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Theorem addsrpr 8665
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
StepHypRef Expression
1 opex 4209 . 2  |-  <. ( A  +P.  C ) ,  ( B  +P.  D
) >.  e.  _V
2 opex 4209 . 2  |-  <. (
a  +P.  g ) ,  ( b  +P.  h ) >.  e.  _V
3 opex 4209 . 2  |-  <. (
c  +P.  t ) ,  ( d  +P.  s ) >.  e.  _V
4 enrex 8660 . 2  |-  ~R  e.  _V
5 enrer 8658 . 2  |-  ~R  Er  ( P.  X.  P. )
6 df-enr 8649 . 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 5801 . . . 4  |-  ( ( z  =  a  /\  u  =  d )  ->  ( z  +P.  u
)  =  ( a  +P.  d ) )
8 oveq12 5801 . . . 4  |-  ( ( w  =  b  /\  v  =  c )  ->  ( w  +P.  v
)  =  ( b  +P.  c ) )
97, 8eqeqan12d 2273 . . 3  |-  ( ( ( z  =  a  /\  u  =  d )  /\  ( w  =  b  /\  v  =  c ) )  ->  ( ( z  +P.  u )  =  ( w  +P.  v
)  <->  ( a  +P.  d )  =  ( b  +P.  c ) ) )
109an42s 803 . 2  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  ( ( z  +P.  u )  =  ( w  +P.  v
)  <->  ( a  +P.  d )  =  ( b  +P.  c ) ) )
11 oveq12 5801 . . . 4  |-  ( ( z  =  g  /\  u  =  s )  ->  ( z  +P.  u
)  =  ( g  +P.  s ) )
12 oveq12 5801 . . . 4  |-  ( ( w  =  h  /\  v  =  t )  ->  ( w  +P.  v
)  =  ( h  +P.  t ) )
1311, 12eqeqan12d 2273 . . 3  |-  ( ( ( z  =  g  /\  u  =  s )  /\  ( w  =  h  /\  v  =  t ) )  ->  ( ( z  +P.  u )  =  ( w  +P.  v
)  <->  ( g  +P.  s )  =  ( h  +P.  t ) ) )
1413an42s 803 . 2  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  ( ( z  +P.  u )  =  ( w  +P.  v
)  <->  ( g  +P.  s )  =  ( h  +P.  t ) ) )
15 df-plpr 8647 . 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 5801 . . . 4  |-  ( ( w  =  a  /\  u  =  g )  ->  ( w  +P.  u
)  =  ( a  +P.  g ) )
17 oveq12 5801 . . . 4  |-  ( ( v  =  b  /\  f  =  h )  ->  ( v  +P.  f
)  =  ( b  +P.  h ) )
18 opeq12 3772 . . . 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 465 . . 3  |-  ( ( ( w  =  a  /\  u  =  g )  /\  ( v  =  b  /\  f  =  h ) )  ->  <. ( w  +P.  u
) ,  ( v  +P.  f ) >.  =  <. ( a  +P.  g ) ,  ( b  +P.  h )
>. )
2019an4s 802 . 2  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  ->  <. ( w  +P.  u
) ,  ( v  +P.  f ) >.  =  <. ( a  +P.  g ) ,  ( b  +P.  h )
>. )
21 oveq12 5801 . . . 4  |-  ( ( w  =  c  /\  u  =  t )  ->  ( w  +P.  u
)  =  ( c  +P.  t ) )
22 oveq12 5801 . . . 4  |-  ( ( v  =  d  /\  f  =  s )  ->  ( v  +P.  f
)  =  ( d  +P.  s ) )
23 opeq12 3772 . . . 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 465 . . 3  |-  ( ( ( w  =  c  /\  u  =  t )  /\  ( v  =  d  /\  f  =  s ) )  ->  <. ( w  +P.  u ) ,  ( v  +P.  f )
>.  =  <. ( c  +P.  t ) ,  ( d  +P.  s
) >. )
2524an4s 802 . 2  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  <. ( w  +P.  u ) ,  ( v  +P.  f )
>.  =  <. ( c  +P.  t ) ,  ( d  +P.  s
) >. )
26 oveq12 5801 . . . 4  |-  ( ( w  =  A  /\  u  =  C )  ->  ( w  +P.  u
)  =  ( A  +P.  C ) )
27 oveq12 5801 . . . 4  |-  ( ( v  =  B  /\  f  =  D )  ->  ( v  +P.  f
)  =  ( B  +P.  D ) )
28 opeq12 3772 . . . 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 465 . . 3  |-  ( ( ( w  =  A  /\  u  =  C )  /\  ( v  =  B  /\  f  =  D ) )  ->  <. ( w  +P.  u
) ,  ( v  +P.  f ) >.  =  <. ( A  +P.  C ) ,  ( B  +P.  D ) >.
)
3029an4s 802 . 2  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  <. ( w  +P.  u
) ,  ( v  +P.  f ) >.  =  <. ( A  +P.  C ) ,  ( B  +P.  D ) >.
)
31 df-plr 8651 . 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 8650 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
33 addcmpblnr 8662 . 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 6736 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 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   <.cop 3617  (class class class)co 5792   [cec 6626   P.cnp 8449    +P. cpp 8451    +pR cplpr 8454    ~R cer 8456   R.cnr 8457    +R cplr 8461
This theorem is referenced by:  addclsr  8673  addcomsr  8677  addasssr  8678  distrsr  8681  m1p1sr  8682  0idsr  8687  ltasr  8690
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-omul 6452  df-er 6628  df-ec 6630  df-qs 6634  df-ni 8464  df-pli 8465  df-mi 8466  df-lti 8467  df-plpq 8500  df-mpq 8501  df-ltpq 8502  df-enq 8503  df-nq 8504  df-erq 8505  df-plq 8506  df-mq 8507  df-1nq 8508  df-rq 8509  df-ltnq 8510  df-np 8573  df-plp 8575  df-ltp 8577  df-plpr 8647  df-enr 8649  df-nr 8650  df-plr 8651
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