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Theorem ecovdi 6787
Description: Lemma used to transfer a distributive law via an equivalence relation. (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 5881 . . 3  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( [ <. x ,  y >. ]  .~  .x.  ( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  ) )  =  ( A  .x.  ( [ <. z ,  w >. ]  .~  .+  [ <. v ,  u >. ]  .~  ) ) )
3 oveq1 5881 . . . 4  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( [ <. x ,  y >. ]  .~  .x. 
[ <. z ,  w >. ]  .~  )  =  ( A  .x.  [ <. z ,  w >. ]  .~  ) )
4 oveq1 5881 . . . 4  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( [ <. x ,  y >. ]  .~  .x. 
[ <. v ,  u >. ]  .~  )  =  ( A  .x.  [ <. v ,  u >. ]  .~  ) )
53, 4oveq12d 5892 . . 3  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( ( [ <. x ,  y >. ]  .~  .x. 
[ <. z ,  w >. ]  .~  )  .+  ( [ <. x ,  y
>. ]  .~  .x.  [ <. v ,  u >. ]  .~  ) )  =  ( ( A  .x.  [ <. z ,  w >. ]  .~  )  .+  ( A  .x.  [
<. v ,  u >. ]  .~  ) ) )
62, 5eqeq12d 2310 . 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 5881 . . . 4  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  )  =  ( B  .+  [ <. v ,  u >. ]  .~  ) )
87oveq2d 5890 . . 3  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( A  .x.  ( [ <. z ,  w >. ]  .~  .+  [ <. v ,  u >. ]  .~  ) )  =  ( A  .x.  ( B  .+  [ <. v ,  u >. ]  .~  )
) )
9 oveq2 5882 . . . 4  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( A  .x.  [ <. z ,  w >. ]  .~  )  =  ( A  .x.  B ) )
109oveq1d 5889 . . 3  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( ( A  .x.  [
<. z ,  w >. ]  .~  )  .+  ( A  .x.  [ <. v ,  u >. ]  .~  )
)  =  ( ( A  .x.  B ) 
.+  ( A  .x.  [
<. v ,  u >. ]  .~  ) ) )
118, 10eqeq12d 2310 . 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 5882 . . . 4  |-  ( [
<. v ,  u >. ]  .~  =  C  -> 
( B  .+  [ <. v ,  u >. ]  .~  )  =  ( B  .+  C ) )
1312oveq2d 5890 . . 3  |-  ( [
<. v ,  u >. ]  .~  =  C  -> 
( A  .x.  ( B  .+  [ <. v ,  u >. ]  .~  )
)  =  ( A 
.x.  ( B  .+  C ) ) )
14 oveq2 5882 . . . 4  |-  ( [
<. v ,  u >. ]  .~  =  C  -> 
( A  .x.  [ <. v ,  u >. ]  .~  )  =  ( A  .x.  C ) )
1514oveq2d 5890 . . 3  |-  ( [
<. v ,  u >. ]  .~  =  C  -> 
( ( A  .x.  B )  .+  ( A  .x.  [ <. v ,  u >. ]  .~  )
)  =  ( ( A  .x.  B ) 
.+  ( A  .x.  C ) ) )
1613, 15eqeq12d 2310 . 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 3814 . . . . 5  |-  ( ( H  =  K  /\  J  =  L )  -> 
<. H ,  J >.  = 
<. K ,  L >. )
20 eceq1 6712 . . . . 5  |-  ( <. H ,  J >.  = 
<. K ,  L >.  ->  [ <. H ,  J >. ]  .~  =  [ <. K ,  L >. ]  .~  )
2119, 20syl 15 . . . 4  |-  ( ( H  =  K  /\  J  =  L )  ->  [ <. H ,  J >. ]  .~  =  [ <. K ,  L >. ]  .~  )
2217, 18, 21mp2an 653 . . 3  |-  [ <. H ,  J >. ]  .~  =  [ <. K ,  L >. ]  .~
23 ecovdi.2 . . . . . . 7  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. v ,  u >. ]  .~  )  =  [ <. M ,  N >. ]  .~  )
2423oveq2d 5890 . . . . . 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 >. ]  .~  ) )
2524adantl 452 . . . . 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 >. ]  .~  ) )
26 ecovdi.7 . . . . . 6  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( M  e.  S  /\  N  e.  S
) )
27 ecovdi.3 . . . . . 6  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( M  e.  S  /\  N  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .x. 
[ <. M ,  N >. ]  .~  )  =  [ <. H ,  J >. ]  .~  )
2826, 27sylan2 460 . . . . 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 >. ]  .~  )
2925, 28eqtrd 2328 . . . 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 >. ]  .~  )
30293impb 1147 . . 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 >. ]  .~  )
31 ecovdi.4 . . . . . 6  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .x. 
[ <. z ,  w >. ]  .~  )  =  [ <. W ,  X >. ]  .~  )
32 ecovdi.5 . . . . . 6  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .x. 
[ <. v ,  u >. ]  .~  )  =  [ <. Y ,  Z >. ]  .~  )
3331, 32oveqan12d 5893 . . . . 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 >. ]  .~  ) )
34 ecovdi.8 . . . . . 6  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( W  e.  S  /\  X  e.  S
) )
35 ecovdi.9 . . . . . 6  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( v  e.  S  /\  u  e.  S ) )  -> 
( Y  e.  S  /\  Z  e.  S
) )
36 ecovdi.6 . . . . . 6  |-  ( ( ( W  e.  S  /\  X  e.  S
)  /\  ( Y  e.  S  /\  Z  e.  S ) )  -> 
( [ <. W ,  X >. ]  .~  .+  [
<. Y ,  Z >. ]  .~  )  =  [ <. K ,  L >. ]  .~  )
3734, 35, 36syl2an 463 . . . . 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 >. ]  .~  )
3833, 37eqtrd 2328 . . . 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 >. ]  .~  )
39383impdi 1237 . . 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 >. ]  .~  )
4022, 30, 393eqtr4a 2354 . 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 >. ]  .~  ) ) )
411, 6, 11, 16, 403ecoptocl 6766 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   <.cop 3656    X. cxp 4703  (class class class)co 5874   [cec 6674   /.cqs 6675
This theorem is referenced by:  distrsr  8729  axdistr  8796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279  df-ov 5877  df-ec 6678  df-qs 6682
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