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Theorem ecovidi 6402
Description: Lemma used to transfer a distributive law via an equivalence relation. (Contributed by Jim Kingdon, 17-Sep-2019.)
Hypotheses
Ref Expression
ecovidi.1  |-  D  =  ( ( S  X.  S ) /.  .~  )
ecovidi.2  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  )  =  [ <. M ,  N >. ]  .~  )
ecovidi.3  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( M  e.  S  /\  N  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .x. 
[ <. M ,  N >. ]  .~  )  =  [ <. H ,  J >. ]  .~  )
ecovidi.4  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .x. 
[ <. z ,  w >. ]  .~  )  =  [ <. W ,  X >. ]  .~  )
ecovidi.5  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .x. 
[ <. v ,  u >. ]  .~  )  =  [ <. Y ,  Z >. ]  .~  )
ecovidi.6  |-  ( ( ( W  e.  S  /\  X  e.  S
)  /\  ( Y  e.  S  /\  Z  e.  S ) )  -> 
( [ <. W ,  X >. ]  .~  .+  [
<. Y ,  Z >. ]  .~  )  =  [ <. K ,  L >. ]  .~  )
ecovidi.7  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( M  e.  S  /\  N  e.  S
) )
ecovidi.8  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( W  e.  S  /\  X  e.  S
) )
ecovidi.9  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( Y  e.  S  /\  Z  e.  S
) )
ecovidi.10  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S )  /\  (
v  e.  S  /\  u  e.  S )
)  ->  H  =  K )
ecovidi.11  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S )  /\  (
v  e.  S  /\  u  e.  S )
)  ->  J  =  L )
Assertion
Ref Expression
ecovidi  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  ->  ( A  .x.  ( B  .+  C ) )  =  ( ( A 
.x.  B )  .+  ( A  .x.  C ) ) )
Distinct variable groups:    x, y, z, w, v, u, A   
z, B, w, v, u    w, C, v, u    x,  .+ , y, z, w, v, u    x,  .~ , y, z, w, v, u    x, S, y, z, w, v, u   
x,  .x. , y, z, w, v, u    z, D, w, v, u
Allowed substitution hints:    B( x, y)    C( x, y, z)    D( x, y)    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)    W( x, y, z, w, v, u)    X( x, y, z, w, v, u)    Y( x, y, z, w, v, u)    Z( x, y, z, w, v, u)

Proof of Theorem ecovidi
StepHypRef Expression
1 ecovidi.1 . 2  |-  D  =  ( ( S  X.  S ) /.  .~  )
2 oveq1 5659 . . 3  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( [ <. x ,  y >. ]  .~  .x.  ( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  ) )  =  ( A  .x.  ( [ <. z ,  w >. ]  .~  .+  [ <. v ,  u >. ]  .~  ) ) )
3 oveq1 5659 . . . 4  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( [ <. x ,  y >. ]  .~  .x. 
[ <. z ,  w >. ]  .~  )  =  ( A  .x.  [ <. z ,  w >. ]  .~  ) )
4 oveq1 5659 . . . 4  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( [ <. x ,  y >. ]  .~  .x. 
[ <. v ,  u >. ]  .~  )  =  ( A  .x.  [ <. v ,  u >. ]  .~  ) )
53, 4oveq12d 5670 . . 3  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( ( [ <. x ,  y >. ]  .~  .x. 
[ <. z ,  w >. ]  .~  )  .+  ( [ <. x ,  y
>. ]  .~  .x.  [ <. v ,  u >. ]  .~  ) )  =  ( ( A  .x.  [ <. z ,  w >. ]  .~  )  .+  ( A  .x.  [
<. v ,  u >. ]  .~  ) ) )
62, 5eqeq12d 2102 . 2  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( ( [ <. x ,  y >. ]  .~  .x.  ( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  ) )  =  ( ( [ <. x ,  y >. ]  .~  .x. 
[ <. z ,  w >. ]  .~  )  .+  ( [ <. x ,  y
>. ]  .~  .x.  [ <. v ,  u >. ]  .~  ) )  <->  ( A  .x.  ( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  ) )  =  ( ( A  .x.  [
<. z ,  w >. ]  .~  )  .+  ( A  .x.  [ <. v ,  u >. ]  .~  )
) ) )
7 oveq1 5659 . . . 4  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  )  =  ( B  .+  [ <. v ,  u >. ]  .~  ) )
87oveq2d 5668 . . 3  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( A  .x.  ( [ <. z ,  w >. ]  .~  .+  [ <. v ,  u >. ]  .~  ) )  =  ( A  .x.  ( B  .+  [ <. v ,  u >. ]  .~  )
) )
9 oveq2 5660 . . . 4  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( A  .x.  [ <. z ,  w >. ]  .~  )  =  ( A  .x.  B ) )
109oveq1d 5667 . . 3  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( ( A  .x.  [
<. z ,  w >. ]  .~  )  .+  ( A  .x.  [ <. v ,  u >. ]  .~  )
)  =  ( ( A  .x.  B ) 
.+  ( A  .x.  [
<. v ,  u >. ]  .~  ) ) )
118, 10eqeq12d 2102 . 2  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( ( A  .x.  ( [ <. z ,  w >. ]  .~  .+  [ <. v ,  u >. ]  .~  ) )  =  ( ( A  .x.  [
<. z ,  w >. ]  .~  )  .+  ( A  .x.  [ <. v ,  u >. ]  .~  )
)  <->  ( A  .x.  ( B  .+  [ <. v ,  u >. ]  .~  ) )  =  ( ( A  .x.  B
)  .+  ( A  .x.  [ <. v ,  u >. ]  .~  ) ) ) )
12 oveq2 5660 . . . 4  |-  ( [
<. v ,  u >. ]  .~  =  C  -> 
( B  .+  [ <. v ,  u >. ]  .~  )  =  ( B  .+  C ) )
1312oveq2d 5668 . . 3  |-  ( [
<. v ,  u >. ]  .~  =  C  -> 
( A  .x.  ( B  .+  [ <. v ,  u >. ]  .~  )
)  =  ( A 
.x.  ( B  .+  C ) ) )
14 oveq2 5660 . . . 4  |-  ( [
<. v ,  u >. ]  .~  =  C  -> 
( A  .x.  [ <. v ,  u >. ]  .~  )  =  ( A  .x.  C ) )
1514oveq2d 5668 . . 3  |-  ( [
<. v ,  u >. ]  .~  =  C  -> 
( ( A  .x.  B )  .+  ( A  .x.  [ <. v ,  u >. ]  .~  )
)  =  ( ( A  .x.  B ) 
.+  ( A  .x.  C ) ) )
1613, 15eqeq12d 2102 . 2  |-  ( [
<. v ,  u >. ]  .~  =  C  -> 
( ( A  .x.  ( B  .+  [ <. v ,  u >. ]  .~  ) )  =  ( ( A  .x.  B
)  .+  ( A  .x.  [ <. v ,  u >. ]  .~  ) )  <-> 
( A  .x.  ( B  .+  C ) )  =  ( ( A 
.x.  B )  .+  ( A  .x.  C ) ) ) )
17 ecovidi.10 . . . 4  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S )  /\  (
v  e.  S  /\  u  e.  S )
)  ->  H  =  K )
18 ecovidi.11 . . . 4  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S )  /\  (
v  e.  S  /\  u  e.  S )
)  ->  J  =  L )
19 opeq12 3624 . . . . 5  |-  ( ( H  =  K  /\  J  =  L )  -> 
<. H ,  J >.  = 
<. K ,  L >. )
2019eceq1d 6326 . . . 4  |-  ( ( H  =  K  /\  J  =  L )  ->  [ <. H ,  J >. ]  .~  =  [ <. K ,  L >. ]  .~  )
2117, 18, 20syl2anc 403 . . 3  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S )  /\  (
v  e.  S  /\  u  e.  S )
)  ->  [ <. H ,  J >. ]  .~  =  [ <. K ,  L >. ]  .~  )
22 ecovidi.2 . . . . . . 7  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  )  =  [ <. M ,  N >. ]  .~  )
2322oveq2d 5668 . . . . . 6  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .x.  ( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  ) )  =  ( [ <. x ,  y >. ]  .~  .x. 
[ <. M ,  N >. ]  .~  ) )
2423adantl 271 . . . . 5  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( (
z  e.  S  /\  w  e.  S )  /\  ( v  e.  S  /\  u  e.  S
) ) )  -> 
( [ <. x ,  y >. ]  .~  .x.  ( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  ) )  =  ( [ <. x ,  y >. ]  .~  .x. 
[ <. M ,  N >. ]  .~  ) )
25 ecovidi.7 . . . . . 6  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( M  e.  S  /\  N  e.  S
) )
26 ecovidi.3 . . . . . 6  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( M  e.  S  /\  N  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .x. 
[ <. M ,  N >. ]  .~  )  =  [ <. H ,  J >. ]  .~  )
2725, 26sylan2 280 . . . . 5  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( (
z  e.  S  /\  w  e.  S )  /\  ( v  e.  S  /\  u  e.  S
) ) )  -> 
( [ <. x ,  y >. ]  .~  .x. 
[ <. M ,  N >. ]  .~  )  =  [ <. H ,  J >. ]  .~  )
2824, 27eqtrd 2120 . . . 4  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( (
z  e.  S  /\  w  e.  S )  /\  ( v  e.  S  /\  u  e.  S
) ) )  -> 
( [ <. x ,  y >. ]  .~  .x.  ( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  ) )  =  [ <. H ,  J >. ]  .~  )
29283impb 1139 . . 3  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S )  /\  (
v  e.  S  /\  u  e.  S )
)  ->  ( [ <. x ,  y >. ]  .~  .x.  ( [ <. z ,  w >. ]  .~  .+  [ <. v ,  u >. ]  .~  ) )  =  [ <. H ,  J >. ]  .~  )
30 ecovidi.4 . . . . . 6  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .x. 
[ <. z ,  w >. ]  .~  )  =  [ <. W ,  X >. ]  .~  )
31 ecovidi.5 . . . . . 6  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .x. 
[ <. v ,  u >. ]  .~  )  =  [ <. Y ,  Z >. ]  .~  )
3230, 31oveqan12d 5671 . . . . 5  |-  ( ( ( ( x  e.  S  /\  y  e.  S )  /\  (
z  e.  S  /\  w  e.  S )
)  /\  ( (
x  e.  S  /\  y  e.  S )  /\  ( v  e.  S  /\  u  e.  S
) ) )  -> 
( ( [ <. x ,  y >. ]  .~  .x. 
[ <. z ,  w >. ]  .~  )  .+  ( [ <. x ,  y
>. ]  .~  .x.  [ <. v ,  u >. ]  .~  ) )  =  ( [ <. W ,  X >. ]  .~  .+  [ <. Y ,  Z >. ]  .~  ) )
33 ecovidi.8 . . . . . 6  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( W  e.  S  /\  X  e.  S
) )
34 ecovidi.9 . . . . . 6  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( Y  e.  S  /\  Z  e.  S
) )
35 ecovidi.6 . . . . . 6  |-  ( ( ( W  e.  S  /\  X  e.  S
)  /\  ( Y  e.  S  /\  Z  e.  S ) )  -> 
( [ <. W ,  X >. ]  .~  .+  [
<. Y ,  Z >. ]  .~  )  =  [ <. K ,  L >. ]  .~  )
3633, 34, 35syl2an 283 . . . . 5  |-  ( ( ( ( x  e.  S  /\  y  e.  S )  /\  (
z  e.  S  /\  w  e.  S )
)  /\  ( (
x  e.  S  /\  y  e.  S )  /\  ( v  e.  S  /\  u  e.  S
) ) )  -> 
( [ <. W ,  X >. ]  .~  .+  [
<. Y ,  Z >. ]  .~  )  =  [ <. K ,  L >. ]  .~  )
3732, 36eqtrd 2120 . . . 4  |-  ( ( ( ( x  e.  S  /\  y  e.  S )  /\  (
z  e.  S  /\  w  e.  S )
)  /\  ( (
x  e.  S  /\  y  e.  S )  /\  ( v  e.  S  /\  u  e.  S
) ) )  -> 
( ( [ <. x ,  y >. ]  .~  .x. 
[ <. z ,  w >. ]  .~  )  .+  ( [ <. x ,  y
>. ]  .~  .x.  [ <. v ,  u >. ]  .~  ) )  =  [ <. K ,  L >. ]  .~  )
38373impdi 1229 . . 3  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S )  /\  (
v  e.  S  /\  u  e.  S )
)  ->  ( ( [ <. x ,  y
>. ]  .~  .x.  [ <. z ,  w >. ]  .~  )  .+  ( [ <. x ,  y >. ]  .~  .x. 
[ <. v ,  u >. ]  .~  ) )  =  [ <. K ,  L >. ]  .~  )
3921, 29, 383eqtr4d 2130 . 2  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S )  /\  (
v  e.  S  /\  u  e.  S )
)  ->  ( [ <. x ,  y >. ]  .~  .x.  ( [ <. z ,  w >. ]  .~  .+  [ <. v ,  u >. ]  .~  ) )  =  ( ( [ <. x ,  y >. ]  .~  .x. 
[ <. z ,  w >. ]  .~  )  .+  ( [ <. x ,  y
>. ]  .~  .x.  [ <. v ,  u >. ]  .~  ) ) )
401, 6, 11, 16, 393ecoptocl 6379 1  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  ->  ( A  .x.  ( B  .+  C ) )  =  ( ( A 
.x.  B )  .+  ( A  .x.  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 924    = wceq 1289    e. wcel 1438   <.cop 3449    X. cxp 4436  (class class class)co 5652   [cec 6288   /.cqs 6289
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-xp 4444  df-cnv 4446  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fv 5023  df-ov 5655  df-ec 6292  df-qs 6296
This theorem is referenced by:  distrnqg  6944  distrsrg  7303  axdistr  7407
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