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Theorem divsaddvallem 13469
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 5555 . . . . . 6  |-  ( Base `  R )  e.  _V
62, 5syl6eqel 2384 . . . . 5  |-  ( ph  ->  V  e.  _V )
7 erex 6700 . . . . 5  |-  (  .~  Er  V  ->  ( V  e.  _V  ->  .~  e.  _V ) )
84, 6, 7sylc 56 . . . 4  |-  ( ph  ->  .~  e.  _V )
9 divsaddf.z . . . 4  |-  ( ph  ->  R  e.  Z )
101, 2, 3, 8, 9divslem 13461 . . 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 13467 . . 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 13447 . 2  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  ( ( F `  X )  .xb  ( F `  Y
) )  =  ( F `  ( X 
.x.  Y ) ) )
1643ad2ant1 976 . . . 4  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  .~  Er  V
)
1763ad2ant1 976 . . . 4  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  V  e.  _V )
1816, 17, 3divsfval 13465 . . 3  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  ( F `  X )  =  [ X ]  .~  )
1916, 17, 3divsfval 13465 . . 3  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  ( F `  Y )  =  [ Y ]  .~  )
2018, 19oveq12d 5892 . 2  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  ( ( F `  X )  .xb  ( F `  Y
) )  =  ( [ X ]  .~  .xb 
[ Y ]  .~  ) )
2116, 17, 3divsfval 13465 . 2  |-  ( (
ph  /\  X  e.  V  /\  Y  e.  V
)  ->  ( F `  ( X  .x.  Y
) )  =  [
( X  .x.  Y
) ]  .~  )
2215, 20, 213eqtr3d 2336 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653   <.cop 3656   U_ciun 3921   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874    Er wer 6673   [cec 6674   /.cqs 6675   Basecbs 13164    /.s cqus 13424
This theorem is referenced by:  divsaddval  13471  divsmulval  13473
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-er 6676  df-ec 6678  df-qs 6682
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