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Theorem m1p1sr 7709
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 7682 . . 3 -1R = [⟨1P, (1P +P 1P)⟩] ~R
2 df-1r 7681 . . 3 1R = [⟨(1P +P 1P), 1P⟩] ~R
31, 2oveq12i 5862 . 2 (-1R +R 1R) = ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
4 df-0r 7680 . . 3 0R = [⟨1P, 1P⟩] ~R
5 1pr 7503 . . . . 5 1PP
6 addclpr 7486 . . . . . 6 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
75, 5, 6mp2an 424 . . . . 5 (1P +P 1P) ∈ P
8 addsrpr 7694 . . . . 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 425 . . . 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 7528 . . . . . . 7 ((1PP ∧ 1PP ∧ 1PP) → ((1P +P 1P) +P 1P) = (1P +P (1P +P 1P)))
115, 5, 5, 10mp3an 1332 . . . . . 6 ((1P +P 1P) +P 1P) = (1P +P (1P +P 1P))
1211oveq2i 5861 . . . . 5 (1P +P ((1P +P 1P) +P 1P)) = (1P +P (1P +P (1P +P 1P)))
13 addclpr 7486 . . . . . . 7 ((1PP ∧ (1P +P 1P) ∈ P) → (1P +P (1P +P 1P)) ∈ P)
145, 7, 13mp2an 424 . . . . . 6 (1P +P (1P +P 1P)) ∈ P
15 addclpr 7486 . . . . . . 7 (((1P +P 1P) ∈ P ∧ 1PP) → ((1P +P 1P) +P 1P) ∈ P)
167, 5, 15mp2an 424 . . . . . 6 ((1P +P 1P) +P 1P) ∈ P
17 enreceq 7685 . . . . . 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 425 . . . . 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 145 . . . 4 [⟨1P, 1P⟩] ~R = [⟨(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)⟩] ~R
209, 19eqtr4i 2194 . . 3 ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨1P, 1P⟩] ~R
214, 20eqtr4i 2194 . 2 0R = ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
223, 21eqtr4i 2194 1 (-1R +R 1R) = 0R
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1348  wcel 2141  cop 3584  (class class class)co 5850  [cec 6507  Pcnp 7240  1Pc1p 7241   +P cpp 7242   ~R cer 7245  0Rc0r 7247  1Rc1r 7248  -1Rcm1r 7249   +R cplr 7250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-eprel 4272  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-irdg 6346  df-1o 6392  df-2o 6393  df-oadd 6396  df-omul 6397  df-er 6509  df-ec 6511  df-qs 6515  df-ni 7253  df-pli 7254  df-mi 7255  df-lti 7256  df-plpq 7293  df-mpq 7294  df-enq 7296  df-nqqs 7297  df-plqqs 7298  df-mqqs 7299  df-1nqqs 7300  df-rq 7301  df-ltnqqs 7302  df-enq0 7373  df-nq0 7374  df-0nq0 7375  df-plq0 7376  df-mq0 7377  df-inp 7415  df-i1p 7416  df-iplp 7417  df-enr 7675  df-nr 7676  df-plr 7677  df-0r 7680  df-1r 7681  df-m1r 7682
This theorem is referenced by:  pn0sr  7720  ltm1sr  7726  caucvgsrlemoffres  7749  caucvgsr  7751  suplocsrlempr  7756  axi2m1  7824
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