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Theorem m1p1sr 8972
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 8946 . . 3  |-  -1R  =  [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R
2 df-1r 8945 . . 3  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
31, 2oveq12i 6096 . 2  |-  ( -1R 
+R  1R )  =  ( [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R  +R  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )
4 df-0r 8944 . . 3  |-  0R  =  [ <. 1P ,  1P >. ]  ~R
5 1pr 8897 . . . . 5  |-  1P  e.  P.
6 addclpr 8900 . . . . . 6  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
75, 5, 6mp2an 655 . . . . 5  |-  ( 1P 
+P.  1P )  e.  P.
8 addsrpr 8955 . . . . 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 656 . . . 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 8904 . . . . . 6  |-  ( ( 1P  +P.  1P )  +P.  1P )  =  ( 1P  +P.  ( 1P  +P.  1P ) )
1110oveq2i 6095 . . . . 5  |-  ( 1P 
+P.  ( ( 1P 
+P.  1P )  +P.  1P ) )  =  ( 1P  +P.  ( 1P 
+P.  ( 1P  +P.  1P ) ) )
12 addclpr 8900 . . . . . . 7  |-  ( ( 1P  e.  P.  /\  ( 1P  +P.  1P )  e.  P. )  -> 
( 1P  +P.  ( 1P  +P.  1P ) )  e.  P. )
135, 7, 12mp2an 655 . . . . . 6  |-  ( 1P 
+P.  ( 1P  +P.  1P ) )  e.  P.
14 addclpr 8900 . . . . . . 7  |-  ( ( ( 1P  +P.  1P )  e.  P.  /\  1P  e.  P. )  ->  (
( 1P  +P.  1P )  +P.  1P )  e. 
P. )
157, 5, 14mp2an 655 . . . . . 6  |-  ( ( 1P  +P.  1P )  +P.  1P )  e. 
P.
16 enreceq 8949 . . . . . 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 656 . . . . 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 202 . . . 4  |-  [ <. 1P ,  1P >. ]  ~R  =  [ <. ( 1P  +P.  ( 1P  +P.  1P ) ) ,  ( ( 1P  +P.  1P )  +P.  1P ) >. ]  ~R
199, 18eqtr4i 2461 . . 3  |-  ( [
<. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  +R  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )  =  [ <. 1P ,  1P >. ]  ~R
204, 19eqtr4i 2461 . 2  |-  0R  =  ( [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R  +R  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )
213, 20eqtr4i 2461 1  |-  ( -1R 
+R  1R )  =  0R
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    e. wcel 1726   <.cop 3819  (class class class)co 6084   [cec 6906   P.cnp 8739   1Pc1p 8740    +P. cpp 8741    ~R cer 8746   0Rc0r 8748   1Rc1r 8749   -1Rcm1r 8750    +R cplr 8751
This theorem is referenced by:  pn0sr  8981  supsrlem  8991  axi2m1  9039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-omul 6732  df-er 6908  df-ec 6910  df-qs 6914  df-ni 8754  df-pli 8755  df-mi 8756  df-lti 8757  df-plpq 8790  df-mpq 8791  df-ltpq 8792  df-enq 8793  df-nq 8794  df-erq 8795  df-plq 8796  df-mq 8797  df-1nq 8798  df-rq 8799  df-ltnq 8800  df-np 8863  df-1p 8864  df-plp 8865  df-ltp 8867  df-plpr 8937  df-enr 8939  df-nr 8940  df-plr 8941  df-0r 8944  df-1r 8945  df-m1r 8946
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