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Theorem ecovass 6975
Description: Lemma used to transfer an associative law via an equivalence relation. (Contributed by NM, 31-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.)
Hypotheses
Ref Expression
ecovass.1  |-  D  =  ( ( S  X.  S ) /.  .~  )
ecovass.2  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  =  [ <. G ,  H >. ]  .~  )
ecovass.3  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  )  =  [ <. N ,  Q >. ]  .~  )
ecovass.4  |-  ( ( ( G  e.  S  /\  H  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( [ <. G ,  H >. ]  .~  .+  [
<. v ,  u >. ]  .~  )  =  [ <. J ,  K >. ]  .~  )
ecovass.5  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( N  e.  S  /\  Q  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .+ 
[ <. N ,  Q >. ]  .~  )  =  [ <. L ,  M >. ]  .~  )
ecovass.6  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( G  e.  S  /\  H  e.  S
) )
ecovass.7  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( N  e.  S  /\  Q  e.  S
) )
ecovass.8  |-  J  =  L
ecovass.9  |-  K  =  M
Assertion
Ref Expression
ecovass  |-  ( ( 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 ecovass
StepHypRef Expression
1 ecovass.1 . 2  |-  D  =  ( ( S  X.  S ) /.  .~  )
2 oveq1 6047 . . . 4  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  =  ( A  .+  [ <. z ,  w >. ]  .~  ) )
32oveq1d 6055 . . 3  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( ( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  .+  [
<. v ,  u >. ]  .~  )  =  ( ( A  .+  [ <. z ,  w >. ]  .~  )  .+  [ <. v ,  u >. ]  .~  ) )
4 oveq1 6047 . . 3  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( [ <. x ,  y >. ]  .~  .+  ( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  ) )  =  ( A  .+  ( [ <. z ,  w >. ]  .~  .+  [ <. v ,  u >. ]  .~  ) ) )
53, 4eqeq12d 2418 . 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 6048 . . . 4  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( A  .+  [ <. z ,  w >. ]  .~  )  =  ( A  .+  B ) )
76oveq1d 6055 . . 3  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( ( A  .+  [
<. z ,  w >. ]  .~  )  .+  [ <. v ,  u >. ]  .~  )  =  ( ( A  .+  B
)  .+  [ <. v ,  u >. ]  .~  )
)
8 oveq1 6047 . . . 4  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  )  =  ( B  .+  [ <. v ,  u >. ]  .~  ) )
98oveq2d 6056 . . 3  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( A  .+  ( [ <. z ,  w >. ]  .~  .+  [ <. v ,  u >. ]  .~  ) )  =  ( A  .+  ( B  .+  [ <. v ,  u >. ]  .~  )
) )
107, 9eqeq12d 2418 . 2  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( ( ( A 
.+  [ <. z ,  w >. ]  .~  )  .+  [ <. v ,  u >. ]  .~  )  =  ( A  .+  ( [ <. z ,  w >. ]  .~  .+  [ <. v ,  u >. ]  .~  ) )  <->  ( ( A  .+  B )  .+  [
<. v ,  u >. ]  .~  )  =  ( A  .+  ( B 
.+  [ <. v ,  u >. ]  .~  )
) ) )
11 oveq2 6048 . . 3  |-  ( [
<. v ,  u >. ]  .~  =  C  -> 
( ( A  .+  B )  .+  [ <. v ,  u >. ]  .~  )  =  ( ( A  .+  B
)  .+  C )
)
12 oveq2 6048 . . . 4  |-  ( [
<. v ,  u >. ]  .~  =  C  -> 
( B  .+  [ <. v ,  u >. ]  .~  )  =  ( B  .+  C ) )
1312oveq2d 6056 . . 3  |-  ( [
<. v ,  u >. ]  .~  =  C  -> 
( A  .+  ( B  .+  [ <. v ,  u >. ]  .~  )
)  =  ( A 
.+  ( B  .+  C ) ) )
1411, 13eqeq12d 2418 . 2  |-  ( [
<. v ,  u >. ]  .~  =  C  -> 
( ( ( A 
.+  B )  .+  [
<. v ,  u >. ]  .~  )  =  ( A  .+  ( B 
.+  [ <. v ,  u >. ]  .~  )
)  <->  ( ( A 
.+  B )  .+  C )  =  ( A  .+  ( B 
.+  C ) ) ) )
15 ecovass.8 . . . 4  |-  J  =  L
16 ecovass.9 . . . 4  |-  K  =  M
17 opeq12 3946 . . . . 5  |-  ( ( J  =  L  /\  K  =  M )  -> 
<. J ,  K >.  = 
<. L ,  M >. )
18 eceq1 6900 . . . . 5  |-  ( <. J ,  K >.  = 
<. L ,  M >.  ->  [ <. J ,  K >. ]  .~  =  [ <. L ,  M >. ]  .~  )
1917, 18syl 16 . . . 4  |-  ( ( J  =  L  /\  K  =  M )  ->  [ <. J ,  K >. ]  .~  =  [ <. L ,  M >. ]  .~  )
2015, 16, 19mp2an 654 . . 3  |-  [ <. J ,  K >. ]  .~  =  [ <. L ,  M >. ]  .~
21 ecovass.2 . . . . . . 7  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  =  [ <. G ,  H >. ]  .~  )
2221oveq1d 6055 . . . . . 6  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( ( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  .+  [
<. v ,  u >. ]  .~  )  =  ( [ <. G ,  H >. ]  .~  .+  [ <. v ,  u >. ]  .~  ) )
2322adantr 452 . . . . 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 >. ]  .~  ) )
24 ecovass.6 . . . . . 6  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( G  e.  S  /\  H  e.  S
) )
25 ecovass.4 . . . . . 6  |-  ( ( ( G  e.  S  /\  H  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( [ <. G ,  H >. ]  .~  .+  [
<. v ,  u >. ]  .~  )  =  [ <. J ,  K >. ]  .~  )
2624, 25sylan 458 . . . . 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 >. ]  .~  )
2723, 26eqtrd 2436 . . . 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 >. ]  .~  )
28273impa 1148 . . 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 >. ]  .~  )
29 ecovass.3 . . . . . . 7  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  )  =  [ <. N ,  Q >. ]  .~  )
3029oveq2d 6056 . . . . . 6  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .+  ( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  ) )  =  ( [ <. x ,  y >. ]  .~  .+ 
[ <. N ,  Q >. ]  .~  ) )
3130adantl 453 . . . . 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 >. ]  .~  ) )
32 ecovass.7 . . . . . 6  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( N  e.  S  /\  Q  e.  S
) )
33 ecovass.5 . . . . . 6  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( N  e.  S  /\  Q  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .+ 
[ <. N ,  Q >. ]  .~  )  =  [ <. L ,  M >. ]  .~  )
3432, 33sylan2 461 . . . . 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 >. ]  .~  )
3531, 34eqtrd 2436 . . . 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 >. ]  .~  )
36353impb 1149 . . 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 >. ]  .~  )
3720, 28, 363eqtr4a 2462 . 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 >. ]  .~  ) ) )
381, 5, 10, 14, 373ecoptocl 6955 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   <.cop 3777    X. cxp 4835  (class class class)co 6040   [cec 6862   /.cqs 6863
This theorem is referenced by:  addasssr  8919  mulasssr  8921  axaddass  8987  axmulass  8988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-xp 4843  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fv 5421  df-ov 6043  df-ec 6866  df-qs 6870
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