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Theorem ecoviass 6216
Description: Lemma used to transfer an associative law via an equivalence relation. (Contributed by Jim Kingdon, 16-Sep-2019.)
Hypotheses
Ref Expression
ecoviass.1  |-  D  =  ( ( S  X.  S ) /.  .~  )
ecoviass.2  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  =  [ <. G ,  H >. ]  .~  )
ecoviass.3  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  )  =  [ <. N ,  Q >. ]  .~  )
ecoviass.4  |-  ( ( ( G  e.  S  /\  H  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( [ <. G ,  H >. ]  .~  .+  [
<. v ,  u >. ]  .~  )  =  [ <. J ,  K >. ]  .~  )
ecoviass.5  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( N  e.  S  /\  Q  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .+ 
[ <. N ,  Q >. ]  .~  )  =  [ <. L ,  M >. ]  .~  )
ecoviass.6  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( G  e.  S  /\  H  e.  S
) )
ecoviass.7  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( N  e.  S  /\  Q  e.  S
) )
ecoviass.8  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S )  /\  (
v  e.  S  /\  u  e.  S )
)  ->  J  =  L )
ecoviass.9  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S )  /\  (
v  e.  S  /\  u  e.  S )
)  ->  K  =  M )
Assertion
Ref Expression
ecoviass  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  ->  ( ( A  .+  B )  .+  C
)  =  ( A 
.+  ( B  .+  C ) ) )
Distinct variable groups:    x, y, z, w, v, u, A   
z, B, w, v, u    x, C, y, z, w, v, u   
x,  .+ , y, z, w, v, u    x,  .~ , y, z, w, v, u   
x, S, y, z, w, v, u    z, D, w, v, u
Allowed substitution hints:    B( x, y)    D( x, y)    Q( x, y, z, w, v, u)    G( x, y, z, w, v, u)    H( x, y, z, w, v, u)    J( x, y, z, w, v, u)    K( x, y, z, w, v, u)    L( x, y, z, w, v, u)    M( x, y, z, w, v, u)    N( x, y, z, w, v, u)

Proof of Theorem ecoviass
StepHypRef Expression
1 ecoviass.1 . 2  |-  D  =  ( ( S  X.  S ) /.  .~  )
2 oveq1 5519 . . . 4  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  =  ( A  .+  [ <. z ,  w >. ]  .~  ) )
32oveq1d 5527 . . 3  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( ( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  .+  [
<. v ,  u >. ]  .~  )  =  ( ( A  .+  [ <. z ,  w >. ]  .~  )  .+  [ <. v ,  u >. ]  .~  ) )
4 oveq1 5519 . . 3  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( [ <. x ,  y >. ]  .~  .+  ( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  ) )  =  ( A  .+  ( [ <. z ,  w >. ]  .~  .+  [ <. v ,  u >. ]  .~  ) ) )
53, 4eqeq12d 2054 . 2  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( ( ( [
<. x ,  y >. ]  .~  .+  [ <. z ,  w >. ]  .~  )  .+  [ <. v ,  u >. ]  .~  )  =  ( [ <. x ,  y >. ]  .~  .+  ( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  ) )  <->  ( ( A  .+  [ <. z ,  w >. ]  .~  )  .+  [ <. v ,  u >. ]  .~  )  =  ( A  .+  ( [ <. z ,  w >. ]  .~  .+  [ <. v ,  u >. ]  .~  ) ) ) )
6 oveq2 5520 . . . 4  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( A  .+  [ <. z ,  w >. ]  .~  )  =  ( A  .+  B ) )
76oveq1d 5527 . . 3  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( ( A  .+  [
<. z ,  w >. ]  .~  )  .+  [ <. v ,  u >. ]  .~  )  =  ( ( A  .+  B
)  .+  [ <. v ,  u >. ]  .~  )
)
8 oveq1 5519 . . . 4  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  )  =  ( B  .+  [ <. v ,  u >. ]  .~  ) )
98oveq2d 5528 . . 3  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( A  .+  ( [ <. z ,  w >. ]  .~  .+  [ <. v ,  u >. ]  .~  ) )  =  ( A  .+  ( B  .+  [ <. v ,  u >. ]  .~  )
) )
107, 9eqeq12d 2054 . 2  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( ( ( A 
.+  [ <. z ,  w >. ]  .~  )  .+  [ <. v ,  u >. ]  .~  )  =  ( A  .+  ( [ <. z ,  w >. ]  .~  .+  [ <. v ,  u >. ]  .~  ) )  <->  ( ( A  .+  B )  .+  [
<. v ,  u >. ]  .~  )  =  ( A  .+  ( B 
.+  [ <. v ,  u >. ]  .~  )
) ) )
11 oveq2 5520 . . 3  |-  ( [
<. v ,  u >. ]  .~  =  C  -> 
( ( A  .+  B )  .+  [ <. v ,  u >. ]  .~  )  =  ( ( A  .+  B
)  .+  C )
)
12 oveq2 5520 . . . 4  |-  ( [
<. v ,  u >. ]  .~  =  C  -> 
( B  .+  [ <. v ,  u >. ]  .~  )  =  ( B  .+  C ) )
1312oveq2d 5528 . . 3  |-  ( [
<. v ,  u >. ]  .~  =  C  -> 
( A  .+  ( B  .+  [ <. v ,  u >. ]  .~  )
)  =  ( A 
.+  ( B  .+  C ) ) )
1411, 13eqeq12d 2054 . 2  |-  ( [
<. v ,  u >. ]  .~  =  C  -> 
( ( ( A 
.+  B )  .+  [
<. v ,  u >. ]  .~  )  =  ( A  .+  ( B 
.+  [ <. v ,  u >. ]  .~  )
)  <->  ( ( A 
.+  B )  .+  C )  =  ( A  .+  ( B 
.+  C ) ) ) )
15 ecoviass.8 . . . 4  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S )  /\  (
v  e.  S  /\  u  e.  S )
)  ->  J  =  L )
16 ecoviass.9 . . . 4  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S )  /\  (
v  e.  S  /\  u  e.  S )
)  ->  K  =  M )
17 opeq12 3551 . . . . 5  |-  ( ( J  =  L  /\  K  =  M )  -> 
<. J ,  K >.  = 
<. L ,  M >. )
1817eceq1d 6142 . . . 4  |-  ( ( J  =  L  /\  K  =  M )  ->  [ <. J ,  K >. ]  .~  =  [ <. L ,  M >. ]  .~  )
1915, 16, 18syl2anc 391 . . 3  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S )  /\  (
v  e.  S  /\  u  e.  S )
)  ->  [ <. J ,  K >. ]  .~  =  [ <. L ,  M >. ]  .~  )
20 ecoviass.2 . . . . . . 7  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  =  [ <. G ,  H >. ]  .~  )
2120oveq1d 5527 . . . . . 6  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( ( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  .+  [
<. v ,  u >. ]  .~  )  =  ( [ <. G ,  H >. ]  .~  .+  [ <. v ,  u >. ]  .~  ) )
2221adantr 261 . . . . 5  |-  ( ( ( ( x  e.  S  /\  y  e.  S )  /\  (
z  e.  S  /\  w  e.  S )
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( ( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  .+  [
<. v ,  u >. ]  .~  )  =  ( [ <. G ,  H >. ]  .~  .+  [ <. v ,  u >. ]  .~  ) )
23 ecoviass.6 . . . . . 6  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( G  e.  S  /\  H  e.  S
) )
24 ecoviass.4 . . . . . 6  |-  ( ( ( G  e.  S  /\  H  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( [ <. G ,  H >. ]  .~  .+  [
<. v ,  u >. ]  .~  )  =  [ <. J ,  K >. ]  .~  )
2523, 24sylan 267 . . . . 5  |-  ( ( ( ( x  e.  S  /\  y  e.  S )  /\  (
z  e.  S  /\  w  e.  S )
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( [ <. G ,  H >. ]  .~  .+  [
<. v ,  u >. ]  .~  )  =  [ <. J ,  K >. ]  .~  )
2622, 25eqtrd 2072 . . . 4  |-  ( ( ( ( x  e.  S  /\  y  e.  S )  /\  (
z  e.  S  /\  w  e.  S )
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( ( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  .+  [
<. v ,  u >. ]  .~  )  =  [ <. J ,  K >. ]  .~  )
27263impa 1099 . . 3  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S )  /\  (
v  e.  S  /\  u  e.  S )
)  ->  ( ( [ <. x ,  y
>. ]  .~  .+  [ <. z ,  w >. ]  .~  )  .+  [ <. v ,  u >. ]  .~  )  =  [ <. J ,  K >. ]  .~  )
28 ecoviass.3 . . . . . . 7  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  )  =  [ <. N ,  Q >. ]  .~  )
2928oveq2d 5528 . . . . . 6  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .+  ( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  ) )  =  ( [ <. x ,  y >. ]  .~  .+ 
[ <. N ,  Q >. ]  .~  ) )
3029adantl 262 . . . . 5  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( (
z  e.  S  /\  w  e.  S )  /\  ( v  e.  S  /\  u  e.  S
) ) )  -> 
( [ <. x ,  y >. ]  .~  .+  ( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  ) )  =  ( [ <. x ,  y >. ]  .~  .+ 
[ <. N ,  Q >. ]  .~  ) )
31 ecoviass.7 . . . . . 6  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( N  e.  S  /\  Q  e.  S
) )
32 ecoviass.5 . . . . . 6  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( N  e.  S  /\  Q  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .+ 
[ <. N ,  Q >. ]  .~  )  =  [ <. L ,  M >. ]  .~  )
3331, 32sylan2 270 . . . . 5  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( (
z  e.  S  /\  w  e.  S )  /\  ( v  e.  S  /\  u  e.  S
) ) )  -> 
( [ <. x ,  y >. ]  .~  .+ 
[ <. N ,  Q >. ]  .~  )  =  [ <. L ,  M >. ]  .~  )
3430, 33eqtrd 2072 . . . 4  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( (
z  e.  S  /\  w  e.  S )  /\  ( v  e.  S  /\  u  e.  S
) ) )  -> 
( [ <. x ,  y >. ]  .~  .+  ( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  ) )  =  [ <. L ,  M >. ]  .~  )
35343impb 1100 . . 3  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S )  /\  (
v  e.  S  /\  u  e.  S )
)  ->  ( [ <. x ,  y >. ]  .~  .+  ( [
<. z ,  w >. ]  .~  .+  [ <. v ,  u >. ]  .~  ) )  =  [ <. L ,  M >. ]  .~  )
3619, 27, 353eqtr4d 2082 . 2  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S )  /\  (
v  e.  S  /\  u  e.  S )
)  ->  ( ( [ <. x ,  y
>. ]  .~  .+  [ <. z ,  w >. ]  .~  )  .+  [ <. v ,  u >. ]  .~  )  =  ( [ <. x ,  y
>. ]  .~  .+  ( [ <. z ,  w >. ]  .~  .+  [ <. v ,  u >. ]  .~  ) ) )
371, 5, 10, 14, 363ecoptocl 6195 1  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  ->  ( ( A  .+  B )  .+  C
)  =  ( A 
.+  ( B  .+  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    /\ w3a 885    = wceq 1243    e. wcel 1393   <.cop 3378    X. cxp 4343  (class class class)co 5512   [cec 6104   /.cqs 6105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fv 4910  df-ov 5515  df-ec 6108  df-qs 6112
This theorem is referenced by:  addassnqg  6480  mulassnqg  6482  addasssrg  6841  mulasssrg  6843  axaddass  6946  axmulass  6947
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