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Theorem divsaddvallem 14508
Description: Value of an operation defined on a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
Hypotheses
Ref Expression
divsaddf.u  |-  ( ph  ->  U  =  ( R 
/.s  .~  ) )
divsaddf.v  |-  ( ph  ->  V  =  ( Base `  R ) )
divsaddf.r  |-  ( ph  ->  .~  Er  V )
divsaddf.z  |-  ( ph  ->  R  e.  Z )
divsaddf.e  |-  ( ph  ->  ( ( a  .~  p  /\  b  .~  q
)  ->  ( a  .x.  b )  .~  (
p  .x.  q )
) )
divsaddf.c  |-  ( (
ph  /\  ( p  e.  V  /\  q  e.  V ) )  -> 
( p  .x.  q
)  e.  V )
divsaddflem.f  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
divsaddflem.g  |-  ( ph  -> 
.xb  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>. } )
Assertion
Ref Expression
divsaddvallem  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  ( [ X ]  .~  .xb  [ Y ]  .~  )  =  [
( X  .x.  Y
) ]  .~  )
Distinct variable groups:    a, b, p, q, x,  .~    F, a, b, p, q    ph, a,
b, p, q, x    V, a, b, p, q, x    R, p, q, x    .x. , p, q, x    X, p, q, x    .xb , a, b, p, q    Y, p, q, x
Allowed substitution hints:    R( a, b)    .xb (
x)    .x. ( a, b)    U( x, q, p, a, b)    F( x)    X( a, b)    Y( a, b)    Z( x, q, p, a, b)

Proof of Theorem divsaddvallem
StepHypRef Expression
1 divsaddf.u . . . 4  |-  ( ph  ->  U  =  ( R 
/.s  .~  ) )
2 divsaddf.v . . . 4  |-  ( ph  ->  V  =  ( Base `  R ) )
3 divsaddflem.f . . . 4  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
4 divsaddf.r . . . . 5  |-  ( ph  ->  .~  Er  V )
5 fvex 5720 . . . . . 6  |-  ( Base `  R )  e.  _V
62, 5syl6eqel 2531 . . . . 5  |-  ( ph  ->  V  e.  _V )
7 erex 7144 . . . . 5  |-  (  .~  Er  V  ->  ( V  e.  _V  ->  .~  e.  _V ) )
84, 6, 7sylc 60 . . . 4  |-  ( ph  ->  .~  e.  _V )
9 divsaddf.z . . . 4  |-  ( ph  ->  R  e.  Z )
101, 2, 3, 8, 9divslem 14500 . . 3  |-  ( ph  ->  F : V -onto-> ( V /.  .~  ) )
11 divsaddf.c . . . 4  |-  ( (
ph  /\  ( p  e.  V  /\  q  e.  V ) )  -> 
( p  .x.  q
)  e.  V )
12 divsaddf.e . . . 4  |-  ( ph  ->  ( ( a  .~  p  /\  b  .~  q
)  ->  ( a  .x.  b )  .~  (
p  .x.  q )
) )
134, 6, 3, 11, 12ercpbl 14506 . . 3  |-  ( (
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 ) ) ) )
14 divsaddflem.g . . 3  |-  ( ph  -> 
.xb  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>. } )
1510, 13, 14imasaddvallem 14486 . 2  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  ( ( F `  X )  .xb  ( F `  Y
) )  =  ( F `  ( X 
.x.  Y ) ) )
1643ad2ant1 1009 . . . 4  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  .~  Er  V
)
1763ad2ant1 1009 . . . 4  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  V  e.  _V )
1816, 17, 3divsfval 14504 . . 3  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  ( F `  X )  =  [ X ]  .~  )
1916, 17, 3divsfval 14504 . . 3  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  ( F `  Y )  =  [ Y ]  .~  )
2018, 19oveq12d 6128 . 2  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  ( ( F `  X )  .xb  ( F `  Y
) )  =  ( [ X ]  .~  .xb 
[ Y ]  .~  ) )
2116, 17, 3divsfval 14504 . 2  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  ( F `  ( X  .x.  Y
) )  =  [
( X  .x.  Y
) ]  .~  )
2215, 20, 213eqtr3d 2483 1  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  ( [ X ]  .~  .xb  [ Y ]  .~  )  =  [
( X  .x.  Y
) ]  .~  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2991   {csn 3896   <.cop 3902   U_ciun 4190   class class class wbr 4311    e. cmpt 4369   ` cfv 5437  (class class class)co 6110    Er wer 7117   [cec 7118   /.cqs 7119   Basecbs 14193    /.s cqus 14462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-fo 5443  df-fv 5445  df-ov 6113  df-er 7120  df-ec 7122  df-qs 7126
This theorem is referenced by:  divsaddval  14510  divsmulval  14512
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