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Theorem ovec 6951
Description: Express an operation on equivalence classes of ordered pairs in terms of equivalence class of operations on ordered pairs. See set.mm for additional comments describing the hypotheses. (Unnecessary distinct variable restrictions were removed by David Abernethy, 4-Jun-2013.) (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 4-Jun-2013.)
Hypotheses
Ref Expression
ovec.1  |-  H  e. 
_V
ovec.2  |-  K  e. 
_V
ovec.3  |-  L  e. 
_V
ovec.4  |-  .~  e.  _V
ovec.5  |-  .~  Er  ( S  X.  S
)
ovec.7  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ph )
) }
ovec.8  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  ( ph  <->  ps )
)
ovec.9  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  ( ph  <->  ch )
)
ovec.10  |-  .+  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( S  X.  S
)  /\  y  e.  ( S  X.  S
) )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  J
) ) }
ovec.11  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  ->  J  =  K )
ovec.12  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  J  =  L )
ovec.13  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  J  =  H )
ovec.14  |-  .+^  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  Q  /\  y  e.  Q )  /\  E. a E. b E. c E. d ( ( x  =  [ <. a ,  b >. ]  .~  /\  y  =  [ <. c ,  d >. ]  .~  )  /\  z  =  [
( <. a ,  b
>.  .+  <. c ,  d
>. ) ]  .~  )
) }
ovec.15  |-  Q  =  ( ( S  X.  S ) /.  .~  )
ovec.16  |-  ( ( ( ( a  e.  S  /\  b  e.  S )  /\  (
c  e.  S  /\  d  e.  S )
)  /\  ( (
g  e.  S  /\  h  e.  S )  /\  ( t  e.  S  /\  s  e.  S
) ) )  -> 
( ( ps  /\  ch )  ->  K  .~  L ) )
Assertion
Ref Expression
ovec  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( [ <. A ,  B >. ]  .~  .+^  [ <. C ,  D >. ]  .~  )  =  [ H ]  .~  )
Distinct variable groups:    a, b,
c, d, f, u, v, w, x, y, z, C    D, a,
b, c, d, f, u, v, w, x, y, z    x, J, y, z    g, a, h, A, b, c, d, f, u, v, w, x, y, z    ch, u, v, w, z   
f, H, u, v, w, x, y, z    B, a, b, c, d, f, g, h, u, v, w, x, y, z    f, K, u, v, w, x, y, z    ps, u, v, w, z    f, L, u, v, w, x, y, z    ph, x, y    s,
a, t, S, b, c, d, f, g, h, u, v, w, x, y, z    .+ , a,
b, c, d, g, h, s, t, x, y, z    .~ , a,
b, c, d, g, h, s, t, x, y, z
Allowed substitution hints:    ph( z, w, v, u, t, f, g, h, s, a, b, c, d)    ps( x, y, t, f, g, h, s, a, b, c, d)    ch( x, y, t, f, g, h, s, a, b, c, d)    A( t, s)    B( t, s)    C( t, g, h, s)    D( t, g, h, s)    .+ ( w, v, u, f)    .+^ ( x, y, z, w, v, u, t, f, g, h, s, a, b, c, d)    Q( x, y, z, w, v, u, t, f, g, h, s, a, b, c, d)    .~ ( w, v, u, f)    H( t, g, h, s, a, b, c, d)    J( w, v, u, t, f, g, h, s, a, b, c, d)    K( t, g, h, s, a, b, c, d)    L( t, g, h, s, a, b, c, d)

Proof of Theorem ovec
StepHypRef Expression
1 ovec.4 . . 3  |-  .~  e.  _V
2 ovec.5 . . 3  |-  .~  Er  ( S  X.  S
)
3 ovec.16 . . . 4  |-  ( ( ( ( a  e.  S  /\  b  e.  S )  /\  (
c  e.  S  /\  d  e.  S )
)  /\  ( (
g  e.  S  /\  h  e.  S )  /\  ( t  e.  S  /\  s  e.  S
) ) )  -> 
( ( ps  /\  ch )  ->  K  .~  L ) )
4 ovec.8 . . . . . 6  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  ( ph  <->  ps )
)
5 ovec.7 . . . . . 6  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ph )
) }
64, 5opbrop 4896 . . . . 5  |-  ( ( ( a  e.  S  /\  b  e.  S
)  /\  ( c  e.  S  /\  d  e.  S ) )  -> 
( <. a ,  b
>.  .~  <. c ,  d
>. 
<->  ps ) )
7 ovec.9 . . . . . 6  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  ( ph  <->  ch )
)
87, 5opbrop 4896 . . . . 5  |-  ( ( ( g  e.  S  /\  h  e.  S
)  /\  ( t  e.  S  /\  s  e.  S ) )  -> 
( <. g ,  h >.  .~  <. t ,  s
>. 
<->  ch ) )
96, 8bi2anan9 844 . . . 4  |-  ( ( ( ( a  e.  S  /\  b  e.  S )  /\  (
c  e.  S  /\  d  e.  S )
)  /\  ( (
g  e.  S  /\  h  e.  S )  /\  ( t  e.  S  /\  s  e.  S
) ) )  -> 
( ( <. a ,  b >.  .~  <. c ,  d >.  /\  <. g ,  h >.  .~  <. t ,  s >. )  <->  ( ps  /\  ch )
) )
10 ovec.2 . . . . . . 7  |-  K  e. 
_V
11 ovec.11 . . . . . . 7  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  ->  J  =  K )
12 ovec.10 . . . . . . 7  |-  .+  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( S  X.  S
)  /\  y  e.  ( S  X.  S
) )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  J
) ) }
1310, 11, 12ov3 6150 . . . . . 6  |-  ( ( ( a  e.  S  /\  b  e.  S
)  /\  ( g  e.  S  /\  h  e.  S ) )  -> 
( <. a ,  b
>.  .+  <. g ,  h >. )  =  K )
14 ovec.3 . . . . . . 7  |-  L  e. 
_V
15 ovec.12 . . . . . . 7  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  J  =  L )
1614, 15, 12ov3 6150 . . . . . 6  |-  ( ( ( c  e.  S  /\  d  e.  S
)  /\  ( t  e.  S  /\  s  e.  S ) )  -> 
( <. c ,  d
>.  .+  <. t ,  s
>. )  =  L
)
1713, 16breqan12d 4169 . . . . 5  |-  ( ( ( ( a  e.  S  /\  b  e.  S )  /\  (
g  e.  S  /\  h  e.  S )
)  /\  ( (
c  e.  S  /\  d  e.  S )  /\  ( t  e.  S  /\  s  e.  S
) ) )  -> 
( ( <. a ,  b >.  .+  <. g ,  h >. )  .~  ( <. c ,  d
>.  .+  <. t ,  s
>. )  <->  K  .~  L ) )
1817an4s 800 . . . 4  |-  ( ( ( ( a  e.  S  /\  b  e.  S )  /\  (
c  e.  S  /\  d  e.  S )
)  /\  ( (
g  e.  S  /\  h  e.  S )  /\  ( t  e.  S  /\  s  e.  S
) ) )  -> 
( ( <. a ,  b >.  .+  <. g ,  h >. )  .~  ( <. c ,  d
>.  .+  <. t ,  s
>. )  <->  K  .~  L ) )
193, 9, 183imtr4d 260 . . 3  |-  ( ( ( ( a  e.  S  /\  b  e.  S )  /\  (
c  e.  S  /\  d  e.  S )
)  /\  ( (
g  e.  S  /\  h  e.  S )  /\  ( t  e.  S  /\  s  e.  S
) ) )  -> 
( ( <. a ,  b >.  .~  <. c ,  d >.  /\  <. g ,  h >.  .~  <. t ,  s >. )  ->  ( <. a ,  b
>.  .+  <. g ,  h >. )  .~  ( <.
c ,  d >.  .+  <. t ,  s
>. ) ) )
20 ovec.14 . . . 4  |-  .+^  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  Q  /\  y  e.  Q )  /\  E. a E. b E. c E. d ( ( x  =  [ <. a ,  b >. ]  .~  /\  y  =  [ <. c ,  d >. ]  .~  )  /\  z  =  [
( <. a ,  b
>.  .+  <. c ,  d
>. ) ]  .~  )
) }
21 ovec.15 . . . . . . . 8  |-  Q  =  ( ( S  X.  S ) /.  .~  )
2221eleq2i 2452 . . . . . . 7  |-  ( x  e.  Q  <->  x  e.  ( ( S  X.  S ) /.  .~  ) )
2321eleq2i 2452 . . . . . . 7  |-  ( y  e.  Q  <->  y  e.  ( ( S  X.  S ) /.  .~  ) )
2422, 23anbi12i 679 . . . . . 6  |-  ( ( x  e.  Q  /\  y  e.  Q )  <->  ( x  e.  ( ( S  X.  S ) /.  .~  )  /\  y  e.  ( ( S  X.  S ) /.  .~  ) ) )
2524anbi1i 677 . . . . 5  |-  ( ( ( x  e.  Q  /\  y  e.  Q
)  /\  E. a E. b E. c E. d ( ( x  =  [ <. a ,  b >. ]  .~  /\  y  =  [ <. c ,  d >. ]  .~  )  /\  z  =  [
( <. a ,  b
>.  .+  <. c ,  d
>. ) ]  .~  )
)  <->  ( ( x  e.  ( ( S  X.  S ) /.  .~  )  /\  y  e.  ( ( S  X.  S ) /.  .~  ) )  /\  E. a E. b E. c E. d ( ( x  =  [ <. a ,  b >. ]  .~  /\  y  =  [ <. c ,  d >. ]  .~  )  /\  z  =  [
( <. a ,  b
>.  .+  <. c ,  d
>. ) ]  .~  )
) )
2625oprabbii 6069 . . . 4  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  Q  /\  y  e.  Q
)  /\  E. a E. b E. c E. d ( ( x  =  [ <. a ,  b >. ]  .~  /\  y  =  [ <. c ,  d >. ]  .~  )  /\  z  =  [
( <. a ,  b
>.  .+  <. c ,  d
>. ) ]  .~  )
) }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( ( S  X.  S ) /.  .~  )  /\  y  e.  ( ( S  X.  S
) /.  .~  )
)  /\  E. a E. b E. c E. d ( ( x  =  [ <. a ,  b >. ]  .~  /\  y  =  [ <. c ,  d >. ]  .~  )  /\  z  =  [
( <. a ,  b
>.  .+  <. c ,  d
>. ) ]  .~  )
) }
2720, 26eqtri 2408 . . 3  |-  .+^  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( ( S  X.  S ) /.  .~  )  /\  y  e.  ( ( S  X.  S
) /.  .~  )
)  /\  E. a E. b E. c E. d ( ( x  =  [ <. a ,  b >. ]  .~  /\  y  =  [ <. c ,  d >. ]  .~  )  /\  z  =  [
( <. a ,  b
>.  .+  <. c ,  d
>. ) ]  .~  )
) }
281, 2, 19, 27th3q 6950 . 2  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( [ <. A ,  B >. ]  .~  .+^  [ <. C ,  D >. ]  .~  )  =  [ ( <. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  )
29 ovec.1 . . . 4  |-  H  e. 
_V
30 ovec.13 . . . 4  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  J  =  H )
3129, 30, 12ov3 6150 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( <. A ,  B >.  .+  <. C ,  D >. )  =  H )
32 eceq1 6878 . . 3  |-  ( (
<. A ,  B >.  .+ 
<. C ,  D >. )  =  H  ->  [ (
<. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  =  [ H ]  .~  )
3331, 32syl 16 . 2  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  ->  [ ( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  =  [ H ]  .~  )
3428, 33eqtrd 2420 1  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( [ <. A ,  B >. ]  .~  .+^  [ <. C ,  D >. ]  .~  )  =  [ H ]  .~  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   _Vcvv 2900   <.cop 3761   class class class wbr 4154   {copab 4207    X. cxp 4817  (class class class)co 6021   {coprab 6022    Er wer 6839   [cec 6840   /.cqs 6841
This theorem is referenced by:  addsrpr  8884  mulsrpr  8885
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-un 4642
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-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fv 5403  df-ov 6024  df-oprab 6025  df-er 6842  df-ec 6844  df-qs 6848
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