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Theorem ecovdi 6276
Description: Lemma used to transfer a distributive law via an equivalence relation. Most likely ecovidi 6277 will be more helpful. (Contributed by NM, 2-Sep-1995.) (Revised by David Abernethy, 4-Jun-2013.)
Hypotheses
Ref Expression
ecovdi.1  |-  D  =  ( ( S  X.  S ) /.  .~  )
ecovdi.2  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  )  =  [ <. M ,  N >. ]  .~  )
ecovdi.3  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( M  e.  S  /\  N  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .x. 
[ <. M ,  N >. ]  .~  )  =  [ <. H ,  J >. ]  .~  )
ecovdi.4  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .x. 
[ <. z ,  w >. ]  .~  )  =  [ <. W ,  X >. ]  .~  )
ecovdi.5  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .x. 
[ <. v ,  u >. ]  .~  )  =  [ <. Y ,  Z >. ]  .~  )
ecovdi.6  |-  ( ( ( W  e.  S  /\  X  e.  S
)  /\  ( Y  e.  S  /\  Z  e.  S ) )  -> 
( [ <. W ,  X >. ]  .~  .+  [
<. Y ,  Z >. ]  .~  )  =  [ <. K ,  L >. ]  .~  )
ecovdi.7  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( M  e.  S  /\  N  e.  S
) )
ecovdi.8  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( W  e.  S  /\  X  e.  S
) )
ecovdi.9  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( Y  e.  S  /\  Z  e.  S
) )
ecovdi.10  |-  H  =  K
ecovdi.11  |-  J  =  L
Assertion
Ref Expression
ecovdi  |-  ( ( 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 ecovdi
StepHypRef Expression
1 ecovdi.1 . 2  |-  D  =  ( ( S  X.  S ) /.  .~  )
2 oveq1 5544 . . 3  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( [ <. x ,  y >. ]  .~  .x.  ( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  ) )  =  ( A  .x.  ( [ <. z ,  w >. ]  .~  .+  [ <. v ,  u >. ]  .~  ) ) )
3 oveq1 5544 . . . 4  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( [ <. x ,  y >. ]  .~  .x. 
[ <. z ,  w >. ]  .~  )  =  ( A  .x.  [ <. z ,  w >. ]  .~  ) )
4 oveq1 5544 . . . 4  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( [ <. x ,  y >. ]  .~  .x. 
[ <. v ,  u >. ]  .~  )  =  ( A  .x.  [ <. v ,  u >. ]  .~  ) )
53, 4oveq12d 5555 . . 3  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( ( [ <. x ,  y >. ]  .~  .x. 
[ <. z ,  w >. ]  .~  )  .+  ( [ <. x ,  y
>. ]  .~  .x.  [ <. v ,  u >. ]  .~  ) )  =  ( ( A  .x.  [ <. z ,  w >. ]  .~  )  .+  ( A  .x.  [
<. v ,  u >. ]  .~  ) ) )
62, 5eqeq12d 2096 . 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 5544 . . . 4  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  )  =  ( B  .+  [ <. v ,  u >. ]  .~  ) )
87oveq2d 5553 . . 3  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( A  .x.  ( [ <. z ,  w >. ]  .~  .+  [ <. v ,  u >. ]  .~  ) )  =  ( A  .x.  ( B  .+  [ <. v ,  u >. ]  .~  )
) )
9 oveq2 5545 . . . 4  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( A  .x.  [ <. z ,  w >. ]  .~  )  =  ( A  .x.  B ) )
109oveq1d 5552 . . 3  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( ( A  .x.  [
<. z ,  w >. ]  .~  )  .+  ( A  .x.  [ <. v ,  u >. ]  .~  )
)  =  ( ( A  .x.  B ) 
.+  ( A  .x.  [
<. v ,  u >. ]  .~  ) ) )
118, 10eqeq12d 2096 . 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 5545 . . . 4  |-  ( [
<. v ,  u >. ]  .~  =  C  -> 
( B  .+  [ <. v ,  u >. ]  .~  )  =  ( B  .+  C ) )
1312oveq2d 5553 . . 3  |-  ( [
<. v ,  u >. ]  .~  =  C  -> 
( A  .x.  ( B  .+  [ <. v ,  u >. ]  .~  )
)  =  ( A 
.x.  ( B  .+  C ) ) )
14 oveq2 5545 . . . 4  |-  ( [
<. v ,  u >. ]  .~  =  C  -> 
( A  .x.  [ <. v ,  u >. ]  .~  )  =  ( A  .x.  C ) )
1514oveq2d 5553 . . 3  |-  ( [
<. v ,  u >. ]  .~  =  C  -> 
( ( A  .x.  B )  .+  ( A  .x.  [ <. v ,  u >. ]  .~  )
)  =  ( ( A  .x.  B ) 
.+  ( A  .x.  C ) ) )
1613, 15eqeq12d 2096 . 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 ecovdi.10 . . . 4  |-  H  =  K
18 ecovdi.11 . . . 4  |-  J  =  L
19 opeq12 3574 . . . . 5  |-  ( ( H  =  K  /\  J  =  L )  -> 
<. H ,  J >.  = 
<. K ,  L >. )
2019eceq1d 6201 . . . 4  |-  ( ( H  =  K  /\  J  =  L )  ->  [ <. H ,  J >. ]  .~  =  [ <. K ,  L >. ]  .~  )
2117, 18, 20mp2an 417 . . 3  |-  [ <. H ,  J >. ]  .~  =  [ <. K ,  L >. ]  .~
22 ecovdi.2 . . . . . . 7  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  )  =  [ <. M ,  N >. ]  .~  )
2322oveq2d 5553 . . . . . 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 ecovdi.7 . . . . . 6  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( M  e.  S  /\  N  e.  S
) )
26 ecovdi.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 2114 . . . 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 1135 . . 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 ecovdi.4 . . . . . 6  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .x. 
[ <. z ,  w >. ]  .~  )  =  [ <. W ,  X >. ]  .~  )
31 ecovdi.5 . . . . . 6  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .x. 
[ <. v ,  u >. ]  .~  )  =  [ <. Y ,  Z >. ]  .~  )
3230, 31oveqan12d 5556 . . . . 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 ecovdi.8 . . . . . 6  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( W  e.  S  /\  X  e.  S
) )
34 ecovdi.9 . . . . . 6  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( Y  e.  S  /\  Z  e.  S
) )
35 ecovdi.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 2114 . . . 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 1225 . . 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, 383eqtr4a 2140 . 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 6254 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 920    = 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: (None)
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