Theorem axrnegex 11056

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 11080. (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

Step	Hyp	Ref	Expression
1		elreal2 11026	. . . . 5 ⊢ (𝐴 ∈ ℝ ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1st ‘𝐴), 0R⟩))
2	1	simplbi 497	. . . 4 ⊢ (𝐴 ∈ ℝ → (1st ‘𝐴) ∈ R)
3		m1r 10976	. . . 4 ⊢ -1R ∈ R
4		mulclsr 10978	. . . 4 ⊢ (((1st ‘𝐴) ∈ R ∧ -1R ∈ R) → ((1st ‘𝐴) ·R -1R) ∈ R)
5	2, 3, 4	sylancl 586	. . 3 ⊢ (𝐴 ∈ ℝ → ((1st ‘𝐴) ·R -1R) ∈ R)
6		opelreal 11024	. . 3 ⊢ (⟨((1st ‘𝐴) ·R -1R), 0R⟩ ∈ ℝ ↔ ((1st ‘𝐴) ·R -1R) ∈ R)
7	5, 6	sylibr 234	. 2 ⊢ (𝐴 ∈ ℝ → ⟨((1st ‘𝐴) ·R -1R), 0R⟩ ∈ ℝ)
8	1	simprbi 496	. . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 = ⟨(1st ‘𝐴), 0R⟩)
9	8	oveq1d 7364	. . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = (⟨(1st ‘𝐴), 0R⟩ + ⟨((1st ‘𝐴) ·R -1R), 0R⟩))
10		addresr 11032	. . . 4 ⊢ (((1st ‘𝐴) ∈ R ∧ ((1st ‘𝐴) ·R -1R) ∈ R) → (⟨(1st ‘𝐴), 0R⟩ + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = ⟨((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R⟩)
11	2, 5, 10	syl2anc 584	. . 3 ⊢ (𝐴 ∈ ℝ → (⟨(1st ‘𝐴), 0R⟩ + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = ⟨((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R⟩)
12		pn0sr 10995	. . . . . 6 ⊢ ((1st ‘𝐴) ∈ R → ((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)) = 0R)
13	12	opeq1d 4830	. . . . 5 ⊢ ((1st ‘𝐴) ∈ R → ⟨((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R⟩ = ⟨0R, 0R⟩)
14		df-0 11016	. . . . 5 ⊢ 0 = ⟨0R, 0R⟩
15	13, 14	eqtr4di 2782	. . . 4 ⊢ ((1st ‘𝐴) ∈ R → ⟨((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R⟩ = 0)
16	2, 15	syl 17	. . 3 ⊢ (𝐴 ∈ ℝ → ⟨((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R⟩ = 0)
17	9, 11, 16	3eqtrd 2768	. 2 ⊢ (𝐴 ∈ ℝ → (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = 0)
18		oveq2 7357	. . . 4 ⊢ (𝑥 = ⟨((1st ‘𝐴) ·R -1R), 0R⟩ → (𝐴 + 𝑥) = (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩))
19	18	eqeq1d 2731	. . 3 ⊢ (𝑥 = ⟨((1st ‘𝐴) ·R -1R), 0R⟩ → ((𝐴 + 𝑥) = 0 ↔ (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = 0))
20	19	rspcev 3577	. 2 ⊢ ((⟨((1st ‘𝐴) ·R -1R), 0R⟩ ∈ ℝ ∧ (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = 0) → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
21	7, 17, 20	syl2anc 584	1 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  ⟨cop 4583  ‘cfv 6482  (class class class)co 7349  1^st c1st 7922  Rcnr 10759  0_Rc0r 10760  -1_Rcm1r 10762   +_R cplr 10763   ·_R cmr 10764  ℝcr 11008  0cc0 11009   + caddc 11012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-oadd 8392  df-omul 8393  df-er 8625  df-ec 8627  df-qs 8631  df-ni 10766  df-pli 10767  df-mi 10768  df-lti 10769  df-plpq 10802  df-mpq 10803  df-ltpq 10804  df-enq 10805  df-nq 10806  df-erq 10807  df-plq 10808  df-mq 10809  df-1nq 10810  df-rq 10811  df-ltnq 10812  df-np 10875  df-1p 10876  df-plp 10877  df-mp 10878  df-ltp 10879  df-enr 10949  df-nr 10950  df-plr 10951  df-mr 10952  df-0r 10954  df-1r 10955  df-m1r 10956  df-c 11015  df-0 11016  df-r 11019  df-add 11020

This theorem is referenced by: (None)