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Theorem ovec 7006
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 4947 . . . . 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 4947 . . . . 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 6202 . . . . . 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 6202 . . . . . 6  |-  ( ( ( c  e.  S  /\  d  e.  S
)  /\  ( t  e.  S  /\  s  e.  S ) )  -> 
( <. c ,  d
>.  .+  <. t ,  s
>. )  =  L
)
1713, 16breqan12d 4219 . . . . 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 2499 . . . . . . 7  |-  ( x  e.  Q  <->  x  e.  ( ( S  X.  S ) /.  .~  ) )
2321eleq2i 2499 . . . . . . 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 6121 . . . 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 2455 . . 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 7005 . 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 6202 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( <. A ,  B >.  .+  <. C ,  D >. )  =  H )
32 eceq1 6933 . . 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 2467 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 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809   class class class wbr 4204   {copab 4257    X. cxp 4868  (class class class)co 6073   {coprab 6074    Er wer 6894   [cec 6895   /.cqs 6896
This theorem is referenced by:  addsrpr  8942  mulsrpr  8943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-er 6897  df-ec 6899  df-qs 6903
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