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Theorem axrnegex 11200
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 11224. (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 11170 . . . . 5 (𝐴 ∈ ℝ ↔ ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))
21simplbi 497 . . . 4 (𝐴 ∈ ℝ → (1st𝐴) ∈ R)
3 m1r 11120 . . . 4 -1RR
4 mulclsr 11122 . . . 4 (((1st𝐴) ∈ R ∧ -1RR) → ((1st𝐴) ·R -1R) ∈ R)
52, 3, 4sylancl 586 . . 3 (𝐴 ∈ ℝ → ((1st𝐴) ·R -1R) ∈ R)
6 opelreal 11168 . . 3 (⟨((1st𝐴) ·R -1R), 0R⟩ ∈ ℝ ↔ ((1st𝐴) ·R -1R) ∈ R)
75, 6sylibr 234 . 2 (𝐴 ∈ ℝ → ⟨((1st𝐴) ·R -1R), 0R⟩ ∈ ℝ)
81simprbi 496 . . . 4 (𝐴 ∈ ℝ → 𝐴 = ⟨(1st𝐴), 0R⟩)
98oveq1d 7446 . . 3 (𝐴 ∈ ℝ → (𝐴 + ⟨((1st𝐴) ·R -1R), 0R⟩) = (⟨(1st𝐴), 0R⟩ + ⟨((1st𝐴) ·R -1R), 0R⟩))
10 addresr 11176 . . . 4 (((1st𝐴) ∈ R ∧ ((1st𝐴) ·R -1R) ∈ R) → (⟨(1st𝐴), 0R⟩ + ⟨((1st𝐴) ·R -1R), 0R⟩) = ⟨((1st𝐴) +R ((1st𝐴) ·R -1R)), 0R⟩)
112, 5, 10syl2anc 584 . . 3 (𝐴 ∈ ℝ → (⟨(1st𝐴), 0R⟩ + ⟨((1st𝐴) ·R -1R), 0R⟩) = ⟨((1st𝐴) +R ((1st𝐴) ·R -1R)), 0R⟩)
12 pn0sr 11139 . . . . . 6 ((1st𝐴) ∈ R → ((1st𝐴) +R ((1st𝐴) ·R -1R)) = 0R)
1312opeq1d 4884 . . . . 5 ((1st𝐴) ∈ R → ⟨((1st𝐴) +R ((1st𝐴) ·R -1R)), 0R⟩ = ⟨0R, 0R⟩)
14 df-0 11160 . . . . 5 0 = ⟨0R, 0R
1513, 14eqtr4di 2793 . . . 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 2779 . 2 (𝐴 ∈ ℝ → (𝐴 + ⟨((1st𝐴) ·R -1R), 0R⟩) = 0)
18 oveq2 7439 . . . 4 (𝑥 = ⟨((1st𝐴) ·R -1R), 0R⟩ → (𝐴 + 𝑥) = (𝐴 + ⟨((1st𝐴) ·R -1R), 0R⟩))
1918eqeq1d 2737 . . 3 (𝑥 = ⟨((1st𝐴) ·R -1R), 0R⟩ → ((𝐴 + 𝑥) = 0 ↔ (𝐴 + ⟨((1st𝐴) ·R -1R), 0R⟩) = 0))
2019rspcev 3622 . 2 ((⟨((1st𝐴) ·R -1R), 0R⟩ ∈ ℝ ∧ (𝐴 + ⟨((1st𝐴) ·R -1R), 0R⟩) = 0) → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
217, 17, 20syl2anc 584 1 (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wrex 3068  cop 4637  cfv 6563  (class class class)co 7431  1st c1st 8011  Rcnr 10903  0Rc0r 10904  -1Rcm1r 10906   +R cplr 10907   ·R cmr 10908  cr 11152  0cc0 11153   + caddc 11156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510  df-er 8744  df-ec 8746  df-qs 8750  df-ni 10910  df-pli 10911  df-mi 10912  df-lti 10913  df-plpq 10946  df-mpq 10947  df-ltpq 10948  df-enq 10949  df-nq 10950  df-erq 10951  df-plq 10952  df-mq 10953  df-1nq 10954  df-rq 10955  df-ltnq 10956  df-np 11019  df-1p 11020  df-plp 11021  df-mp 11022  df-ltp 11023  df-enr 11093  df-nr 11094  df-plr 11095  df-mr 11096  df-0r 11098  df-1r 11099  df-m1r 11100  df-c 11159  df-0 11160  df-r 11163  df-add 11164
This theorem is referenced by: (None)
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