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Theorem ecovcom 6769
Description: Lemma used to transfer a commutative law via an equivalence relation. (Contributed by NM, 29-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.)
Hypotheses
Ref Expression
ecovcom.1  |-  C  =  ( ( S  X.  S ) /.  .~  )
ecovcom.2  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  =  [ <. D ,  G >. ]  .~  )
ecovcom.3  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. x ,  y >. ]  .~  )  =  [ <. H ,  J >. ]  .~  )
ecovcom.4  |-  D  =  H
ecovcom.5  |-  G  =  J
Assertion
Ref Expression
ecovcom  |-  ( ( A  e.  C  /\  B  e.  C )  ->  ( A  .+  B
)  =  ( B 
.+  A ) )
Distinct variable groups:    x, y,
z, w, A    z, B, w    x,  .+ , y,
z, w    x,  .~ , y, z, w    x, S, y, z, w    z, C, w
Allowed substitution hints:    B( x, y)    C( x, y)    D( x, y, z, w)    G( x, y, z, w)    H( x, y, z, w)    J( x, y, z, w)

Proof of Theorem ecovcom
StepHypRef Expression
1 ecovcom.1 . 2  |-  C  =  ( ( S  X.  S ) /.  .~  )
2 oveq1 5865 . . 3  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  =  ( A  .+  [ <. z ,  w >. ]  .~  ) )
3 oveq2 5866 . . 3  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. x ,  y >. ]  .~  )  =  ( [ <. z ,  w >. ]  .~  .+  A
) )
42, 3eqeq12d 2297 . 2  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( ( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  =  ( [ <. z ,  w >. ]  .~  .+  [
<. x ,  y >. ]  .~  )  <->  ( A  .+  [ <. z ,  w >. ]  .~  )  =  ( [ <. z ,  w >. ]  .~  .+  A ) ) )
5 oveq2 5866 . . 3  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( A  .+  [ <. z ,  w >. ]  .~  )  =  ( A  .+  B ) )
6 oveq1 5865 . . 3  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( [ <. z ,  w >. ]  .~  .+  A )  =  ( B  .+  A ) )
75, 6eqeq12d 2297 . 2  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( ( A  .+  [
<. z ,  w >. ]  .~  )  =  ( [ <. z ,  w >. ]  .~  .+  A
)  <->  ( A  .+  B )  =  ( B  .+  A ) ) )
8 ecovcom.4 . . . 4  |-  D  =  H
9 ecovcom.5 . . . 4  |-  G  =  J
10 opeq12 3798 . . . . 5  |-  ( ( D  =  H  /\  G  =  J )  -> 
<. D ,  G >.  = 
<. H ,  J >. )
11 eceq1 6696 . . . . 5  |-  ( <. D ,  G >.  = 
<. H ,  J >.  ->  [ <. D ,  G >. ]  .~  =  [ <. H ,  J >. ]  .~  )
1210, 11syl 15 . . . 4  |-  ( ( D  =  H  /\  G  =  J )  ->  [ <. D ,  G >. ]  .~  =  [ <. H ,  J >. ]  .~  )
138, 9, 12mp2an 653 . . 3  |-  [ <. D ,  G >. ]  .~  =  [ <. H ,  J >. ]  .~
14 ecovcom.2 . . 3  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  =  [ <. D ,  G >. ]  .~  )
15 ecovcom.3 . . . 4  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. x ,  y >. ]  .~  )  =  [ <. H ,  J >. ]  .~  )
1615ancoms 439 . . 3  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. x ,  y >. ]  .~  )  =  [ <. H ,  J >. ]  .~  )
1713, 14, 163eqtr4a 2341 . 2  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  =  ( [ <. z ,  w >. ]  .~  .+  [
<. x ,  y >. ]  .~  ) )
181, 4, 7, 172ecoptocl 6749 1  |-  ( ( A  e.  C  /\  B  e.  C )  ->  ( A  .+  B
)  =  ( B 
.+  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643    X. cxp 4687  (class class class)co 5858   [cec 6658   /.cqs 6659
This theorem is referenced by:  addcomsr  8709  mulcomsr  8711  axmulcom  8777
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-ov 5861  df-ec 6662  df-qs 6666
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