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Theorem divsaddvallem 13778
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 5744 . . . . . 6  |-  ( Base `  R )  e.  _V
62, 5syl6eqel 2526 . . . . 5  |-  ( ph  ->  V  e.  _V )
7 erex 6931 . . . . 5  |-  (  .~  Er  V  ->  ( V  e.  _V  ->  .~  e.  _V ) )
84, 6, 7sylc 59 . . . 4  |-  ( ph  ->  .~  e.  _V )
9 divsaddf.z . . . 4  |-  ( ph  ->  R  e.  Z )
101, 2, 3, 8, 9divslem 13770 . . 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 13776 . . 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 13756 . 2  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  ( ( F `  X )  .xb  ( F `  Y
) )  =  ( F `  ( X 
.x.  Y ) ) )
1643ad2ant1 979 . . . 4  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  .~  Er  V
)
1763ad2ant1 979 . . . 4  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  V  e.  _V )
1816, 17, 3divsfval 13774 . . 3  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  ( F `  X )  =  [ X ]  .~  )
1916, 17, 3divsfval 13774 . . 3  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  ( F `  Y )  =  [ Y ]  .~  )
2018, 19oveq12d 6101 . 2  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  ( ( F `  X )  .xb  ( F `  Y
) )  =  ( [ X ]  .~  .xb 
[ Y ]  .~  ) )
2116, 17, 3divsfval 13774 . 2  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  ( F `  ( X  .x.  Y
) )  =  [
( X  .x.  Y
) ]  .~  )
2215, 20, 213eqtr3d 2478 1  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  ( [ X ]  .~  .xb  [ Y ]  .~  )  =  [
( X  .x.  Y
) ]  .~  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   _Vcvv 2958   {csn 3816   <.cop 3819   U_ciun 4095   class class class wbr 4214    e. cmpt 4268   ` cfv 5456  (class class class)co 6083    Er wer 6904   [cec 6905   /.cqs 6906   Basecbs 13471    /.s cqus 13733
This theorem is referenced by:  divsaddval  13780  divsmulval  13782
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fo 5462  df-fv 5464  df-ov 6086  df-er 6907  df-ec 6909  df-qs 6913
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