Theorem axrnegex 11154

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 11178. (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 11124	. . . . 5 ⊢ (𝐴 ∈ ℝ ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1st ‘𝐴), 0R⟩))
2	1	simplbi 497	. . . 4 ⊢ (𝐴 ∈ ℝ → (1st ‘𝐴) ∈ R)
3		m1r 11074	. . . 4 ⊢ -1R ∈ R
4		mulclsr 11076	. . . 4 ⊢ (((1st ‘𝐴) ∈ R ∧ -1R ∈ R) → ((1st ‘𝐴) ·R -1R) ∈ R)
5	2, 3, 4	sylancl 585	. . 3 ⊢ (𝐴 ∈ ℝ → ((1st ‘𝐴) ·R -1R) ∈ R)
6		opelreal 11122	. . 3 ⊢ (⟨((1st ‘𝐴) ·R -1R), 0R⟩ ∈ ℝ ↔ ((1st ‘𝐴) ·R -1R) ∈ R)
7	5, 6	sylibr 233	. 2 ⊢ (𝐴 ∈ ℝ → ⟨((1st ‘𝐴) ·R -1R), 0R⟩ ∈ ℝ)
8	1	simprbi 496	. . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 = ⟨(1st ‘𝐴), 0R⟩)
9	8	oveq1d 7417	. . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = (⟨(1st ‘𝐴), 0R⟩ + ⟨((1st ‘𝐴) ·R -1R), 0R⟩))
10		addresr 11130	. . . 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 583	. . 3 ⊢ (𝐴 ∈ ℝ → (⟨(1st ‘𝐴), 0R⟩ + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = ⟨((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R⟩)
12		pn0sr 11093	. . . . . 6 ⊢ ((1st ‘𝐴) ∈ R → ((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)) = 0R)
13	12	opeq1d 4872	. . . . 5 ⊢ ((1st ‘𝐴) ∈ R → ⟨((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R⟩ = ⟨0R, 0R⟩)
14		df-0 11114	. . . . 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 7410	. . . 4 ⊢ (𝑥 = ⟨((1st ‘𝐴) ·R -1R), 0R⟩ → (𝐴 + 𝑥) = (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩))
19	18	eqeq1d 2726	. . 3 ⊢ (𝑥 = ⟨((1st ‘𝐴) ·R -1R), 0R⟩ → ((𝐴 + 𝑥) = 0 ↔ (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = 0))
20	19	rspcev 3604	. 2 ⊢ ((⟨((1st ‘𝐴) ·R -1R), 0R⟩ ∈ ℝ ∧ (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = 0) → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
21	7, 17, 20	syl2anc 583	1 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  ∃wrex 3062  ⟨cop 4627  ‘cfv 6534  (class class class)co 7402  1^st c1st 7967  Rcnr 10857  0_Rc0r 10858  -1_Rcm1r 10860   +_R cplr 10861   ·_R cmr 10862  ℝcr 11106  0cc0 11107   + caddc 11110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-inf2 9633

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-oadd 8466  df-omul 8467  df-er 8700  df-ec 8702  df-qs 8706  df-ni 10864  df-pli 10865  df-mi 10866  df-lti 10867  df-plpq 10900  df-mpq 10901  df-ltpq 10902  df-enq 10903  df-nq 10904  df-erq 10905  df-plq 10906  df-mq 10907  df-1nq 10908  df-rq 10909  df-ltnq 10910  df-np 10973  df-1p 10974  df-plp 10975  df-mp 10976  df-ltp 10977  df-enr 11047  df-nr 11048  df-plr 11049  df-mr 11050  df-0r 11052  df-1r 11053  df-m1r 11054  df-c 11113  df-0 11114  df-r 11117  df-add 11118

This theorem is referenced by: (None)