Theorem axrnegex 8034

Description: Existence of negative of real number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rnegex 8076. (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 7985	. . . . 5 ⊢ (𝐴 ∈ ℝ ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1st ‘𝐴), 0R⟩))
2	1	simplbi 274	. . . 4 ⊢ (𝐴 ∈ ℝ → (1st ‘𝐴) ∈ R)
3		m1r 7907	. . . 4 ⊢ -1R ∈ R
4		mulclsr 7909	. . . 4 ⊢ (((1st ‘𝐴) ∈ R ∧ -1R ∈ R) → ((1st ‘𝐴) ·R -1R) ∈ R)
5	2, 3, 4	sylancl 413	. . 3 ⊢ (𝐴 ∈ ℝ → ((1st ‘𝐴) ·R -1R) ∈ R)
6		opelreal 7982	. . 3 ⊢ (⟨((1st ‘𝐴) ·R -1R), 0R⟩ ∈ ℝ ↔ ((1st ‘𝐴) ·R -1R) ∈ R)
7	5, 6	sylibr 134	. 2 ⊢ (𝐴 ∈ ℝ → ⟨((1st ‘𝐴) ·R -1R), 0R⟩ ∈ ℝ)
8	1	simprbi 275	. . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 = ⟨(1st ‘𝐴), 0R⟩)
9	8	oveq1d 5989	. . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = (⟨(1st ‘𝐴), 0R⟩ + ⟨((1st ‘𝐴) ·R -1R), 0R⟩))
10		addresr 7992	. . . 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 411	. . 3 ⊢ (𝐴 ∈ ℝ → (⟨(1st ‘𝐴), 0R⟩ + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = ⟨((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R⟩)
12		pn0sr 7926	. . . . . 6 ⊢ ((1st ‘𝐴) ∈ R → ((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)) = 0R)
13	12	opeq1d 3842	. . . . 5 ⊢ ((1st ‘𝐴) ∈ R → ⟨((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R⟩ = ⟨0R, 0R⟩)
14		df-0 7974	. . . . 5 ⊢ 0 = ⟨0R, 0R⟩
15	13, 14	eqtr4di 2260	. . . 4 ⊢ ((1st ‘𝐴) ∈ R → ⟨((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R⟩ = 0)
16	2, 15	syl 14	. . 3 ⊢ (𝐴 ∈ ℝ → ⟨((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R⟩ = 0)
17	9, 11, 16	3eqtrd 2246	. 2 ⊢ (𝐴 ∈ ℝ → (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = 0)
18		oveq2 5982	. . . 4 ⊢ (𝑥 = ⟨((1st ‘𝐴) ·R -1R), 0R⟩ → (𝐴 + 𝑥) = (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩))
19	18	eqeq1d 2218	. . 3 ⊢ (𝑥 = ⟨((1st ‘𝐴) ·R -1R), 0R⟩ → ((𝐴 + 𝑥) = 0 ↔ (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = 0))
20	19	rspcev 2887	. 2 ⊢ ((⟨((1st ‘𝐴) ·R -1R), 0R⟩ ∈ ℝ ∧ (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = 0) → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
21	7, 17, 20	syl2anc 411	1 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)

Syntax hints:   → wi 4   = wceq 1375   ∈ wcel 2180  ∃wrex 2489  ⟨cop 3649  ‘cfv 5294  (class class class)co 5974  1^st c1st 6254  Rcnr 7452  0_Rc0r 7453  -1_Rcm1r 7455   +_R cplr 7456   ·_R cmr 7457  ℝcr 7966  0cc0 7967   + caddc 7970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-eprel 4357  df-id 4361  df-po 4364  df-iso 4365  df-iord 4434  df-on 4436  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-irdg 6486  df-1o 6532  df-2o 6533  df-oadd 6536  df-omul 6537  df-er 6650  df-ec 6652  df-qs 6656  df-ni 7459  df-pli 7460  df-mi 7461  df-lti 7462  df-plpq 7499  df-mpq 7500  df-enq 7502  df-nqqs 7503  df-plqqs 7504  df-mqqs 7505  df-1nqqs 7506  df-rq 7507  df-ltnqqs 7508  df-enq0 7579  df-nq0 7580  df-0nq0 7581  df-plq0 7582  df-mq0 7583  df-inp 7621  df-i1p 7622  df-iplp 7623  df-imp 7624  df-enr 7881  df-nr 7882  df-plr 7883  df-mr 7884  df-0r 7886  df-1r 7887  df-m1r 7888  df-c 7973  df-0 7974  df-r 7977  df-add 7978

This theorem is referenced by: (None)