Theorem ax0id 7907

Description: 0 is an identity element for real addition. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-0id 7949.

In the Metamath Proof Explorer this is not a complex number axiom but is instead proved from other axioms. That proof relies on excluded middle and it is not known whether it is possible to prove this from the other axioms without excluded middle. (Contributed by Jim Kingdon, 16-Jan-2020.) (New usage is discouraged.)

Assertion

Ref	Expression
ax0id	⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)

Proof of Theorem ax0id

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-c 7847	. 2 ⊢ ℂ = (R × R)
2		oveq1 5903	. . 3 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (⟨𝑥, 𝑦⟩ + 0) = (𝐴 + 0))
3		id 19	. . 3 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → ⟨𝑥, 𝑦⟩ = 𝐴)
4	2, 3	eqeq12d 2204	. 2 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → ((⟨𝑥, 𝑦⟩ + 0) = ⟨𝑥, 𝑦⟩ ↔ (𝐴 + 0) = 𝐴))
5		0r 7779	. . . 4 ⊢ 0R ∈ R
6		addcnsr 7863	. . . 4 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (0R ∈ R ∧ 0R ∈ R)) → (⟨𝑥, 𝑦⟩ + ⟨0R, 0R⟩) = ⟨(𝑥 +R 0R), (𝑦 +R 0R)⟩)
7	5, 5, 6	mpanr12 439	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (⟨𝑥, 𝑦⟩ + ⟨0R, 0R⟩) = ⟨(𝑥 +R 0R), (𝑦 +R 0R)⟩)
8		df-0 7848	. . . . . 6 ⊢ 0 = ⟨0R, 0R⟩
9	8	eqcomi 2193	. . . . 5 ⊢ ⟨0R, 0R⟩ = 0
10	9	a1i 9	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → ⟨0R, 0R⟩ = 0)
11	10	oveq2d 5912	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (⟨𝑥, 𝑦⟩ + ⟨0R, 0R⟩) = (⟨𝑥, 𝑦⟩ + 0))
12		0idsr 7796	. . . . 5 ⊢ (𝑥 ∈ R → (𝑥 +R 0R) = 𝑥)
13	12	adantr 276	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 +R 0R) = 𝑥)
14		0idsr 7796	. . . . 5 ⊢ (𝑦 ∈ R → (𝑦 +R 0R) = 𝑦)
15	14	adantl 277	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑦 +R 0R) = 𝑦)
16	13, 15	opeq12d 3801	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → ⟨(𝑥 +R 0R), (𝑦 +R 0R)⟩ = ⟨𝑥, 𝑦⟩)
17	7, 11, 16	3eqtr3d 2230	. 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (⟨𝑥, 𝑦⟩ + 0) = ⟨𝑥, 𝑦⟩)
18	1, 4, 17	optocl 4720	1 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2160  ⟨cop 3610  (class class class)co 5896  Rcnr 7326  0_Rc0r 7327   +_R cplr 7330  ℂcc 7839  0cc0 7841   + caddc 7844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-irdg 6395  df-1o 6441  df-2o 6442  df-oadd 6445  df-omul 6446  df-er 6559  df-ec 6561  df-qs 6565  df-ni 7333  df-pli 7334  df-mi 7335  df-lti 7336  df-plpq 7373  df-mpq 7374  df-enq 7376  df-nqqs 7377  df-plqqs 7378  df-mqqs 7379  df-1nqqs 7380  df-rq 7381  df-ltnqqs 7382  df-enq0 7453  df-nq0 7454  df-0nq0 7455  df-plq0 7456  df-mq0 7457  df-inp 7495  df-i1p 7496  df-iplp 7497  df-enr 7755  df-nr 7756  df-plr 7757  df-0r 7760  df-c 7847  df-0 7848  df-add 7852

This theorem is referenced by: (None)