Theorem axrnegex 11085

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 11109. (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 11055	. . . . 5 ⊢ (𝐴 ∈ ℝ ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1st ‘𝐴), 0R⟩))
2	1	simplbi 496	. . . 4 ⊢ (𝐴 ∈ ℝ → (1st ‘𝐴) ∈ R)
3		m1r 11005	. . . 4 ⊢ -1R ∈ R
4		mulclsr 11007	. . . 4 ⊢ (((1st ‘𝐴) ∈ R ∧ -1R ∈ R) → ((1st ‘𝐴) ·R -1R) ∈ R)
5	2, 3, 4	sylancl 587	. . 3 ⊢ (𝐴 ∈ ℝ → ((1st ‘𝐴) ·R -1R) ∈ R)
6		opelreal 11053	. . 3 ⊢ (⟨((1st ‘𝐴) ·R -1R), 0R⟩ ∈ ℝ ↔ ((1st ‘𝐴) ·R -1R) ∈ R)
7	5, 6	sylibr 234	. 2 ⊢ (𝐴 ∈ ℝ → ⟨((1st ‘𝐴) ·R -1R), 0R⟩ ∈ ℝ)
8	1	simprbi 497	. . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 = ⟨(1st ‘𝐴), 0R⟩)
9	8	oveq1d 7382	. . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = (⟨(1st ‘𝐴), 0R⟩ + ⟨((1st ‘𝐴) ·R -1R), 0R⟩))
10		addresr 11061	. . . 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 585	. . 3 ⊢ (𝐴 ∈ ℝ → (⟨(1st ‘𝐴), 0R⟩ + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = ⟨((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R⟩)
12		pn0sr 11024	. . . . . 6 ⊢ ((1st ‘𝐴) ∈ R → ((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)) = 0R)
13	12	opeq1d 4822	. . . . 5 ⊢ ((1st ‘𝐴) ∈ R → ⟨((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R⟩ = ⟨0R, 0R⟩)
14		df-0 11045	. . . . 5 ⊢ 0 = ⟨0R, 0R⟩
15	13, 14	eqtr4di 2789	. . . 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 2775	. 2 ⊢ (𝐴 ∈ ℝ → (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = 0)
18		oveq2 7375	. . . 4 ⊢ (𝑥 = ⟨((1st ‘𝐴) ·R -1R), 0R⟩ → (𝐴 + 𝑥) = (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩))
19	18	eqeq1d 2738	. . 3 ⊢ (𝑥 = ⟨((1st ‘𝐴) ·R -1R), 0R⟩ → ((𝐴 + 𝑥) = 0 ↔ (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = 0))
20	19	rspcev 3564	. 2 ⊢ ((⟨((1st ‘𝐴) ·R -1R), 0R⟩ ∈ ℝ ∧ (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = 0) → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
21	7, 17, 20	syl2anc 585	1 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∃wrex 3061  ⟨cop 4573  ‘cfv 6498  (class class class)co 7367  1^st c1st 7940  Rcnr 10788  0_Rc0r 10789  -1_Rcm1r 10791   +_R cplr 10792   ·_R cmr 10793  ℝcr 11037  0cc0 11038   + caddc 11041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-oadd 8409  df-omul 8410  df-er 8643  df-ec 8645  df-qs 8649  df-ni 10795  df-pli 10796  df-mi 10797  df-lti 10798  df-plpq 10831  df-mpq 10832  df-ltpq 10833  df-enq 10834  df-nq 10835  df-erq 10836  df-plq 10837  df-mq 10838  df-1nq 10839  df-rq 10840  df-ltnq 10841  df-np 10904  df-1p 10905  df-plp 10906  df-mp 10907  df-ltp 10908  df-enr 10978  df-nr 10979  df-plr 10980  df-mr 10981  df-0r 10983  df-1r 10984  df-m1r 10985  df-c 11044  df-0 11045  df-r 11048  df-add 11049

This theorem is referenced by: (None)