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Theorem m1p1sr 6845
 Description: Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.)
Assertion
Ref Expression
m1p1sr (-1R +R 1R) = 0R

Proof of Theorem m1p1sr
StepHypRef Expression
1 df-m1r 6818 . . 3 -1R = [⟨1P, (1P +P 1P)⟩] ~R
2 df-1r 6817 . . 3 1R = [⟨(1P +P 1P), 1P⟩] ~R
31, 2oveq12i 5524 . 2 (-1R +R 1R) = ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
4 df-0r 6816 . . 3 0R = [⟨1P, 1P⟩] ~R
5 1pr 6652 . . . . 5 1PP
6 addclpr 6635 . . . . . 6 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
75, 5, 6mp2an 402 . . . . 5 (1P +P 1P) ∈ P
8 addsrpr 6830 . . . . 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 403 . . . 4 ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)⟩] ~R
10 addassprg 6677 . . . . . . 7 ((1PP ∧ 1PP ∧ 1PP) → ((1P +P 1P) +P 1P) = (1P +P (1P +P 1P)))
115, 5, 5, 10mp3an 1232 . . . . . 6 ((1P +P 1P) +P 1P) = (1P +P (1P +P 1P))
1211oveq2i 5523 . . . . 5 (1P +P ((1P +P 1P) +P 1P)) = (1P +P (1P +P (1P +P 1P)))
13 addclpr 6635 . . . . . . 7 ((1PP ∧ (1P +P 1P) ∈ P) → (1P +P (1P +P 1P)) ∈ P)
145, 7, 13mp2an 402 . . . . . 6 (1P +P (1P +P 1P)) ∈ P
15 addclpr 6635 . . . . . . 7 (((1P +P 1P) ∈ P ∧ 1PP) → ((1P +P 1P) +P 1P) ∈ P)
167, 5, 15mp2an 402 . . . . . 6 ((1P +P 1P) +P 1P) ∈ P
17 enreceq 6821 . . . . . 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)))))
185, 5, 14, 16, 17mp4an 403 . . . . 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))))
1912, 18mpbir 134 . . . 4 [⟨1P, 1P⟩] ~R = [⟨(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)⟩] ~R
209, 19eqtr4i 2063 . . 3 ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨1P, 1P⟩] ~R
214, 20eqtr4i 2063 . 2 0R = ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
223, 21eqtr4i 2063 1 (-1R +R 1R) = 0R
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ⟨cop 3378  (class class class)co 5512  [cec 6104  Pcnp 6389  1Pc1p 6390   +P cpp 6391   ~R cer 6394  0Rc0r 6396  1Rc1r 6397  -1Rcm1r 6398   +R cplr 6399 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-enr 6811  df-nr 6812  df-plr 6813  df-0r 6816  df-1r 6817  df-m1r 6818 This theorem is referenced by:  pn0sr  6856  caucvgsrlemoffres  6884  caucvgsr  6886  axi2m1  6949
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