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Theorem mapdhval0 35364
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
StepHypRef 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 5889 . . . . 5  |-  ( 0g
`  U )  e. 
_V
75, 6eqeltri 2545 . . . 4  |-  .0.  e.  _V
87a1i 11 . . 3  |-  ( ph  ->  .0.  e.  _V )
91, 2, 3, 4, 8mapdhval 35363 . 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 2471 . . 3  |-  .0.  =  .0.
1110iftruei 3879 . 2  |-  if (  .0.  =  .0.  ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( N `
 {  .0.  }
) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  .0.  ) } ) )  =  ( J `  { ( F R h ) } ) ) ) )  =  Q
129, 11syl6eq 2521 1  |-  ( ph  ->  ( I `  <. X ,  F ,  .0.  >.
)  =  Q )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031   ifcif 3872   {csn 3959   <.cotp 3967    |-> cmpt 4454   ` cfv 5589   iota_crio 6269  (class class class)co 6308   1stc1st 6810   2ndc2nd 6811   0gc0g 15416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-ot 3968  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fv 5597  df-riota 6270  df-ov 6311  df-1st 6812  df-2nd 6813
This theorem is referenced by:  mapdhcl  35366  mapdh6bN  35376  mapdh6cN  35377  mapdh6dN  35378  mapdh8  35428
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