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Theorem axrnegex 7159
Description: Existence of negative of real number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rnegex 7199. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
Assertion
Ref Expression
axrnegex (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
Distinct variable group:   𝑥,𝐴

Proof of Theorem axrnegex
StepHypRef Expression
1 elreal2 7113 . . . . 5 (𝐴 ∈ ℝ ↔ ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))
21simplbi 268 . . . 4 (𝐴 ∈ ℝ → (1st𝐴) ∈ R)
3 m1r 7043 . . . 4 -1RR
4 mulclsr 7045 . . . 4 (((1st𝐴) ∈ R ∧ -1RR) → ((1st𝐴) ·R -1R) ∈ R)
52, 3, 4sylancl 404 . . 3 (𝐴 ∈ ℝ → ((1st𝐴) ·R -1R) ∈ R)
6 opelreal 7110 . . 3 (⟨((1st𝐴) ·R -1R), 0R⟩ ∈ ℝ ↔ ((1st𝐴) ·R -1R) ∈ R)
75, 6sylibr 132 . 2 (𝐴 ∈ ℝ → ⟨((1st𝐴) ·R -1R), 0R⟩ ∈ ℝ)
81simprbi 269 . . . 4 (𝐴 ∈ ℝ → 𝐴 = ⟨(1st𝐴), 0R⟩)
98oveq1d 5578 . . 3 (𝐴 ∈ ℝ → (𝐴 + ⟨((1st𝐴) ·R -1R), 0R⟩) = (⟨(1st𝐴), 0R⟩ + ⟨((1st𝐴) ·R -1R), 0R⟩))
10 addresr 7119 . . . 4 (((1st𝐴) ∈ R ∧ ((1st𝐴) ·R -1R) ∈ R) → (⟨(1st𝐴), 0R⟩ + ⟨((1st𝐴) ·R -1R), 0R⟩) = ⟨((1st𝐴) +R ((1st𝐴) ·R -1R)), 0R⟩)
112, 5, 10syl2anc 403 . . 3 (𝐴 ∈ ℝ → (⟨(1st𝐴), 0R⟩ + ⟨((1st𝐴) ·R -1R), 0R⟩) = ⟨((1st𝐴) +R ((1st𝐴) ·R -1R)), 0R⟩)
12 pn0sr 7062 . . . . . 6 ((1st𝐴) ∈ R → ((1st𝐴) +R ((1st𝐴) ·R -1R)) = 0R)
1312opeq1d 3596 . . . . 5 ((1st𝐴) ∈ R → ⟨((1st𝐴) +R ((1st𝐴) ·R -1R)), 0R⟩ = ⟨0R, 0R⟩)
14 df-0 7102 . . . . 5 0 = ⟨0R, 0R
1513, 14syl6eqr 2133 . . . 4 ((1st𝐴) ∈ R → ⟨((1st𝐴) +R ((1st𝐴) ·R -1R)), 0R⟩ = 0)
162, 15syl 14 . . 3 (𝐴 ∈ ℝ → ⟨((1st𝐴) +R ((1st𝐴) ·R -1R)), 0R⟩ = 0)
179, 11, 163eqtrd 2119 . 2 (𝐴 ∈ ℝ → (𝐴 + ⟨((1st𝐴) ·R -1R), 0R⟩) = 0)
18 oveq2 5571 . . . 4 (𝑥 = ⟨((1st𝐴) ·R -1R), 0R⟩ → (𝐴 + 𝑥) = (𝐴 + ⟨((1st𝐴) ·R -1R), 0R⟩))
1918eqeq1d 2091 . . 3 (𝑥 = ⟨((1st𝐴) ·R -1R), 0R⟩ → ((𝐴 + 𝑥) = 0 ↔ (𝐴 + ⟨((1st𝐴) ·R -1R), 0R⟩) = 0))
2019rspcev 2710 . 2 ((⟨((1st𝐴) ·R -1R), 0R⟩ ∈ ℝ ∧ (𝐴 + ⟨((1st𝐴) ·R -1R), 0R⟩) = 0) → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
217, 17, 20syl2anc 403 1 (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wcel 1434  wrex 2354  cop 3419  cfv 4952  (class class class)co 5563  1st c1st 5816  Rcnr 6601  0Rc0r 6602  -1Rcm1r 6604   +R cplr 6605   ·R cmr 6606  cr 7094  0cc0 7095   + caddc 7098
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-eprel 4072  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-irdg 6039  df-1o 6085  df-2o 6086  df-oadd 6089  df-omul 6090  df-er 6193  df-ec 6195  df-qs 6199  df-ni 6608  df-pli 6609  df-mi 6610  df-lti 6611  df-plpq 6648  df-mpq 6649  df-enq 6651  df-nqqs 6652  df-plqqs 6653  df-mqqs 6654  df-1nqqs 6655  df-rq 6656  df-ltnqqs 6657  df-enq0 6728  df-nq0 6729  df-0nq0 6730  df-plq0 6731  df-mq0 6732  df-inp 6770  df-i1p 6771  df-iplp 6772  df-imp 6773  df-enr 7017  df-nr 7018  df-plr 7019  df-mr 7020  df-0r 7022  df-1r 7023  df-m1r 7024  df-c 7101  df-0 7102  df-r 7105  df-add 7106
This theorem is referenced by: (None)
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