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

Theorem ovec 6768
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 4767 . . . . 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 4767 . . . . 5  |-  ( ( ( g  e.  S  /\  h  e.  S
)  /\  ( t  e.  S  /\  s  e.  S ) )  -> 
( <. g ,  h >.  .~  <. t ,  s
>. 
<->  ch ) )
96, 8bi2anan9 843 . . . 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 5984 . . . . . 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 5984 . . . . . 6  |-  ( ( ( c  e.  S  /\  d  e.  S
)  /\  ( t  e.  S  /\  s  e.  S ) )  -> 
( <. c ,  d
>.  .+  <. t ,  s
>. )  =  L
)
1713, 16breqan12d 4038 . . . . 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 799 . . . 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 259 . . 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 2347 . . . . . . 7  |-  ( x  e.  Q  <->  x  e.  ( ( S  X.  S ) /.  .~  ) )
2321eleq2i 2347 . . . . . . 7  |-  ( y  e.  Q  <->  y  e.  ( ( S  X.  S ) /.  .~  ) )
2422, 23anbi12i 678 . . . . . 6  |-  ( ( x  e.  Q  /\  y  e.  Q )  <->  ( x  e.  ( ( S  X.  S ) /.  .~  )  /\  y  e.  ( ( S  X.  S ) /.  .~  ) ) )
2524anbi1i 676 . . . . 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 5903 . . . 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 2303 . . 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 6767 . 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 5984 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( <. A ,  B >.  .+  <. C ,  D >. )  =  H )
32 eceq1 6696 . . 3  |-  ( (
<. A ,  B >.  .+ 
<. C ,  D >. )  =  H  ->  [ (
<. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  =  [ H ]  .~  )
3331, 32syl 15 . 2  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  ->  [ ( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  =  [ H ]  .~  )
3428, 33eqtrd 2315 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 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   class class class wbr 4023   {copab 4076    X. cxp 4687  (class class class)co 5858   {coprab 5859    Er wer 6657   [cec 6658   /.cqs 6659
This theorem is referenced by:  addsrpr  8697  mulsrpr  8698
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-er 6660  df-ec 6662  df-qs 6666
  Copyright terms: Public domain W3C validator