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Theorem m1p1sr 8714
Description: Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
Assertion
Ref Expression
m1p1sr  |-  ( -1R 
+R  1R )  =  0R

Proof of Theorem m1p1sr
StepHypRef Expression
1 df-m1r 8688 . . 3  |-  -1R  =  [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R
2 df-1r 8687 . . 3  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
31, 2oveq12i 5870 . 2  |-  ( -1R 
+R  1R )  =  ( [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R  +R  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )
4 df-0r 8686 . . 3  |-  0R  =  [ <. 1P ,  1P >. ]  ~R
5 1pr 8639 . . . . 5  |-  1P  e.  P.
6 addclpr 8642 . . . . . 6  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
75, 5, 6mp2an 653 . . . . 5  |-  ( 1P 
+P.  1P )  e.  P.
8 addsrpr 8697 . . . . 5  |-  ( ( ( 1P  e.  P.  /\  ( 1P  +P.  1P )  e.  P. )  /\  ( ( 1P  +P.  1P )  e.  P.  /\  1P  e.  P. ) )  ->  ( [ <. 1P ,  ( 1P  +P.  1P ) >. ]  ~R  +R  [
<. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )  =  [ <. ( 1P  +P.  ( 1P  +P.  1P ) ) ,  ( ( 1P  +P.  1P )  +P.  1P ) >. ]  ~R  )
95, 7, 7, 5, 8mp4an 654 . . . 4  |-  ( [
<. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  +R  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )  =  [ <. ( 1P  +P.  ( 1P  +P.  1P ) ) ,  ( ( 1P 
+P.  1P )  +P.  1P ) >. ]  ~R
10 addasspr 8646 . . . . . 6  |-  ( ( 1P  +P.  1P )  +P.  1P )  =  ( 1P  +P.  ( 1P  +P.  1P ) )
1110oveq2i 5869 . . . . 5  |-  ( 1P 
+P.  ( ( 1P 
+P.  1P )  +P.  1P ) )  =  ( 1P  +P.  ( 1P 
+P.  ( 1P  +P.  1P ) ) )
12 addclpr 8642 . . . . . . 7  |-  ( ( 1P  e.  P.  /\  ( 1P  +P.  1P )  e.  P. )  -> 
( 1P  +P.  ( 1P  +P.  1P ) )  e.  P. )
135, 7, 12mp2an 653 . . . . . 6  |-  ( 1P 
+P.  ( 1P  +P.  1P ) )  e.  P.
14 addclpr 8642 . . . . . . 7  |-  ( ( ( 1P  +P.  1P )  e.  P.  /\  1P  e.  P. )  ->  (
( 1P  +P.  1P )  +P.  1P )  e. 
P. )
157, 5, 14mp2an 653 . . . . . 6  |-  ( ( 1P  +P.  1P )  +P.  1P )  e. 
P.
16 enreceq 8691 . . . . . 6  |-  ( ( ( 1P  e.  P.  /\  1P  e.  P. )  /\  ( ( 1P  +P.  ( 1P  +P.  1P ) )  e.  P.  /\  ( ( 1P  +P.  1P )  +P.  1P )  e.  P. ) )  ->  ( [ <. 1P ,  1P >. ]  ~R  =  [ <. ( 1P  +P.  ( 1P  +P.  1P ) ) ,  ( ( 1P  +P.  1P )  +P.  1P ) >. ]  ~R  <->  ( 1P  +P.  ( ( 1P  +P.  1P )  +P.  1P ) )  =  ( 1P 
+P.  ( 1P  +P.  ( 1P  +P.  1P ) ) ) ) )
175, 5, 13, 15, 16mp4an 654 . . . . 5  |-  ( [
<. 1P ,  1P >. ]  ~R  =  [ <. ( 1P  +P.  ( 1P 
+P.  1P ) ) ,  ( ( 1P  +P.  1P )  +P.  1P )
>. ]  ~R  <->  ( 1P  +P.  ( ( 1P  +P.  1P )  +P.  1P ) )  =  ( 1P 
+P.  ( 1P  +P.  ( 1P  +P.  1P ) ) ) )
1811, 17mpbir 200 . . . 4  |-  [ <. 1P ,  1P >. ]  ~R  =  [ <. ( 1P  +P.  ( 1P  +P.  1P ) ) ,  ( ( 1P  +P.  1P )  +P.  1P ) >. ]  ~R
199, 18eqtr4i 2306 . . 3  |-  ( [
<. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  +R  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )  =  [ <. 1P ,  1P >. ]  ~R
204, 19eqtr4i 2306 . 2  |-  0R  =  ( [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R  +R  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )
213, 20eqtr4i 2306 1  |-  ( -1R 
+R  1R )  =  0R
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = 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   0Rc0r 8490   1Rc1r 8491   -1Rcm1r 8492    +R cplr 8493
This theorem is referenced by:  pn0sr  8723  supsrlem  8733  axi2m1  8781
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  df-1r 8687  df-m1r 8688
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