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Theorem erov2 7005
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 7003 1  |-  ( (
ph  /\  P  e.  A  /\  Q  e.  A
)  ->  ( [ P ]  .~  .+^  [ Q ]  .~  )  =  [
( P  .+  Q
) ]  .~  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2708    C_ wss 3322   class class class wbr 4214    X. cxp 4878   -->wf 5452  (class class class)co 6083   {coprab 6084    Er wer 6904   [cec 6905   /.cqs 6906
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-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-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  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-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-er 6907  df-ec 6909  df-qs 6913
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