Theorem axrnegex 5255

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 5222	. 2 ⊢ (A ∈ ℝ ↔ ∃y(y ∈ R ⋀ ⟨y, 0R⟩ = A))
2		opreq1 3953	. . . 4 ⊢ (⟨y, 0R⟩ = A → (⟨y, 0R⟩ + x) = (A + x))
3	2	eqeq1d 1475	. . 3 ⊢ (⟨y, 0R⟩ = A → ((⟨y, 0R⟩ + x) = 0 ↔ (A + x) = 0))
4	3	rexbidv 1656	. 2 ⊢ (⟨y, 0R⟩ = A → (∃x ∈ ℝ (⟨y, 0R⟩ + x) = 0 ↔ ∃x ∈ ℝ (A + x) = 0))
5		negexsr 5183	. . 3 ⊢ (y ∈ R → ∃z(z ∈ R ⋀ (y +R z) = 0R))
6		addresr 5228	. . . . . . . 8 ⊢ ((y ∈ R ⋀ z ∈ R) → (⟨y, 0R⟩ + ⟨z, 0R⟩) = ⟨(y +R z), 0R⟩)
7	6	eqeq1d 1475	. . . . . . 7 ⊢ ((y ∈ R ⋀ z ∈ R) → ((⟨y, 0R⟩ + ⟨z, 0R⟩) = 0 ↔ ⟨(y +R z), 0R⟩ = 0))
8		df-0 5213	. . . . . . . . 9 ⊢ 0 = ⟨0R, 0R⟩
9	8	eqeq2i 1477	. . . . . . . 8 ⊢ (⟨(y +R z), 0R⟩ = 0 ↔ ⟨(y +R z), 0R⟩ = ⟨0R, 0R⟩)
10		oprex 3968	. . . . . . . . 9 ⊢ (y +R z) ∈ V
11	10	eqresr 5227	. . . . . . . 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 534	. . . . . 6 ⊢ ((y ∈ R ⋀ z ∈ R) → ((⟨y, 0R⟩ + ⟨z, 0R⟩) = 0 ↔ (y +R z) = 0R))
14	13	pm5.32da 647	. . . . 5 ⊢ (y ∈ R → ((z ∈ R ⋀ (⟨y, 0R⟩ + ⟨z, 0R⟩) = 0) ↔ (z ∈ R ⋀ (y +R z) = 0R)))
15		opex 2772	. . . . . . . 8 ⊢ ⟨z, 0R⟩ ∈ V
16		eleq1 1526	. . . . . . . . 9 ⊢ (x = ⟨z, 0R⟩ → (x ∈ ℝ ↔ ⟨z, 0R⟩ ∈ ℝ))
17		opreq2 3954	. . . . . . . . . 10 ⊢ (x = ⟨z, 0R⟩ → (⟨y, 0R⟩ + x) = (⟨y, 0R⟩ + ⟨z, 0R⟩))
18	17	eqeq1d 1475	. . . . . . . . 9 ⊢ (x = ⟨z, 0R⟩ → ((⟨y, 0R⟩ + x) = 0 ↔ (⟨y, 0R⟩ + ⟨z, 0R⟩) = 0))
19	16, 18	anbi12d 626	. . . . . . . 8 ⊢ (x = ⟨z, 0R⟩ → ((x ∈ ℝ ⋀ (⟨y, 0R⟩ + x) = 0) ↔ (⟨z, 0R⟩ ∈ ℝ ⋀ (⟨y, 0R⟩ + ⟨z, 0R⟩) = 0)))
20	15, 19	cla4ev 1860	. . . . . . 7 ⊢ ((⟨z, 0R⟩ ∈ ℝ ⋀ (⟨y, 0R⟩ + ⟨z, 0R⟩) = 0) → ∃x(x ∈ ℝ ⋀ (⟨y, 0R⟩ + x) = 0))
21		opelreal 5221	. . . . . . 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 1642	. . . . . 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 1209	. . 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 1819	1 ⊢ (A ∈ ℝ → ∃x ∈ ℝ (A + x) = 0)

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 953   ∈ wcel 955  ∃wex 977  ∃wrex 1638  ⟨cop 2401  (class class class)co 3948  Rcnr 4965  0_Rc0r 4966   +_R cplr 4969  ℝcr 5205  0cc0 5206   + caddc 5209

This theorem is referenced by:  cnegextlem1 5317  cnegextlem2 5318  cnegextlem3 5319  cnegext 5320  renegcl 5388  0re 5412

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-r 5216  df-plus 5217

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