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Theorem imasaddvallem 13754
Description: The operation of an image structure is defined to distribute over the mapping function. (Contributed by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
imasaddf.f  |-  ( ph  ->  F : V -onto-> B
)
imasaddf.e  |-  ( (
ph  /\  ( a  e.  V  /\  b  e.  V )  /\  (
p  e.  V  /\  q  e.  V )
)  ->  ( (
( F `  a
)  =  ( F `
 p )  /\  ( F `  b )  =  ( F `  q ) )  -> 
( F `  (
a  .x.  b )
)  =  ( F `
 ( p  .x.  q ) ) ) )
imasaddflem.a  |-  ( ph  -> 
.xb  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>. } )
Assertion
Ref Expression
imasaddvallem  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  ( ( F `  X )  .xb  ( F `  Y
) )  =  ( F `  ( X 
.x.  Y ) ) )
Distinct variable groups:    q, p, B    a, b, p, q, V    .x. , p, q    X, p    F, a, b, p, q    ph, a, b, p, q    .xb , a, b, p, q    Y, p, q
Allowed substitution hints:    B( a, b)    .x. ( a, b)    X( q, a, b)    Y( a, b)

Proof of Theorem imasaddvallem
StepHypRef Expression
1 df-ov 6084 . 2  |-  ( ( F `  X ) 
.xb  ( F `  Y ) )  =  (  .xb  `  <. ( F `  X ) ,  ( F `  Y ) >. )
2 imasaddf.f . . . . . 6  |-  ( ph  ->  F : V -onto-> B
)
3 imasaddf.e . . . . . 6  |-  ( (
ph  /\  ( a  e.  V  /\  b  e.  V )  /\  (
p  e.  V  /\  q  e.  V )
)  ->  ( (
( F `  a
)  =  ( F `
 p )  /\  ( F `  b )  =  ( F `  q ) )  -> 
( F `  (
a  .x.  b )
)  =  ( F `
 ( p  .x.  q ) ) ) )
4 imasaddflem.a . . . . . 6  |-  ( ph  -> 
.xb  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>. } )
52, 3, 4imasaddfnlem 13753 . . . . 5  |-  ( ph  -> 
.xb  Fn  ( B  X.  B ) )
6 fnfun 5542 . . . . 5  |-  (  .xb  Fn  ( B  X.  B
)  ->  Fun  .xb  )
75, 6syl 16 . . . 4  |-  ( ph  ->  Fun  .xb  )
873ad2ant1 978 . . 3  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  Fun  .xb  )
9 fveq2 5728 . . . . . . . . . . 11  |-  ( p  =  X  ->  ( F `  p )  =  ( F `  X ) )
109opeq1d 3990 . . . . . . . . . 10  |-  ( p  =  X  ->  <. ( F `  p ) ,  ( F `  Y ) >.  =  <. ( F `  X ) ,  ( F `  Y ) >. )
11 oveq1 6088 . . . . . . . . . . 11  |-  ( p  =  X  ->  (
p  .x.  Y )  =  ( X  .x.  Y ) )
1211fveq2d 5732 . . . . . . . . . 10  |-  ( p  =  X  ->  ( F `  ( p  .x.  Y ) )  =  ( F `  ( X  .x.  Y ) ) )
1310, 12opeq12d 3992 . . . . . . . . 9  |-  ( p  =  X  ->  <. <. ( F `  p ) ,  ( F `  Y ) >. ,  ( F `  ( p 
.x.  Y ) )
>.  =  <. <. ( F `  X ) ,  ( F `  Y ) >. ,  ( F `  ( X 
.x.  Y ) )
>. )
1413sneqd 3827 . . . . . . . 8  |-  ( p  =  X  ->  { <. <.
( F `  p
) ,  ( F `
 Y ) >. ,  ( F `  ( p  .x.  Y ) ) >. }  =  { <. <. ( F `  X ) ,  ( F `  Y )
>. ,  ( F `  ( X  .x.  Y
) ) >. } )
1514ssiun2s 4135 . . . . . . 7  |-  ( X  e.  V  ->  { <. <.
( F `  X
) ,  ( F `
 Y ) >. ,  ( F `  ( X  .x.  Y ) ) >. }  C_  U_ p  e.  V  { <. <. ( F `  p ) ,  ( F `  Y ) >. ,  ( F `  ( p 
.x.  Y ) )
>. } )
16153ad2ant2 979 . . . . . 6  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  { <. <. ( F `  X ) ,  ( F `  Y ) >. ,  ( F `  ( X 
.x.  Y ) )
>. }  C_  U_ p  e.  V  { <. <. ( F `  p ) ,  ( F `  Y ) >. ,  ( F `  ( p 
.x.  Y ) )
>. } )
17 fveq2 5728 . . . . . . . . . . . . 13  |-  ( q  =  Y  ->  ( F `  q )  =  ( F `  Y ) )
1817opeq2d 3991 . . . . . . . . . . . 12  |-  ( q  =  Y  ->  <. ( F `  p ) ,  ( F `  q ) >.  =  <. ( F `  p ) ,  ( F `  Y ) >. )
19 oveq2 6089 . . . . . . . . . . . . 13  |-  ( q  =  Y  ->  (
p  .x.  q )  =  ( p  .x.  Y ) )
2019fveq2d 5732 . . . . . . . . . . . 12  |-  ( q  =  Y  ->  ( F `  ( p  .x.  q ) )  =  ( F `  (
p  .x.  Y )
) )
2118, 20opeq12d 3992 . . . . . . . . . . 11  |-  ( q  =  Y  ->  <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>.  =  <. <. ( F `  p ) ,  ( F `  Y ) >. ,  ( F `  ( p 
.x.  Y ) )
>. )
2221sneqd 3827 . . . . . . . . . 10  |-  ( q  =  Y  ->  { <. <.
( F `  p
) ,  ( F `
 q ) >. ,  ( F `  ( p  .x.  q ) ) >. }  =  { <. <. ( F `  p ) ,  ( F `  Y )
>. ,  ( F `  ( p  .x.  Y
) ) >. } )
2322ssiun2s 4135 . . . . . . . . 9  |-  ( Y  e.  V  ->  { <. <.
( F `  p
) ,  ( F `
 Y ) >. ,  ( F `  ( p  .x.  Y ) ) >. }  C_  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>. } )
2423ralrimivw 2790 . . . . . . . 8  |-  ( Y  e.  V  ->  A. p  e.  V  { <. <. ( F `  p ) ,  ( F `  Y ) >. ,  ( F `  ( p 
.x.  Y ) )
>. }  C_  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>. } )
25 ss2iun 4108 . . . . . . . 8  |-  ( A. p  e.  V  { <. <. ( F `  p ) ,  ( F `  Y )
>. ,  ( F `  ( p  .x.  Y
) ) >. }  C_  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. }  ->  U_ p  e.  V  { <. <. ( F `  p ) ,  ( F `  Y )
>. ,  ( F `  ( p  .x.  Y
) ) >. }  C_  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. } )
2624, 25syl 16 . . . . . . 7  |-  ( Y  e.  V  ->  U_ p  e.  V  { <. <. ( F `  p ) ,  ( F `  Y ) >. ,  ( F `  ( p 
.x.  Y ) )
>. }  C_  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>. } )
27263ad2ant3 980 . . . . . 6  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  U_ p  e.  V  { <. <. ( F `  p ) ,  ( F `  Y ) >. ,  ( F `  ( p 
.x.  Y ) )
>. }  C_  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>. } )
2816, 27sstrd 3358 . . . . 5  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  { <. <. ( F `  X ) ,  ( F `  Y ) >. ,  ( F `  ( X 
.x.  Y ) )
>. }  C_  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>. } )
2943ad2ant1 978 . . . . 5  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  .xb  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>. } )
3028, 29sseqtr4d 3385 . . . 4  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  { <. <. ( F `  X ) ,  ( F `  Y ) >. ,  ( F `  ( X 
.x.  Y ) )
>. }  C_  .xb  )
31 opex 4427 . . . . 5  |-  <. <. ( F `  X ) ,  ( F `  Y ) >. ,  ( F `  ( X 
.x.  Y ) )
>.  e.  _V
3231snss 3926 . . . 4  |-  ( <. <. ( F `  X
) ,  ( F `
 Y ) >. ,  ( F `  ( X  .x.  Y ) ) >.  e.  .xb  <->  { <. <. ( F `  X ) ,  ( F `  Y ) >. ,  ( F `  ( X 
.x.  Y ) )
>. }  C_  .xb  )
3330, 32sylibr 204 . . 3  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  <. <. ( F `  X ) ,  ( F `  Y ) >. ,  ( F `  ( X 
.x.  Y ) )
>.  e.  .xb  )
34 funopfv 5766 . . 3  |-  ( Fun  .xb  ->  ( <. <. ( F `  X ) ,  ( F `  Y ) >. ,  ( F `  ( X 
.x.  Y ) )
>.  e.  .xb  ->  (  .xb  ` 
<. ( F `  X
) ,  ( F `
 Y ) >.
)  =  ( F `
 ( X  .x.  Y ) ) ) )
358, 33, 34sylc 58 . 2  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  (  .xb  ` 
<. ( F `  X
) ,  ( F `
 Y ) >.
)  =  ( F `
 ( X  .x.  Y ) ) )
361, 35syl5eq 2480 1  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  ( ( F `  X )  .xb  ( F `  Y
) )  =  ( F `  ( X 
.x.  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705    C_ wss 3320   {csn 3814   <.cop 3817   U_ciun 4093    X. cxp 4876   Fun wfun 5448    Fn wfn 5449   -onto->wfo 5452   ` cfv 5454  (class class class)co 6081
This theorem is referenced by:  imasaddval  13757  imasmulval  13760  divsaddvallem  13776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-ov 6084
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