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Theorem addsrpr 8697
Description: Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
Assertion
Ref Expression
addsrpr  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( [ <. A ,  B >. ]  ~R  +R  [ <. C ,  D >. ]  ~R  )  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D
) >. ]  ~R  )

Proof of Theorem addsrpr
Dummy variables  x  y  z  w  v  u  t  s  f 
g  h  a  b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 4237 . 2  |-  <. ( A  +P.  C ) ,  ( B  +P.  D
) >.  e.  _V
2 opex 4237 . 2  |-  <. (
a  +P.  g ) ,  ( b  +P.  h ) >.  e.  _V
3 opex 4237 . 2  |-  <. (
c  +P.  t ) ,  ( d  +P.  s ) >.  e.  _V
4 enrex 8692 . 2  |-  ~R  e.  _V
5 enrer 8690 . 2  |-  ~R  Er  ( P.  X.  P. )
6 df-enr 8681 . 2  |-  ~R  =  { <. x ,  y
>.  |  ( (
x  e.  ( P. 
X.  P. )  /\  y  e.  ( P.  X.  P. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  +P.  u
)  =  ( w  +P.  v ) ) ) }
7 oveq12 5867 . . . 4  |-  ( ( z  =  a  /\  u  =  d )  ->  ( z  +P.  u
)  =  ( a  +P.  d ) )
8 oveq12 5867 . . . 4  |-  ( ( w  =  b  /\  v  =  c )  ->  ( w  +P.  v
)  =  ( b  +P.  c ) )
97, 8eqeqan12d 2298 . . 3  |-  ( ( ( z  =  a  /\  u  =  d )  /\  ( w  =  b  /\  v  =  c ) )  ->  ( ( z  +P.  u )  =  ( w  +P.  v
)  <->  ( a  +P.  d )  =  ( b  +P.  c ) ) )
109an42s 800 . 2  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  ( ( z  +P.  u )  =  ( w  +P.  v
)  <->  ( a  +P.  d )  =  ( b  +P.  c ) ) )
11 oveq12 5867 . . . 4  |-  ( ( z  =  g  /\  u  =  s )  ->  ( z  +P.  u
)  =  ( g  +P.  s ) )
12 oveq12 5867 . . . 4  |-  ( ( w  =  h  /\  v  =  t )  ->  ( w  +P.  v
)  =  ( h  +P.  t ) )
1311, 12eqeqan12d 2298 . . 3  |-  ( ( ( z  =  g  /\  u  =  s )  /\  ( w  =  h  /\  v  =  t ) )  ->  ( ( z  +P.  u )  =  ( w  +P.  v
)  <->  ( g  +P.  s )  =  ( h  +P.  t ) ) )
1413an42s 800 . 2  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  ( ( z  +P.  u )  =  ( w  +P.  v
)  <->  ( g  +P.  s )  =  ( h  +P.  t ) ) )
15 df-plpr 8679 . 2  |-  +pR  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( P.  X.  P. )  /\  y  e.  ( P.  X.  P. )
)  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( w  +P.  u ) ,  ( v  +P.  f ) >. )
) }
16 oveq12 5867 . . . 4  |-  ( ( w  =  a  /\  u  =  g )  ->  ( w  +P.  u
)  =  ( a  +P.  g ) )
17 oveq12 5867 . . . 4  |-  ( ( v  =  b  /\  f  =  h )  ->  ( v  +P.  f
)  =  ( b  +P.  h ) )
18 opeq12 3798 . . . 4  |-  ( ( ( w  +P.  u
)  =  ( a  +P.  g )  /\  ( v  +P.  f
)  =  ( b  +P.  h ) )  ->  <. ( w  +P.  u ) ,  ( v  +P.  f )
>.  =  <. ( a  +P.  g ) ,  ( b  +P.  h
) >. )
1916, 17, 18syl2an 463 . . 3  |-  ( ( ( w  =  a  /\  u  =  g )  /\  ( v  =  b  /\  f  =  h ) )  ->  <. ( w  +P.  u
) ,  ( v  +P.  f ) >.  =  <. ( a  +P.  g ) ,  ( b  +P.  h )
>. )
2019an4s 799 . 2  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  ->  <. ( w  +P.  u
) ,  ( v  +P.  f ) >.  =  <. ( a  +P.  g ) ,  ( b  +P.  h )
>. )
21 oveq12 5867 . . . 4  |-  ( ( w  =  c  /\  u  =  t )  ->  ( w  +P.  u
)  =  ( c  +P.  t ) )
22 oveq12 5867 . . . 4  |-  ( ( v  =  d  /\  f  =  s )  ->  ( v  +P.  f
)  =  ( d  +P.  s ) )
23 opeq12 3798 . . . 4  |-  ( ( ( w  +P.  u
)  =  ( c  +P.  t )  /\  ( v  +P.  f
)  =  ( d  +P.  s ) )  ->  <. ( w  +P.  u ) ,  ( v  +P.  f )
>.  =  <. ( c  +P.  t ) ,  ( d  +P.  s
) >. )
2421, 22, 23syl2an 463 . . 3  |-  ( ( ( w  =  c  /\  u  =  t )  /\  ( v  =  d  /\  f  =  s ) )  ->  <. ( w  +P.  u ) ,  ( v  +P.  f )
>.  =  <. ( c  +P.  t ) ,  ( d  +P.  s
) >. )
2524an4s 799 . 2  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  <. ( w  +P.  u ) ,  ( v  +P.  f )
>.  =  <. ( c  +P.  t ) ,  ( d  +P.  s
) >. )
26 oveq12 5867 . . . 4  |-  ( ( w  =  A  /\  u  =  C )  ->  ( w  +P.  u
)  =  ( A  +P.  C ) )
27 oveq12 5867 . . . 4  |-  ( ( v  =  B  /\  f  =  D )  ->  ( v  +P.  f
)  =  ( B  +P.  D ) )
28 opeq12 3798 . . . 4  |-  ( ( ( w  +P.  u
)  =  ( A  +P.  C )  /\  ( v  +P.  f
)  =  ( B  +P.  D ) )  ->  <. ( w  +P.  u ) ,  ( v  +P.  f )
>.  =  <. ( A  +P.  C ) ,  ( B  +P.  D
) >. )
2926, 27, 28syl2an 463 . . 3  |-  ( ( ( w  =  A  /\  u  =  C )  /\  ( v  =  B  /\  f  =  D ) )  ->  <. ( w  +P.  u
) ,  ( v  +P.  f ) >.  =  <. ( A  +P.  C ) ,  ( B  +P.  D ) >.
)
3029an4s 799 . 2  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  <. ( w  +P.  u
) ,  ( v  +P.  f ) >.  =  <. ( A  +P.  C ) ,  ( B  +P.  D ) >.
)
31 df-plr 8683 . 2  |-  +R  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e. 
R.  /\  y  e.  R. )  /\  E. a E. b E. c E. d ( ( x  =  [ <. a ,  b >. ]  ~R  /\  y  =  [ <. c ,  d >. ]  ~R  )  /\  z  =  [
( <. a ,  b
>.  +pR  <. c ,  d
>. ) ]  ~R  )
) }
32 df-nr 8682 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
33 addcmpblnr 8694 . 2  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  (
c  e.  P.  /\  d  e.  P. )
)  /\  ( (
g  e.  P.  /\  h  e.  P. )  /\  ( t  e.  P.  /\  s  e.  P. )
) )  ->  (
( ( a  +P.  d )  =  ( b  +P.  c )  /\  ( g  +P.  s )  =  ( h  +P.  t ) )  ->  <. ( a  +P.  g ) ,  ( b  +P.  h
) >.  ~R  <. ( c  +P.  t ) ,  ( d  +P.  s
) >. ) )
341, 2, 3, 4, 5, 6, 10, 14, 15, 20, 25, 30, 31, 32, 33ovec 6768 1  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( [ <. A ,  B >. ]  ~R  +R  [ <. C ,  D >. ]  ~R  )  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D
) >. ]  ~R  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643  (class class class)co 5858   [cec 6658   P.cnp 8481    +P. cpp 8483    +pR cplpr 8486    ~R cer 8488   R.cnr 8489    +R cplr 8493
This theorem is referenced by:  addclsr  8705  addcomsr  8709  addasssr  8710  distrsr  8713  m1p1sr  8714  0idsr  8719  ltasr  8722
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-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-er 6660  df-ec 6662  df-qs 6666  df-ni 8496  df-pli 8497  df-mi 8498  df-lti 8499  df-plpq 8532  df-mpq 8533  df-ltpq 8534  df-enq 8535  df-nq 8536  df-erq 8537  df-plq 8538  df-mq 8539  df-1nq 8540  df-rq 8541  df-ltnq 8542  df-np 8605  df-plp 8607  df-ltp 8609  df-plpr 8679  df-enr 8681  df-nr 8682  df-plr 8683
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