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Theorem m1p1sr 9371
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 9345 . . 3  |-  -1R  =  [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R
2 df-1r 9344 . . 3  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
31, 2oveq12i 6213 . 2  |-  ( -1R 
+R  1R )  =  ( [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R  +R  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )
4 df-0r 9343 . . 3  |-  0R  =  [ <. 1P ,  1P >. ]  ~R
5 1pr 9296 . . . . 5  |-  1P  e.  P.
6 addclpr 9299 . . . . . 6  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
75, 5, 6mp2an 672 . . . . 5  |-  ( 1P 
+P.  1P )  e.  P.
8 addsrpr 9354 . . . . 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 673 . . . 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 9303 . . . . . 6  |-  ( ( 1P  +P.  1P )  +P.  1P )  =  ( 1P  +P.  ( 1P  +P.  1P ) )
1110oveq2i 6212 . . . . 5  |-  ( 1P 
+P.  ( ( 1P 
+P.  1P )  +P.  1P ) )  =  ( 1P  +P.  ( 1P 
+P.  ( 1P  +P.  1P ) ) )
12 addclpr 9299 . . . . . . 7  |-  ( ( 1P  e.  P.  /\  ( 1P  +P.  1P )  e.  P. )  -> 
( 1P  +P.  ( 1P  +P.  1P ) )  e.  P. )
135, 7, 12mp2an 672 . . . . . 6  |-  ( 1P 
+P.  ( 1P  +P.  1P ) )  e.  P.
14 addclpr 9299 . . . . . . 7  |-  ( ( ( 1P  +P.  1P )  e.  P.  /\  1P  e.  P. )  ->  (
( 1P  +P.  1P )  +P.  1P )  e. 
P. )
157, 5, 14mp2an 672 . . . . . 6  |-  ( ( 1P  +P.  1P )  +P.  1P )  e. 
P.
16 enreceq 9348 . . . . . 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 673 . . . . 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 209 . . . 4  |-  [ <. 1P ,  1P >. ]  ~R  =  [ <. ( 1P  +P.  ( 1P  +P.  1P ) ) ,  ( ( 1P  +P.  1P )  +P.  1P ) >. ]  ~R
199, 18eqtr4i 2486 . . 3  |-  ( [
<. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  +R  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )  =  [ <. 1P ,  1P >. ]  ~R
204, 19eqtr4i 2486 . 2  |-  0R  =  ( [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R  +R  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )
213, 20eqtr4i 2486 1  |-  ( -1R 
+R  1R )  =  0R
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1370    e. wcel 1758   <.cop 3992  (class class class)co 6201   [cec 7210   P.cnp 9138   1Pc1p 9139    +P. cpp 9140    ~R cer 9145   0Rc0r 9147   1Rc1r 9148   -1Rcm1r 9149    +R cplr 9150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-omul 7036  df-er 7212  df-ec 7214  df-qs 7218  df-ni 9153  df-pli 9154  df-mi 9155  df-lti 9156  df-plpq 9189  df-mpq 9190  df-ltpq 9191  df-enq 9192  df-nq 9193  df-erq 9194  df-plq 9195  df-mq 9196  df-1nq 9197  df-rq 9198  df-ltnq 9199  df-np 9262  df-1p 9263  df-plp 9264  df-ltp 9266  df-plpr 9336  df-enr 9338  df-nr 9339  df-plr 9340  df-0r 9343  df-1r 9344  df-m1r 9345
This theorem is referenced by:  pn0sr  9380  supsrlem  9390  axi2m1  9438
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