Theorem axrnegex 11250

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 11274. (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 11220	. . . . 5 ⊢ (𝐴 ∈ ℝ ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1st ‘𝐴), 0R⟩))
2	1	simplbi 502	. . . 4 ⊢ (𝐴 ∈ ℝ → (1st ‘𝐴) ∈ R)
3		m1r 11170	. . . 4 ⊢ -1R ∈ R
4		mulclsr 11172	. . . 4 ⊢ (((1st ‘𝐴) ∈ R ∧ -1R ∈ R) → ((1st ‘𝐴) ·R -1R) ∈ R)
5	2, 3, 4	sylancl 598	. . 3 ⊢ (𝐴 ∈ ℝ → ((1st ‘𝐴) ·R -1R) ∈ R)
6		opelreal 11218	. . 3 ⊢ (⟨((1st ‘𝐴) ·R -1R), 0R⟩ ∈ ℝ ↔ ((1st ‘𝐴) ·R -1R) ∈ R)
7	5, 6	sylibr 237	. 2 ⊢ (𝐴 ∈ ℝ → ⟨((1st ‘𝐴) ·R -1R), 0R⟩ ∈ ℝ)
8	1	simprbi 503	. . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 = ⟨(1st ‘𝐴), 0R⟩)
9	8	oveq1d 7435	. . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = (⟨(1st ‘𝐴), 0R⟩ + ⟨((1st ‘𝐴) ·R -1R), 0R⟩))
10		addresr 11226	. . . 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 596	. . 3 ⊢ (𝐴 ∈ ℝ → (⟨(1st ‘𝐴), 0R⟩ + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = ⟨((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R⟩)
12		pn0sr 11189	. . . . . 6 ⊢ ((1st ‘𝐴) ∈ R → ((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)) = 0R)
13	12	opeq1d 4845	. . . . 5 ⊢ ((1st ‘𝐴) ∈ R → ⟨((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R⟩ = ⟨0R, 0R⟩)
14		df-0 11210	. . . . 5 ⊢ 0 = ⟨0R, 0R⟩
15	13, 14	eqtr4di 2813	. . . 4 ⊢ ((1st ‘𝐴) ∈ R → ⟨((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R⟩ = 0)
16	2, 15	syl 18	. . 3 ⊢ (𝐴 ∈ ℝ → ⟨((1st ‘𝐴) +R ((1st ‘𝐴) ·R -1R)), 0R⟩ = 0)
17	9, 11, 16	3eqtrd 2799	. 2 ⊢ (𝐴 ∈ ℝ → (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = 0)
18		oveq2 7428	. . . 4 ⊢ (𝑥 = ⟨((1st ‘𝐴) ·R -1R), 0R⟩ → (𝐴 + 𝑥) = (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩))
19	18	eqeq1d 2762	. . 3 ⊢ (𝑥 = ⟨((1st ‘𝐴) ·R -1R), 0R⟩ → ((𝐴 + 𝑥) = 0 ↔ (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = 0))
20	19	rspcev 3576	. 2 ⊢ ((⟨((1st ‘𝐴) ·R -1R), 0R⟩ ∈ ℝ ∧ (𝐴 + ⟨((1st ‘𝐴) ·R -1R), 0R⟩) = 0) → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
21	7, 17, 20	syl2anc 596	1 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  ⟨cop 4834  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  Rcnr 10953  0_Rc0r 10954  -1_Rcm1r 10956   +_R cplr 10957   ·_R cmr 10958  ℝcr 11202  0cc0 11203   + caddc 11206

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9645

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-oadd 8483  df-omul 8484  df-er 8720  df-ec 8722  df-qs 8726  df-ni 10960  df-pli 10961  df-mi 10962  df-lti 10963  df-plpq 10996  df-mpq 10997  df-ltpq 10998  df-enq 10999  df-nq 11000  df-erq 11001  df-plq 11002  df-mq 11003  df-1nq 11004  df-rq 11005  df-ltnq 11006  df-np 11069  df-1p 11070  df-plp 11071  df-mp 11072  df-ltp 11073  df-enr 11143  df-nr 11144  df-plr 11145  df-mr 11146  df-0r 11148  df-1r 11149  df-m1r 11150  df-c 11209  df-0 11210  df-r 11213  df-add 11214

This theorem is used by: (None)