MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eroprf2 Structured version   Unicode version

Theorem eroprf2 7004
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 7002 1  |-  ( ph  -> 
.+^  : ( J  X.  J ) --> J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706    C_ wss 3320   class class class wbr 4212    X. cxp 4876   -->wf 5450  (class class class)co 6081   {coprab 6082    Er wer 6902   [cec 6903   /.cqs 6904
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-er 6905  df-ec 6907  df-qs 6911
  Copyright terms: Public domain W3C validator