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Theorem imasaddvallem 14773
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 6278 . 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 14772 . . . . 5  |-  ( ph  -> 
.xb  Fn  ( B  X.  B ) )
6 fnfun 5669 . . . . 5  |-  (  .xb  Fn  ( B  X.  B
)  ->  Fun  .xb  )
75, 6syl 16 . . . 4  |-  ( ph  ->  Fun  .xb  )
873ad2ant1 1012 . . 3  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  Fun  .xb  )
9 fveq2 5857 . . . . . . . . . . 11  |-  ( p  =  X  ->  ( F `  p )  =  ( F `  X ) )
109opeq1d 4212 . . . . . . . . . 10  |-  ( p  =  X  ->  <. ( F `  p ) ,  ( F `  Y ) >.  =  <. ( F `  X ) ,  ( F `  Y ) >. )
11 oveq1 6282 . . . . . . . . . . 11  |-  ( p  =  X  ->  (
p  .x.  Y )  =  ( X  .x.  Y ) )
1211fveq2d 5861 . . . . . . . . . 10  |-  ( p  =  X  ->  ( F `  ( p  .x.  Y ) )  =  ( F `  ( X  .x.  Y ) ) )
1310, 12opeq12d 4214 . . . . . . . . 9  |-  ( p  =  X  ->  <. <. ( F `  p ) ,  ( F `  Y ) >. ,  ( F `  ( p 
.x.  Y ) )
>.  =  <. <. ( F `  X ) ,  ( F `  Y ) >. ,  ( F `  ( X 
.x.  Y ) )
>. )
1413sneqd 4032 . . . . . . . 8  |-  ( p  =  X  ->  { <. <.
( F `  p
) ,  ( F `
 Y ) >. ,  ( F `  ( p  .x.  Y ) ) >. }  =  { <. <. ( F `  X ) ,  ( F `  Y )
>. ,  ( F `  ( X  .x.  Y
) ) >. } )
1514ssiun2s 4362 . . . . . . 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 1013 . . . . . 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 5857 . . . . . . . . . . . . 13  |-  ( q  =  Y  ->  ( F `  q )  =  ( F `  Y ) )
1817opeq2d 4213 . . . . . . . . . . . 12  |-  ( q  =  Y  ->  <. ( F `  p ) ,  ( F `  q ) >.  =  <. ( F `  p ) ,  ( F `  Y ) >. )
19 oveq2 6283 . . . . . . . . . . . . 13  |-  ( q  =  Y  ->  (
p  .x.  q )  =  ( p  .x.  Y ) )
2019fveq2d 5861 . . . . . . . . . . . 12  |-  ( q  =  Y  ->  ( F `  ( p  .x.  q ) )  =  ( F `  (
p  .x.  Y )
) )
2118, 20opeq12d 4214 . . . . . . . . . . 11  |-  ( q  =  Y  ->  <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>.  =  <. <. ( F `  p ) ,  ( F `  Y ) >. ,  ( F `  ( p 
.x.  Y ) )
>. )
2221sneqd 4032 . . . . . . . . . 10  |-  ( q  =  Y  ->  { <. <.
( F `  p
) ,  ( F `
 q ) >. ,  ( F `  ( p  .x.  q ) ) >. }  =  { <. <. ( F `  p ) ,  ( F `  Y )
>. ,  ( F `  ( p  .x.  Y
) ) >. } )
2322ssiun2s 4362 . . . . . . . . 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 2872 . . . . . . . 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 4334 . . . . . . . 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 1014 . . . . . 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 3507 . . . . 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 1012 . . . . 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 3534 . . . 4  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  { <. <. ( F `  X ) ,  ( F `  Y ) >. ,  ( F `  ( X 
.x.  Y ) )
>. }  C_  .xb  )
31 opex 4704 . . . . 5  |-  <. <. ( F `  X ) ,  ( F `  Y ) >. ,  ( F `  ( X 
.x.  Y ) )
>.  e.  _V
3231snss 4144 . . . 4  |-  ( <. <. ( F `  X
) ,  ( F `
 Y ) >. ,  ( F `  ( X  .x.  Y ) ) >.  e.  .xb  <->  { <. <. ( F `  X ) ,  ( F `  Y ) >. ,  ( F `  ( X 
.x.  Y ) )
>. }  C_  .xb  )
3330, 32sylibr 212 . . 3  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  <. <. ( F `  X ) ,  ( F `  Y ) >. ,  ( F `  ( X 
.x.  Y ) )
>.  e.  .xb  )
34 funopfv 5898 . . 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 60 . 2  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  (  .xb  ` 
<. ( F `  X
) ,  ( F `
 Y ) >.
)  =  ( F `
 ( X  .x.  Y ) ) )
361, 35syl5eq 2513 1  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  ( ( F `  X )  .xb  ( F `  Y
) )  =  ( F `  ( X 
.x.  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807    C_ wss 3469   {csn 4020   <.cop 4026   U_ciun 4318    X. cxp 4990   Fun wfun 5573    Fn wfn 5574   -onto->wfo 5577   ` cfv 5579  (class class class)co 6275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fo 5585  df-fv 5587  df-ov 6278
This theorem is referenced by:  imasaddval  14776  imasmulval  14779  divsaddvallem  14795
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