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

Theorem 0idsr 8719
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 8682 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 oveq1 5865 . . 3  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  +R  0R )  =  ( A  +R  0R ) )
3 id 19 . . 3  |-  ( [
<. x ,  y >. ]  ~R  =  A  ->  [ <. x ,  y
>. ]  ~R  =  A )
42, 3eqeq12d 2297 . 2  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( ( [ <. x ,  y >. ]  ~R  +R  0R )  =  [ <. x ,  y >. ]  ~R  <->  ( A  +R  0R )  =  A
) )
5 df-0r 8686 . . . 4  |-  0R  =  [ <. 1P ,  1P >. ]  ~R
65oveq2i 5869 . . 3  |-  ( [
<. x ,  y >. ]  ~R  +R  0R )  =  ( [ <. x ,  y >. ]  ~R  +R  [ <. 1P ,  1P >. ]  ~R  )
7 1pr 8639 . . . . 5  |-  1P  e.  P.
8 addsrpr 8697 . . . . 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 666 . . . 4  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  +R  [ <. 1P ,  1P >. ]  ~R  )  =  [ <. ( x  +P.  1P ) ,  ( y  +P.  1P ) >. ]  ~R  )
10 addclpr 8642 . . . . . . 7  |-  ( ( x  e.  P.  /\  1P  e.  P. )  -> 
( x  +P.  1P )  e.  P. )
117, 10mpan2 652 . . . . . 6  |-  ( x  e.  P.  ->  (
x  +P.  1P )  e.  P. )
12 addclpr 8642 . . . . . . 7  |-  ( ( y  e.  P.  /\  1P  e.  P. )  -> 
( y  +P.  1P )  e.  P. )
137, 12mpan2 652 . . . . . 6  |-  ( y  e.  P.  ->  (
y  +P.  1P )  e.  P. )
1411, 13anim12i 549 . . . . 5  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( ( x  +P.  1P )  e.  P.  /\  ( y  +P.  1P )  e.  P. )
)
15 vex 2791 . . . . . . 7  |-  x  e. 
_V
16 vex 2791 . . . . . . 7  |-  y  e. 
_V
177elexi 2797 . . . . . . 7  |-  1P  e.  _V
18 addcompr 8645 . . . . . . 7  |-  ( z  +P.  w )  =  ( w  +P.  z
)
19 addasspr 8646 . . . . . . 7  |-  ( ( z  +P.  w )  +P.  v )  =  ( z  +P.  (
w  +P.  v )
)
2015, 16, 17, 18, 19caov12 6048 . . . . . 6  |-  ( x  +P.  ( y  +P. 
1P ) )  =  ( y  +P.  (
x  +P.  1P )
)
21 enreceq 8691 . . . . . 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 224 . . . . 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 649 . . . 4  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  [ <. x ,  y
>. ]  ~R  =  [ <. ( x  +P.  1P ) ,  ( y  +P.  1P ) >. ]  ~R  )
249, 23eqtr4d 2318 . . 3  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  +R  [ <. 1P ,  1P >. ]  ~R  )  =  [ <. x ,  y
>. ]  ~R  )
256, 24syl5eq 2327 . 2  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  +R  0R )  =  [ <. x ,  y >. ]  ~R  )
261, 4, 25ecoptocl 6748 1  |-  ( A  e.  R.  ->  ( A  +R  0R )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643  (class class class)co 5858   [cec 6658   P.cnp 8481   1Pc1p 8482    +P. cpp 8483    ~R cer 8488   R.cnr 8489   0Rc0r 8490    +R cplr 8493
This theorem is referenced by:  addgt0sr  8726  sqgt0sr  8728  map2psrpr  8732  supsrlem  8733  addresr  8760  mulresr  8761  axi2m1  8781  axcnre  8786
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-1p 8606  df-plp 8607  df-ltp 8609  df-plpr 8679  df-enr 8681  df-nr 8682  df-plr 8683  df-0r 8686
  Copyright terms: Public domain W3C validator