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Theorem erov2 6962
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 6960 1  |-  ( (
ph  /\  P  e.  A  /\  Q  e.  A
)  ->  ( [ P ]  .~  .+^  [ Q ]  .~  )  =  [
( P  .+  Q
) ]  .~  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2667    C_ wss 3280   class class class wbr 4172    X. cxp 4835   -->wf 5409  (class class class)co 6040   {coprab 6041    Er wer 6861   [cec 6862   /.cqs 6863
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-er 6864  df-ec 6866  df-qs 6870
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