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Theorem ecovicom 6273
Description: Lemma used to transfer a commutative law via an equivalence relation. (Contributed by Jim Kingdon, 15-Sep-2019.)
Hypotheses
Ref Expression
ecovicom.1  |-  C  =  ( ( S  X.  S ) /.  .~  )
ecovicom.2  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  =  [ <. D ,  G >. ]  .~  )
ecovicom.3  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. x ,  y >. ]  .~  )  =  [ <. H ,  J >. ]  .~  )
ecovicom.4  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  ->  D  =  H )
ecovicom.5  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  ->  G  =  J )
Assertion
Ref Expression
ecovicom  |-  ( ( 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 ecovicom
StepHypRef Expression
1 ecovicom.1 . 2  |-  C  =  ( ( S  X.  S ) /.  .~  )
2 oveq1 5544 . . 3  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  =  ( A  .+  [ <. z ,  w >. ]  .~  ) )
3 oveq2 5545 . . 3  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. x ,  y >. ]  .~  )  =  ( [ <. z ,  w >. ]  .~  .+  A
) )
42, 3eqeq12d 2096 . 2  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( ( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  =  ( [ <. z ,  w >. ]  .~  .+  [
<. x ,  y >. ]  .~  )  <->  ( A  .+  [ <. z ,  w >. ]  .~  )  =  ( [ <. z ,  w >. ]  .~  .+  A ) ) )
5 oveq2 5545 . . 3  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( A  .+  [ <. z ,  w >. ]  .~  )  =  ( A  .+  B ) )
6 oveq1 5544 . . 3  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( [ <. z ,  w >. ]  .~  .+  A )  =  ( B  .+  A ) )
75, 6eqeq12d 2096 . 2  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( ( A  .+  [
<. z ,  w >. ]  .~  )  =  ( [ <. z ,  w >. ]  .~  .+  A
)  <->  ( A  .+  B )  =  ( B  .+  A ) ) )
8 ecovicom.4 . . . 4  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  ->  D  =  H )
9 ecovicom.5 . . . 4  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  ->  G  =  J )
10 opeq12 3574 . . . . 5  |-  ( ( D  =  H  /\  G  =  J )  -> 
<. D ,  G >.  = 
<. H ,  J >. )
1110eceq1d 6201 . . . 4  |-  ( ( D  =  H  /\  G  =  J )  ->  [ <. D ,  G >. ]  .~  =  [ <. H ,  J >. ]  .~  )
128, 9, 11syl2anc 403 . . 3  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  ->  [ <. D ,  G >. ]  .~  =  [ <. H ,  J >. ]  .~  )
13 ecovicom.2 . . 3  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  =  [ <. D ,  G >. ]  .~  )
14 ecovicom.3 . . . 4  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. x ,  y >. ]  .~  )  =  [ <. H ,  J >. ]  .~  )
1514ancoms 264 . . 3  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. x ,  y >. ]  .~  )  =  [ <. H ,  J >. ]  .~  )
1612, 13, 153eqtr4d 2124 . 2  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  =  ( [ <. z ,  w >. ]  .~  .+  [
<. x ,  y >. ]  .~  ) )
171, 4, 7, 162ecoptocl 6253 1  |-  ( ( A  e.  C  /\  B  e.  C )  ->  ( A  .+  B
)  =  ( B 
.+  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   <.cop 3403    X. cxp 4363  (class class class)co 5537   [cec 6163   /.cqs 6164
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-xp 4371  df-cnv 4373  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fv 4934  df-ov 5540  df-ec 6167  df-qs 6171
This theorem is referenced by:  addcomnqg  6622  mulcomnqg  6624  addcomsrg  6983  mulcomsrg  6985  axmulcom  7088
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