Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  m1p1sr Structured version   Visualization version   GIF version

Theorem m1p1sr 10115
 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 10086 . . 3 -1R = [⟨1P, (1P +P 1P)⟩] ~R
2 df-1r 10085 . . 3 1R = [⟨(1P +P 1P), 1P⟩] ~R
31, 2oveq12i 6805 . 2 (-1R +R 1R) = ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
4 df-0r 10084 . . 3 0R = [⟨1P, 1P⟩] ~R
5 1pr 10039 . . . . 5 1PP
6 addclpr 10042 . . . . . 6 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
75, 5, 6mp2an 672 . . . . 5 (1P +P 1P) ∈ P
8 addsrpr 10098 . . . . 5 (((1PP ∧ (1P +P 1P) ∈ P) ∧ ((1P +P 1P) ∈ P ∧ 1PP)) → ([⟨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 10046 . . . . . 6 ((1P +P 1P) +P 1P) = (1P +P (1P +P 1P))
1110oveq2i 6804 . . . . 5 (1P +P ((1P +P 1P) +P 1P)) = (1P +P (1P +P (1P +P 1P)))
12 addclpr 10042 . . . . . . 7 ((1PP ∧ (1P +P 1P) ∈ P) → (1P +P (1P +P 1P)) ∈ P)
135, 7, 12mp2an 672 . . . . . 6 (1P +P (1P +P 1P)) ∈ P
14 addclpr 10042 . . . . . . 7 (((1P +P 1P) ∈ P ∧ 1PP) → ((1P +P 1P) +P 1P) ∈ P)
157, 5, 14mp2an 672 . . . . . 6 ((1P +P 1P) +P 1P) ∈ P
16 enreceq 10089 . . . . . 6 (((1PP ∧ 1PP) ∧ ((1P +P (1P +P 1P)) ∈ P ∧ ((1P +P 1P) +P 1P) ∈ 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 221 . . . 4 [⟨1P, 1P⟩] ~R = [⟨(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)⟩] ~R
199, 18eqtr4i 2796 . . 3 ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨1P, 1P⟩] ~R
204, 19eqtr4i 2796 . 2 0R = ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
213, 20eqtr4i 2796 1 (-1R +R 1R) = 0R
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   = wceq 1631   ∈ wcel 2145  ⟨cop 4322  (class class class)co 6793  [cec 7894  Pcnp 9883  1Pc1p 9884   +P cpp 9885   ~R cer 9888  0Rc0r 9890  1Rc1r 9891  -1Rcm1r 9892   +R cplr 9893 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-omul 7718  df-er 7896  df-ec 7898  df-qs 7902  df-ni 9896  df-pli 9897  df-mi 9898  df-lti 9899  df-plpq 9932  df-mpq 9933  df-ltpq 9934  df-enq 9935  df-nq 9936  df-erq 9937  df-plq 9938  df-mq 9939  df-1nq 9940  df-rq 9941  df-ltnq 9942  df-np 10005  df-1p 10006  df-plp 10007  df-ltp 10009  df-enr 10079  df-nr 10080  df-plr 10081  df-0r 10084  df-1r 10085  df-m1r 10086 This theorem is referenced by:  pn0sr  10124  supsrlem  10134  axi2m1  10182
 Copyright terms: Public domain W3C validator