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Theorem m1p1sr 8804
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 8778 . . 3  |-  -1R  =  [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R
2 df-1r 8777 . . 3  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
31, 2oveq12i 5957 . 2  |-  ( -1R 
+R  1R )  =  ( [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R  +R  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )
4 df-0r 8776 . . 3  |-  0R  =  [ <. 1P ,  1P >. ]  ~R
5 1pr 8729 . . . . 5  |-  1P  e.  P.
6 addclpr 8732 . . . . . 6  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
75, 5, 6mp2an 653 . . . . 5  |-  ( 1P 
+P.  1P )  e.  P.
8 addsrpr 8787 . . . . 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 8736 . . . . . 6  |-  ( ( 1P  +P.  1P )  +P.  1P )  =  ( 1P  +P.  ( 1P  +P.  1P ) )
1110oveq2i 5956 . . . . 5  |-  ( 1P 
+P.  ( ( 1P 
+P.  1P )  +P.  1P ) )  =  ( 1P  +P.  ( 1P 
+P.  ( 1P  +P.  1P ) ) )
12 addclpr 8732 . . . . . . 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 8732 . . . . . . 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 8781 . . . . . 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 2381 . . 3  |-  ( [
<. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  +R  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )  =  [ <. 1P ,  1P >. ]  ~R
204, 19eqtr4i 2381 . 2  |-  0R  =  ( [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R  +R  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )
213, 20eqtr4i 2381 1  |-  ( -1R 
+R  1R )  =  0R
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1642    e. wcel 1710   <.cop 3719  (class class class)co 5945   [cec 6745   P.cnp 8571   1Pc1p 8572    +P. cpp 8573    ~R cer 8578   0Rc0r 8580   1Rc1r 8581   -1Rcm1r 8582    +R cplr 8583
This theorem is referenced by:  pn0sr  8813  supsrlem  8823  axi2m1  8871
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-omul 6571  df-er 6747  df-ec 6749  df-qs 6753  df-ni 8586  df-pli 8587  df-mi 8588  df-lti 8589  df-plpq 8622  df-mpq 8623  df-ltpq 8624  df-enq 8625  df-nq 8626  df-erq 8627  df-plq 8628  df-mq 8629  df-1nq 8630  df-rq 8631  df-ltnq 8632  df-np 8695  df-1p 8696  df-plp 8697  df-ltp 8699  df-plpr 8769  df-enr 8771  df-nr 8772  df-plr 8773  df-0r 8776  df-1r 8777  df-m1r 8778
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