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Theorem eroprf2 6491
Description: Functionality 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
eroprf2  |-  ( ph  -> 
.+^  : ( J  X.  J ) --> J )
Distinct variable groups:    q, p, r, s, t, u, x, y, z, A    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
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 eroprf2
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, 2, 1eroprf 6490 1  |-  ( ph  -> 
.+^  : ( J  X.  J ) --> J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   E.wrex 2394    C_ wss 3041   class class class wbr 3899    X. cxp 4507   -->wf 5089  (class class class)co 5742   {coprab 5743    Er wer 6394   [cec 6395   /.cqs 6396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-er 6397  df-ec 6399  df-qs 6403
This theorem is referenced by: (None)
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