MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0idsr Unicode version

Theorem 0idsr 8907
Description: The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.)
Assertion
Ref Expression
0idsr  |-  ( A  e.  R.  ->  ( A  +R  0R )  =  A )

Proof of Theorem 0idsr
Dummy variables  x  y  z  w  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 8870 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 oveq1 6029 . . 3  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  +R  0R )  =  ( A  +R  0R ) )
3 id 20 . . 3  |-  ( [
<. x ,  y >. ]  ~R  =  A  ->  [ <. x ,  y
>. ]  ~R  =  A )
42, 3eqeq12d 2403 . 2  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( ( [ <. x ,  y >. ]  ~R  +R  0R )  =  [ <. x ,  y >. ]  ~R  <->  ( A  +R  0R )  =  A
) )
5 df-0r 8874 . . . 4  |-  0R  =  [ <. 1P ,  1P >. ]  ~R
65oveq2i 6033 . . 3  |-  ( [
<. x ,  y >. ]  ~R  +R  0R )  =  ( [ <. x ,  y >. ]  ~R  +R  [ <. 1P ,  1P >. ]  ~R  )
7 1pr 8827 . . . . 5  |-  1P  e.  P.
8 addsrpr 8885 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( 1P  e.  P.  /\  1P  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  +R  [ <. 1P ,  1P >. ]  ~R  )  =  [ <. (
x  +P.  1P ) ,  ( y  +P. 
1P ) >. ]  ~R  )
97, 7, 8mpanr12 667 . . . 4  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  +R  [ <. 1P ,  1P >. ]  ~R  )  =  [ <. ( x  +P.  1P ) ,  ( y  +P.  1P ) >. ]  ~R  )
10 addclpr 8830 . . . . . . 7  |-  ( ( x  e.  P.  /\  1P  e.  P. )  -> 
( x  +P.  1P )  e.  P. )
117, 10mpan2 653 . . . . . 6  |-  ( x  e.  P.  ->  (
x  +P.  1P )  e.  P. )
12 addclpr 8830 . . . . . . 7  |-  ( ( y  e.  P.  /\  1P  e.  P. )  -> 
( y  +P.  1P )  e.  P. )
137, 12mpan2 653 . . . . . 6  |-  ( y  e.  P.  ->  (
y  +P.  1P )  e.  P. )
1411, 13anim12i 550 . . . . 5  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( ( x  +P.  1P )  e.  P.  /\  ( y  +P.  1P )  e.  P. )
)
15 vex 2904 . . . . . . 7  |-  x  e. 
_V
16 vex 2904 . . . . . . 7  |-  y  e. 
_V
177elexi 2910 . . . . . . 7  |-  1P  e.  _V
18 addcompr 8833 . . . . . . 7  |-  ( z  +P.  w )  =  ( w  +P.  z
)
19 addasspr 8834 . . . . . . 7  |-  ( ( z  +P.  w )  +P.  v )  =  ( z  +P.  (
w  +P.  v )
)
2015, 16, 17, 18, 19caov12 6216 . . . . . 6  |-  ( x  +P.  ( y  +P. 
1P ) )  =  ( y  +P.  (
x  +P.  1P )
)
21 enreceq 8879 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( ( x  +P.  1P )  e.  P.  /\  ( y  +P.  1P )  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  =  [ <. ( x  +P.  1P ) ,  ( y  +P. 
1P ) >. ]  ~R  <->  ( x  +P.  ( y  +P.  1P ) )  =  ( y  +P.  ( x  +P.  1P ) ) ) )
2220, 21mpbiri 225 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( ( x  +P.  1P )  e.  P.  /\  ( y  +P.  1P )  e.  P. )
)  ->  [ <. x ,  y >. ]  ~R  =  [ <. ( x  +P.  1P ) ,  ( y  +P.  1P ) >. ]  ~R  )
2314, 22mpdan 650 . . . 4  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  [ <. x ,  y
>. ]  ~R  =  [ <. ( x  +P.  1P ) ,  ( y  +P.  1P ) >. ]  ~R  )
249, 23eqtr4d 2424 . . 3  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  +R  [ <. 1P ,  1P >. ]  ~R  )  =  [ <. x ,  y
>. ]  ~R  )
256, 24syl5eq 2433 . 2  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  +R  0R )  =  [ <. x ,  y >. ]  ~R  )
261, 4, 25ecoptocl 6932 1  |-  ( A  e.  R.  ->  ( A  +R  0R )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   <.cop 3762  (class class class)co 6022   [cec 6841   P.cnp 8669   1Pc1p 8670    +P. cpp 8671    ~R cer 8676   R.cnr 8677   0Rc0r 8678    +R cplr 8681
This theorem is referenced by:  addgt0sr  8914  sqgt0sr  8916  map2psrpr  8920  supsrlem  8921  addresr  8948  mulresr  8949  axi2m1  8969  axcnre  8974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-omul 6667  df-er 6843  df-ec 6845  df-qs 6849  df-ni 8684  df-pli 8685  df-mi 8686  df-lti 8687  df-plpq 8720  df-mpq 8721  df-ltpq 8722  df-enq 8723  df-nq 8724  df-erq 8725  df-plq 8726  df-mq 8727  df-1nq 8728  df-rq 8729  df-ltnq 8730  df-np 8793  df-1p 8794  df-plp 8795  df-ltp 8797  df-plpr 8867  df-enr 8869  df-nr 8870  df-plr 8871  df-0r 8874
  Copyright terms: Public domain W3C validator