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Theorem axrnegex 10655
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 10679. (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 10625 . . . . 5 (𝐴 ∈ ℝ ↔ ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))
21simplbi 501 . . . 4 (𝐴 ∈ ℝ → (1st𝐴) ∈ R)
3 m1r 10575 . . . 4 -1RR
4 mulclsr 10577 . . . 4 (((1st𝐴) ∈ R ∧ -1RR) → ((1st𝐴) ·R -1R) ∈ R)
52, 3, 4sylancl 589 . . 3 (𝐴 ∈ ℝ → ((1st𝐴) ·R -1R) ∈ R)
6 opelreal 10623 . . 3 (⟨((1st𝐴) ·R -1R), 0R⟩ ∈ ℝ ↔ ((1st𝐴) ·R -1R) ∈ R)
75, 6sylibr 237 . 2 (𝐴 ∈ ℝ → ⟨((1st𝐴) ·R -1R), 0R⟩ ∈ ℝ)
81simprbi 500 . . . 4 (𝐴 ∈ ℝ → 𝐴 = ⟨(1st𝐴), 0R⟩)
98oveq1d 7179 . . 3 (𝐴 ∈ ℝ → (𝐴 + ⟨((1st𝐴) ·R -1R), 0R⟩) = (⟨(1st𝐴), 0R⟩ + ⟨((1st𝐴) ·R -1R), 0R⟩))
10 addresr 10631 . . . 4 (((1st𝐴) ∈ R ∧ ((1st𝐴) ·R -1R) ∈ R) → (⟨(1st𝐴), 0R⟩ + ⟨((1st𝐴) ·R -1R), 0R⟩) = ⟨((1st𝐴) +R ((1st𝐴) ·R -1R)), 0R⟩)
112, 5, 10syl2anc 587 . . 3 (𝐴 ∈ ℝ → (⟨(1st𝐴), 0R⟩ + ⟨((1st𝐴) ·R -1R), 0R⟩) = ⟨((1st𝐴) +R ((1st𝐴) ·R -1R)), 0R⟩)
12 pn0sr 10594 . . . . . 6 ((1st𝐴) ∈ R → ((1st𝐴) +R ((1st𝐴) ·R -1R)) = 0R)
1312opeq1d 4764 . . . . 5 ((1st𝐴) ∈ R → ⟨((1st𝐴) +R ((1st𝐴) ·R -1R)), 0R⟩ = ⟨0R, 0R⟩)
14 df-0 10615 . . . . 5 0 = ⟨0R, 0R
1513, 14eqtr4di 2791 . . . 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 2777 . 2 (𝐴 ∈ ℝ → (𝐴 + ⟨((1st𝐴) ·R -1R), 0R⟩) = 0)
18 oveq2 7172 . . . 4 (𝑥 = ⟨((1st𝐴) ·R -1R), 0R⟩ → (𝐴 + 𝑥) = (𝐴 + ⟨((1st𝐴) ·R -1R), 0R⟩))
1918eqeq1d 2740 . . 3 (𝑥 = ⟨((1st𝐴) ·R -1R), 0R⟩ → ((𝐴 + 𝑥) = 0 ↔ (𝐴 + ⟨((1st𝐴) ·R -1R), 0R⟩) = 0))
2019rspcev 3524 . 2 ((⟨((1st𝐴) ·R -1R), 0R⟩ ∈ ℝ ∧ (𝐴 + ⟨((1st𝐴) ·R -1R), 0R⟩) = 0) → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
217, 17, 20syl2anc 587 1 (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2113  wrex 3054  cop 4519  cfv 6333  (class class class)co 7164  1st c1st 7705  Rcnr 10358  0Rc0r 10359  -1Rcm1r 10361   +R cplr 10362   ·R cmr 10363  cr 10607  0cc0 10608   + caddc 10611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473  ax-inf2 9170
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-int 4834  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169  df-om 7594  df-1st 7707  df-2nd 7708  df-wrecs 7969  df-recs 8030  df-rdg 8068  df-1o 8124  df-oadd 8128  df-omul 8129  df-er 8313  df-ec 8315  df-qs 8319  df-ni 10365  df-pli 10366  df-mi 10367  df-lti 10368  df-plpq 10401  df-mpq 10402  df-ltpq 10403  df-enq 10404  df-nq 10405  df-erq 10406  df-plq 10407  df-mq 10408  df-1nq 10409  df-rq 10410  df-ltnq 10411  df-np 10474  df-1p 10475  df-plp 10476  df-mp 10477  df-ltp 10478  df-enr 10548  df-nr 10549  df-plr 10550  df-mr 10551  df-0r 10553  df-1r 10554  df-m1r 10555  df-c 10614  df-0 10615  df-r 10618  df-add 10619
This theorem is referenced by: (None)
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