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Theorem erov2 6773
Description: The value of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
eropr2.1  |-  J  =  ( A /.  .~  )
eropr2.2  |-  .+^  =  { <. <. x ,  y
>. ,  z >.  |  E. p  e.  A  E. q  e.  A  ( ( x  =  [ p ]  .~  /\  y  =  [ q ]  .~  )  /\  z  =  [ (
p  .+  q ) ]  .~  ) }
eropr2.3  |-  ( ph  ->  .~  e.  X )
eropr2.4  |-  ( ph  ->  .~  Er  U )
eropr2.5  |-  ( ph  ->  A  C_  U )
eropr2.6  |-  ( ph  ->  .+  : ( A  X.  A ) --> A )
eropr2.7  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  A  /\  u  e.  A
) ) )  -> 
( ( r  .~  s  /\  t  .~  u
)  ->  ( r  .+  t )  .~  (
s  .+  u )
) )
Assertion
Ref Expression
erov2  |-  ( (
ph  /\  P  e.  A  /\  Q  e.  A
)  ->  ( [ P ]  .~  .+^  [ Q ]  .~  )  =  [
( P  .+  Q
) ]  .~  )
Distinct variable groups:    q, p, r, s, t, u, x, y, z, A    P, p, q, r, s, t, u, x, y, z    X, p, q, r, s, t, u, z    .+ , p, q, r, s, t, u, x, y, z    .~ , p, q, r, s, t, u, x, y, z    J, p, q, x, y, z    ph, p, q, r, s, t, u, x, y, z    Q, p, q, r, s, t, u, x, y, z
Allowed substitution hints:    .+^ ( x, y,
z, u, t, s, r, q, p)    U( x, y, z, u, t, s, r, q, p)    J( u, t, s, r)    X( x, y)

Proof of Theorem erov2
StepHypRef Expression
1 eropr2.1 . 2  |-  J  =  ( A /.  .~  )
2 eropr2.3 . 2  |-  ( ph  ->  .~  e.  X )
3 eropr2.4 . 2  |-  ( ph  ->  .~  Er  U )
4 eropr2.5 . 2  |-  ( ph  ->  A  C_  U )
5 eropr2.6 . 2  |-  ( ph  ->  .+  : ( A  X.  A ) --> A )
6 eropr2.7 . 2  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  A  /\  u  e.  A
) ) )  -> 
( ( r  .~  s  /\  t  .~  u
)  ->  ( r  .+  t )  .~  (
s  .+  u )
) )
7 eropr2.2 . 2  |-  .+^  =  { <. <. x ,  y
>. ,  z >.  |  E. p  e.  A  E. q  e.  A  ( ( x  =  [ p ]  .~  /\  y  =  [ q ]  .~  )  /\  z  =  [ (
p  .+  q ) ]  .~  ) }
81, 1, 2, 3, 3, 3, 4, 4, 4, 5, 6, 7, 2, 2erov 6771 1  |-  ( (
ph  /\  P  e.  A  /\  Q  e.  A
)  ->  ( [ P ]  .~  .+^  [ Q ]  .~  )  =  [
( P  .+  Q
) ]  .~  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   class class class wbr 4039    X. cxp 4703   -->wf 5267  (class class class)co 5874   {coprab 5875    Er wer 6673   [cec 6674   /.cqs 6675
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-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-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  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-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-er 6676  df-ec 6678  df-qs 6682
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