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Theorem axrnegex 10185
 Description: Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rnegex 10209. (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 10155 . . . . 5 (𝐴 ∈ ℝ ↔ ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))
21simplbi 485 . . . 4 (𝐴 ∈ ℝ → (1st𝐴) ∈ R)
3 m1r 10105 . . . 4 -1RR
4 mulclsr 10107 . . . 4 (((1st𝐴) ∈ R ∧ -1RR) → ((1st𝐴) ·R -1R) ∈ R)
52, 3, 4sylancl 574 . . 3 (𝐴 ∈ ℝ → ((1st𝐴) ·R -1R) ∈ R)
6 opelreal 10153 . . 3 (⟨((1st𝐴) ·R -1R), 0R⟩ ∈ ℝ ↔ ((1st𝐴) ·R -1R) ∈ R)
75, 6sylibr 224 . 2 (𝐴 ∈ ℝ → ⟨((1st𝐴) ·R -1R), 0R⟩ ∈ ℝ)
81simprbi 484 . . . 4 (𝐴 ∈ ℝ → 𝐴 = ⟨(1st𝐴), 0R⟩)
98oveq1d 6808 . . 3 (𝐴 ∈ ℝ → (𝐴 + ⟨((1st𝐴) ·R -1R), 0R⟩) = (⟨(1st𝐴), 0R⟩ + ⟨((1st𝐴) ·R -1R), 0R⟩))
10 addresr 10161 . . . 4 (((1st𝐴) ∈ R ∧ ((1st𝐴) ·R -1R) ∈ R) → (⟨(1st𝐴), 0R⟩ + ⟨((1st𝐴) ·R -1R), 0R⟩) = ⟨((1st𝐴) +R ((1st𝐴) ·R -1R)), 0R⟩)
112, 5, 10syl2anc 573 . . 3 (𝐴 ∈ ℝ → (⟨(1st𝐴), 0R⟩ + ⟨((1st𝐴) ·R -1R), 0R⟩) = ⟨((1st𝐴) +R ((1st𝐴) ·R -1R)), 0R⟩)
12 pn0sr 10124 . . . . . 6 ((1st𝐴) ∈ R → ((1st𝐴) +R ((1st𝐴) ·R -1R)) = 0R)
1312opeq1d 4545 . . . . 5 ((1st𝐴) ∈ R → ⟨((1st𝐴) +R ((1st𝐴) ·R -1R)), 0R⟩ = ⟨0R, 0R⟩)
14 df-0 10145 . . . . 5 0 = ⟨0R, 0R
1513, 14syl6eqr 2823 . . . 4 ((1st𝐴) ∈ R → ⟨((1st𝐴) +R ((1st𝐴) ·R -1R)), 0R⟩ = 0)
162, 15syl 17 . . 3 (𝐴 ∈ ℝ → ⟨((1st𝐴) +R ((1st𝐴) ·R -1R)), 0R⟩ = 0)
179, 11, 163eqtrd 2809 . 2 (𝐴 ∈ ℝ → (𝐴 + ⟨((1st𝐴) ·R -1R), 0R⟩) = 0)
18 oveq2 6801 . . . 4 (𝑥 = ⟨((1st𝐴) ·R -1R), 0R⟩ → (𝐴 + 𝑥) = (𝐴 + ⟨((1st𝐴) ·R -1R), 0R⟩))
1918eqeq1d 2773 . . 3 (𝑥 = ⟨((1st𝐴) ·R -1R), 0R⟩ → ((𝐴 + 𝑥) = 0 ↔ (𝐴 + ⟨((1st𝐴) ·R -1R), 0R⟩) = 0))
2019rspcev 3460 . 2 ((⟨((1st𝐴) ·R -1R), 0R⟩ ∈ ℝ ∧ (𝐴 + ⟨((1st𝐴) ·R -1R), 0R⟩) = 0) → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
217, 17, 20syl2anc 573 1 (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1631   ∈ wcel 2145  ∃wrex 3062  ⟨cop 4322  ‘cfv 6031  (class class class)co 6793  1st c1st 7313  Rcnr 9889  0Rc0r 9890  -1Rcm1r 9892   +R cplr 9893   ·R cmr 9894  ℝcr 10137  0cc0 10138   + caddc 10141 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-mp 10008  df-ltp 10009  df-enr 10079  df-nr 10080  df-plr 10081  df-mr 10082  df-0r 10084  df-1r 10085  df-m1r 10086  df-c 10144  df-0 10145  df-r 10148  df-add 10149 This theorem is referenced by: (None)
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