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Theorem mapdhval 32536
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 4254 . . 3  |-  <. X ,  F ,  Y >.  e. 
_V
2 fveq2 5541 . . . . . 6  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( 2nd `  x )  =  ( 2nd `  <. X ,  F ,  Y >. ) )
32eqeq1d 2304 . . . . 5  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( ( 2nd `  x )  =  .0.  <->  ( 2nd ` 
<. X ,  F ,  Y >. )  =  .0.  ) )
42sneqd 3666 . . . . . . . . . 10  |-  ( x  =  <. X ,  F ,  Y >.  ->  { ( 2nd `  x ) }  =  { ( 2nd `  <. X ,  F ,  Y >. ) } )
54fveq2d 5545 . . . . . . . . 9  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( N `
 { ( 2nd `  x ) } )  =  ( N `  { ( 2nd `  <. X ,  F ,  Y >. ) } ) )
65fveq2d 5545 . . . . . . . 8  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( M `
 ( N `  { ( 2nd `  x
) } ) )  =  ( M `  ( N `  { ( 2nd `  <. X ,  F ,  Y >. ) } ) ) )
76eqeq1d 2304 . . . . . . 7  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( ( M `  ( N `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  <->  ( M `  ( N `  { ( 2nd `  <. X ,  F ,  Y >. ) } ) )  =  ( J `  {
h } ) ) )
8 fveq2 5541 . . . . . . . . . . . . 13  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( 1st `  x )  =  ( 1st `  <. X ,  F ,  Y >. ) )
98fveq2d 5545 . . . . . . . . . . . 12  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( 1st `  ( 1st `  x
) )  =  ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) ) )
109, 2oveq12d 5892 . . . . . . . . . . 11  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( ( 1st `  ( 1st `  x ) )  .-  ( 2nd `  x ) )  =  ( ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) )
1110sneqd 3666 . . . . . . . . . 10  |-  ( x  =  <. X ,  F ,  Y >.  ->  { ( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) }  =  { ( ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) } )
1211fveq2d 5545 . . . . . . . . 9  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( N `
 { ( ( 1st `  ( 1st `  x ) )  .-  ( 2nd `  x ) ) } )  =  ( N `  {
( ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) } ) )
1312fveq2d 5545 . . . . . . . 8  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( M `
 ( N `  { ( ( 1st `  ( 1st `  x
) )  .-  ( 2nd `  x ) ) } ) )  =  ( M `  ( N `  { (
( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) } ) ) )
148fveq2d 5545 . . . . . . . . . . 11  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( 2nd `  ( 1st `  x
) )  =  ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) )
1514oveq1d 5889 . . . . . . . . . 10  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( ( 2nd `  ( 1st `  x ) ) R h )  =  ( ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) R h ) )
1615sneqd 3666 . . . . . . . . 9  |-  ( x  =  <. X ,  F ,  Y >.  ->  { ( ( 2nd `  ( 1st `  x ) ) R h ) }  =  { ( ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) R h ) } )
1716fveq2d 5545 . . . . . . . 8  |-  ( x  =  <. X ,  F ,  Y >.  ->  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } )  =  ( J `  { ( ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) R h ) } ) )
1813, 17eqeq12d 2310 . . . . . . 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 691 . . . . . 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 6322 . . . . 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 3598 . . . 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 5555 . . . . . 6  |-  ( 0g
`  C )  e. 
_V
2523, 24eqeltri 2366 . . . . 5  |-  Q  e. 
_V
26 riotaex 6324 . . . . 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 3636 . . . 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 5618 . . 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 11 . 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 6152 . . . . 5  |-  ( Y  e.  E  ->  ( 2nd `  <. X ,  F ,  Y >. )  =  Y )
3230, 31syl 15 . . . 4  |-  ( ph  ->  ( 2nd `  <. X ,  F ,  Y >. )  =  Y )
3332eqeq1d 2304 . . 3  |-  ( ph  ->  ( ( 2nd `  <. X ,  F ,  Y >. )  =  .0.  <->  Y  =  .0.  ) )
3432sneqd 3666 . . . . . . . 8  |-  ( ph  ->  { ( 2nd `  <. X ,  F ,  Y >. ) }  =  { Y } )
3534fveq2d 5545 . . . . . . 7  |-  ( ph  ->  ( N `  {
( 2nd `  <. X ,  F ,  Y >. ) } )  =  ( N `  { Y } ) )
3635fveq2d 5545 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( 2nd `  <. X ,  F ,  Y >. ) } ) )  =  ( M `
 ( N `  { Y } ) ) )
3736eqeq1d 2304 . . . . 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 6150 . . . . . . . . . . 11  |-  ( ( X  e.  A  /\  F  e.  B  /\  Y  e.  E )  ->  ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  =  X )
4138, 39, 30, 40syl3anc 1182 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  =  X )
4241, 32oveq12d 5892 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) )  =  ( X 
.-  Y ) )
4342sneqd 3666 . . . . . . . 8  |-  ( ph  ->  { ( ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) }  =  { ( X 
.-  Y ) } )
4443fveq2d 5545 . . . . . . 7  |-  ( ph  ->  ( N `  {
( ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) } )  =  ( N `  {
( X  .-  Y
) } ) )
4544fveq2d 5545 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { (
( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) } ) )  =  ( M `  ( N `  { ( X  .-  Y ) } ) ) )
46 ot2ndg 6151 . . . . . . . . . 10  |-  ( ( X  e.  A  /\  F  e.  B  /\  Y  e.  E )  ->  ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) )  =  F )
4738, 39, 30, 46syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) )  =  F )
4847oveq1d 5889 . . . . . . . 8  |-  ( ph  ->  ( ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) R h )  =  ( F R h ) )
4948sneqd 3666 . . . . . . 7  |-  ( ph  ->  { ( ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) R h ) }  =  {
( F R h ) } )
5049fveq2d 5545 . . . . . 6  |-  ( ph  ->  ( J `  {
( ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) R h ) } )  =  ( J `  { ( F R h ) } ) )
5145, 50eqeq12d 2310 . . . . 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 691 . . . 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 6322 . . 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 3598 . 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 2328 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 set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   ifcif 3578   {csn 3653   <.cotp 3657    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   iota_crio 6313   0gc0g 13416
This theorem is referenced by:  mapdhval0  32537  mapdhval2  32538  hdmap1valc  32616
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-ot 3663  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-riota 6320
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