif 3939   {csn 4027   <.cotp 4035    |-> cmpt 4505   ` cfv 5586   iota_crio 6242  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780   0gc0g 14688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-ot 4036  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fv 5594  df-riota 6243  df-ov 6285  df-1st 6781  df-2nd 6782
This theorem is referenced by:  mapdhval0  36522  mapdhval2  36523  hdmap1valc  36601
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