MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovec Structured version   Unicode version

Theorem ovec 7215
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 4921 . . . . 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 4921 . . . . 5  |-  ( ( ( g  e.  S  /\  h  e.  S
)  /\  ( t  e.  S  /\  s  e.  S ) )  -> 
( <. g ,  h >.  .~  <. t ,  s
>. 
<->  ch ) )
96, 8bi2anan9 868 . . . 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 6232 . . . . . 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 6232 . . . . . 6  |-  ( ( ( c  e.  S  /\  d  e.  S
)  /\  ( t  e.  S  /\  s  e.  S ) )  -> 
( <. c ,  d
>.  .+  <. t ,  s
>. )  =  L
)
1713, 16breqan12d 4312 . . . . 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 822 . . . 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 268 . . 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 2507 . . . . . . 7  |-  ( x  e.  Q  <->  x  e.  ( ( S  X.  S ) /.  .~  ) )
2321eleq2i 2507 . . . . . . 7  |-  ( y  e.  Q  <->  y  e.  ( ( S  X.  S ) /.  .~  ) )
2422, 23anbi12i 697 . . . . . 6  |-  ( ( x  e.  Q  /\  y  e.  Q )  <->  ( x  e.  ( ( S  X.  S ) /.  .~  )  /\  y  e.  ( ( S  X.  S ) /.  .~  ) ) )
2524anbi1i 695 . . . . 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 6146 . . . 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 2463 . . 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 7214 . 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 6232 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( <. A ,  B >.  .+  <. C ,  D >. )  =  H )
32 eceq1 7142 . . 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 2475 1  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( [ <. A ,  B >. ]  .~  .+^  [ <. C ,  D >. ]  .~  )  =  [ H ]  .~  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   _Vcvv 2977   <.cop 3888   class class class wbr 4297   {copab 4354    X. cxp 4843  (class class class)co 6096   {coprab 6097    Er wer 7103   [cec 7104   /.cqs 7105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fv 5431  df-ov 6099  df-oprab 6100  df-er 7106  df-ec 7108  df-qs 7112
This theorem is referenced by:  addsrpr  9247  mulsrpr  9248
  Copyright terms: Public domain W3C validator