Theorem axrnegex 5266

Description: Existence of negative of real number. Axiom 17 of 25 for real and complex numbers, derived from ZF set theory.

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		elreal 5233	. 2 ⊢ (A ∈ ℝ ↔ ∃y(y ∈ R ⋀ ⟨y, 0R⟩ = A))
2		opreq1 3963	. . . 4 ⊢ (⟨y, 0R⟩ = A → (⟨y, 0R⟩ + x) = (A + x))
3	2	eqeq1d 1481	. . 3 ⊢ (⟨y, 0R⟩ = A → ((⟨y, 0R⟩ + x) = 0 ↔ (A + x) = 0))
4	3	rexbidv 1662	. 2 ⊢ (⟨y, 0R⟩ = A → (∃x ∈ ℝ (⟨y, 0R⟩ + x) = 0 ↔ ∃x ∈ ℝ (A + x) = 0))
5		negexsr 5194	. . 3 ⊢ (y ∈ R → ∃z(z ∈ R ⋀ (y +R z) = 0R))
6		addresr 5239	. . . . . . . 8 ⊢ ((y ∈ R ⋀ z ∈ R) → (⟨y, 0R⟩ + ⟨z, 0R⟩) = ⟨(y +R z), 0R⟩)
7	6	eqeq1d 1481	. . . . . . 7 ⊢ ((y ∈ R ⋀ z ∈ R) → ((⟨y, 0R⟩ + ⟨z, 0R⟩) = 0 ↔ ⟨(y +R z), 0R⟩ = 0))
8		df-0 5224	. . . . . . . . 9 ⊢ 0 = ⟨0R, 0R⟩
9	8	eqeq2i 1483	. . . . . . . 8 ⊢ (⟨(y +R z), 0R⟩ = 0 ↔ ⟨(y +R z), 0R⟩ = ⟨0R, 0R⟩)
10		oprex 3978	. . . . . . . . 9 ⊢ (y +R z) ∈ V
11	10	eqresr 5238	. . . . . . . 8 ⊢ (⟨(y +R z), 0R⟩ = ⟨0R, 0R⟩ ↔ (y +R z) = 0R)
12	9, 11	bitr 173	. . . . . . 7 ⊢ (⟨(y +R z), 0R⟩ = 0 ↔ (y +R z) = 0R)
13	7, 12	syl6bb 535	. . . . . 6 ⊢ ((y ∈ R ⋀ z ∈ R) → ((⟨y, 0R⟩ + ⟨z, 0R⟩) = 0 ↔ (y +R z) = 0R))
14	13	pm5.32da 648	. . . . 5 ⊢ (y ∈ R → ((z ∈ R ⋀ (⟨y, 0R⟩ + ⟨z, 0R⟩) = 0) ↔ (z ∈ R ⋀ (y +R z) = 0R)))
15		opex 2778	. . . . . . . 8 ⊢ ⟨z, 0R⟩ ∈ V
16		eleq1 1532	. . . . . . . . 9 ⊢ (x = ⟨z, 0R⟩ → (x ∈ ℝ ↔ ⟨z, 0R⟩ ∈ ℝ))
17		opreq2 3964	. . . . . . . . . 10 ⊢ (x = ⟨z, 0R⟩ → (⟨y, 0R⟩ + x) = (⟨y, 0R⟩ + ⟨z, 0R⟩))
18	17	eqeq1d 1481	. . . . . . . . 9 ⊢ (x = ⟨z, 0R⟩ → ((⟨y, 0R⟩ + x) = 0 ↔ (⟨y, 0R⟩ + ⟨z, 0R⟩) = 0))
19	16, 18	anbi12d 627	. . . . . . . 8 ⊢ (x = ⟨z, 0R⟩ → ((x ∈ ℝ ⋀ (⟨y, 0R⟩ + x) = 0) ↔ (⟨z, 0R⟩ ∈ ℝ ⋀ (⟨y, 0R⟩ + ⟨z, 0R⟩) = 0)))
20	15, 19	cla4ev 1866	. . . . . . 7 ⊢ ((⟨z, 0R⟩ ∈ ℝ ⋀ (⟨y, 0R⟩ + ⟨z, 0R⟩) = 0) → ∃x(x ∈ ℝ ⋀ (⟨y, 0R⟩ + x) = 0))
21		opelreal 5232	. . . . . . 7 ⊢ (⟨z, 0R⟩ ∈ ℝ ↔ z ∈ R)
22	20, 21	sylanbr 450	. . . . . 6 ⊢ ((z ∈ R ⋀ (⟨y, 0R⟩ + ⟨z, 0R⟩) = 0) → ∃x(x ∈ ℝ ⋀ (⟨y, 0R⟩ + x) = 0))
23		df-rex 1648	. . . . . 6 ⊢ (∃x ∈ ℝ (⟨y, 0R⟩ + x) = 0 ↔ ∃x(x ∈ ℝ ⋀ (⟨y, 0R⟩ + x) = 0))
24	22, 23	sylibr 200	. . . . 5 ⊢ ((z ∈ R ⋀ (⟨y, 0R⟩ + ⟨z, 0R⟩) = 0) → ∃x ∈ ℝ (⟨y, 0R⟩ + x) = 0)
25	14, 24	syl6bir 215	. . . 4 ⊢ (y ∈ R → ((z ∈ R ⋀ (y +R z) = 0R) → ∃x ∈ ℝ (⟨y, 0R⟩ + x) = 0))
26	25	19.23adv 1213	. . 3 ⊢ (y ∈ R → (∃z(z ∈ R ⋀ (y +R z) = 0R) → ∃x ∈ ℝ (⟨y, 0R⟩ + x) = 0))
27	5, 26	mpd 26	. 2 ⊢ (y ∈ R → ∃x ∈ ℝ (⟨y, 0R⟩ + x) = 0)
28	1, 4, 27	gencl 1825	1 ⊢ (A ∈ ℝ → ∃x ∈ ℝ (A + x) = 0)

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 955   ∈ wcel 957  ∃wex 979  ∃wrex 1644  ⟨cop 2408  (class class class)co 3958  Rcnr 4976  0_Rc0r 4977   +_R cplr 4980  ℝcr 5216  0cc0 5217   + caddc 5220

This theorem is referenced by:  cnegextlem1 5328  cnegextlem2 5329  cnegextlem3 5330  cnegext 5331  renegcl 5399  0re 5423

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-inf2 4608

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1st 4072  df-2nd 4073  df-1o 4126  df-oadd 4128  df-omul 4129  df-er 4254  df-ec 4256  df-qs 4259  df-ni 4983  df-pli 4984  df-mi 4985  df-lti 4986  df-plpq 5018  df-mpq 5019  df-enq 5020  df-nq 5021  df-plq 5022  df-mq 5023  df-rq 5024  df-ltq 5025  df-1q 5026  df-np 5069  df-1p 5070  df-plp 5071  df-mp 5072  df-ltp 5073  df-plpr 5147  df-mpr 5148  df-enr 5149  df-nr 5150  df-plr 5151  df-mr 5152  df-0r 5154  df-1r 5155  df-m1r 5156  df-c 5223  df-0 5224  df-r 5227  df-plus 5228

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