Theorem axrnegex 4080

Description: Existence of negative of real number. One of the 28 axioms for real and complex numbers, derived from ZF set theory.

Assertion

Ref	Expression
axrnegex	⊢ (A ∈ ℝ → ∃x ∈ ℝ (A + x) = 0)

Distinct variable group(s):   x,A

Proof of Theorem axrnegex

Step	Hyp	Ref	Expression
1		elreal 4044	. 2 ⊢ (A ∈ ℝ ↔ ∃y(y ∈ R ∧ ⟨y, 0R⟩ = A))
2		opreq1 3006	. . . 4 ⊢ (⟨y, 0R⟩ = A → (⟨y, 0R⟩ + x) = (A + x))
3	2	cleq1d 1109	. . 3 ⊢ (⟨y, 0R⟩ = A → ((⟨y, 0R⟩ + x) = 0 ↔ (A + x) = 0))
4	3	birexdv 1220	. 2 ⊢ (⟨y, 0R⟩ = A → (∃x ∈ ℝ (⟨y, 0R⟩ + x) = 0 ↔ ∃x ∈ ℝ (A + x) = 0))
5		negexsr 4005	. . 3 ⊢ (y ∈ R → ∃z(z ∈ R ∧ (y +R z) = 0R))
6		addresr 4050	. . . . . . . . 9 ⊢ ((y ∈ R ∧ z ∈ R) → (⟨y, 0R⟩ + ⟨z, 0R⟩) = ⟨(y +R z), 0R⟩)
7	6	cleq1d 1109	. . . . . . . 8 ⊢ ((y ∈ R ∧ z ∈ R) → ((⟨y, 0R⟩ + ⟨z, 0R⟩) = 0 ↔ ⟨(y +R z), 0R⟩ = 0))
8		df-0 4035	. . . . . . . . . 10 ⊢ 0 = ⟨0R, 0R⟩
9	8	cleq2i 1111	. . . . . . . . 9 ⊢ (⟨(y +R z), 0R⟩ = 0 ↔ ⟨(y +R z), 0R⟩ = ⟨0R, 0R⟩)
10		oprex 3018	. . . . . . . . . 10 ⊢ (y +R z) ∈ V
11	10	eqresr 4049	. . . . . . . . 9 ⊢ (⟨(y +R z), 0R⟩ = ⟨0R, 0R⟩ ↔ (y +R z) = 0R)
12	9, 11	bitr 151	. . . . . . . 8 ⊢ (⟨(y +R z), 0R⟩ = 0 ↔ (y +R z) = 0R)
13	7, 12	syl6bb 414	. . . . . . 7 ⊢ ((y ∈ R ∧ z ∈ R) → ((⟨y, 0R⟩ + ⟨z, 0R⟩) = 0 ↔ (y +R z) = 0R))
14	13	exp 291	. . . . . 6 ⊢ (y ∈ R → (z ∈ R → ((⟨y, 0R⟩ + ⟨z, 0R⟩) = 0 ↔ (y +R z) = 0R)))
15	14	pm5.32d 491	. . . . 5 ⊢ (y ∈ R → ((z ∈ R ∧ (⟨y, 0R⟩ + ⟨z, 0R⟩) = 0) ↔ (z ∈ R ∧ (y +R z) = 0R)))
16		opex 1893	. . . . . . . 8 ⊢ ⟨z, 0R⟩ ∈ V
17		eleq1 1149	. . . . . . . . 9 ⊢ (x = ⟨z, 0R⟩ → (x ∈ ℝ ↔ ⟨z, 0R⟩ ∈ ℝ))
18		opreq2 3007	. . . . . . . . . 10 ⊢ (x = ⟨z, 0R⟩ → (⟨y, 0R⟩ + x) = (⟨y, 0R⟩ + ⟨z, 0R⟩))
19	18	cleq1d 1109	. . . . . . . . 9 ⊢ (x = ⟨z, 0R⟩ → ((⟨y, 0R⟩ + x) = 0 ↔ (⟨y, 0R⟩ + ⟨z, 0R⟩) = 0))
20	17, 19	anbi12d 476	. . . . . . . 8 ⊢ (x = ⟨z, 0R⟩ → ((x ∈ ℝ ∧ (⟨y, 0R⟩ + x) = 0) ↔ (⟨z, 0R⟩ ∈ ℝ ∧ (⟨y, 0R⟩ + ⟨z, 0R⟩) = 0)))
21	16, 20	cla4ev 1401	. . . . . . 7 ⊢ ((⟨z, 0R⟩ ∈ ℝ ∧ (⟨y, 0R⟩ + ⟨z, 0R⟩) = 0) → ∃x(x ∈ ℝ ∧ (⟨y, 0R⟩ + x) = 0))
22		opelreal 4043	. . . . . . 7 ⊢ (⟨z, 0R⟩ ∈ ℝ ↔ z ∈ R)
23	21, 22	sylanbr 345	. . . . . 6 ⊢ ((z ∈ R ∧ (⟨y, 0R⟩ + ⟨z, 0R⟩) = 0) → ∃x(x ∈ ℝ ∧ (⟨y, 0R⟩ + x) = 0))
24		df-rex 1206	. . . . . 6 ⊢ (∃x ∈ ℝ (⟨y, 0R⟩ + x) = 0 ↔ ∃x(x ∈ ℝ ∧ (⟨y, 0R⟩ + x) = 0))
25	23, 24	sylibr 175	. . . . 5 ⊢ ((z ∈ R ∧ (⟨y, 0R⟩ + ⟨z, 0R⟩) = 0) → ∃x ∈ ℝ (⟨y, 0R⟩ + x) = 0)
26	15, 25	syl6bir 188	. . . 4 ⊢ (y ∈ R → ((z ∈ R ∧ (y +R z) = 0R) → ∃x ∈ ℝ (⟨y, 0R⟩ + x) = 0))
27	26	19.23adv 954	. . 3 ⊢ (y ∈ R → (∃z(z ∈ R ∧ (y +R z) = 0R) → ∃x ∈ ℝ (⟨y, 0R⟩ + x) = 0))
28	5, 27	mpd 46	. 2 ⊢ (y ∈ R → ∃x ∈ ℝ (⟨y, 0R⟩ + x) = 0)
29	1, 4, 28	gencl 1365	1 ⊢ (A ∈ ℝ → ∃x ∈ ℝ (A + x) = 0)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  ⟨cop 1810  (class class class)co 3001  Rcnr 3787  0_Rc0r 3788   +_R cplr 3791  ℝcr 4027  0cc0 4028   + caddc 4031

This theorem is referenced by:  renegcl 4171

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-r 4038  df-plus 4039

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