Theorem ax0id 8098

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 8140.

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 8038	. 2 ⊢ ℂ = (R × R)
2		oveq1 6025	. . 3 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (⟨𝑥, 𝑦⟩ + 0) = (𝐴 + 0))
3		id 19	. . 3 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → ⟨𝑥, 𝑦⟩ = 𝐴)
4	2, 3	eqeq12d 2246	. 2 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → ((⟨𝑥, 𝑦⟩ + 0) = ⟨𝑥, 𝑦⟩ ↔ (𝐴 + 0) = 𝐴))
5		0r 7970	. . . 4 ⊢ 0R ∈ R
6		addcnsr 8054	. . . 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 8039	. . . . . 6 ⊢ 0 = ⟨0R, 0R⟩
9	8	eqcomi 2235	. . . . 5 ⊢ ⟨0R, 0R⟩ = 0
10	9	a1i 9	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → ⟨0R, 0R⟩ = 0)
11	10	oveq2d 6034	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (⟨𝑥, 𝑦⟩ + ⟨0R, 0R⟩) = (⟨𝑥, 𝑦⟩ + 0))
12		0idsr 7987	. . . . 5 ⊢ (𝑥 ∈ R → (𝑥 +R 0R) = 𝑥)
13	12	adantr 276	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 +R 0R) = 𝑥)
14		0idsr 7987	. . . . 5 ⊢ (𝑦 ∈ R → (𝑦 +R 0R) = 𝑦)
15	14	adantl 277	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑦 +R 0R) = 𝑦)
16	13, 15	opeq12d 3870	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → ⟨(𝑥 +R 0R), (𝑦 +R 0R)⟩ = ⟨𝑥, 𝑦⟩)
17	7, 11, 16	3eqtr3d 2272	. 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (⟨𝑥, 𝑦⟩ + 0) = ⟨𝑥, 𝑦⟩)
18	1, 4, 17	optocl 4802	1 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202  ⟨cop 3672  (class class class)co 6018  Rcnr 7517  0_Rc0r 7518   +_R cplr 7521  ℂcc 8030  0cc0 8032   + caddc 8035

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-1o 6582  df-2o 6583  df-oadd 6586  df-omul 6587  df-er 6702  df-ec 6704  df-qs 6708  df-ni 7524  df-pli 7525  df-mi 7526  df-lti 7527  df-plpq 7564  df-mpq 7565  df-enq 7567  df-nqqs 7568  df-plqqs 7569  df-mqqs 7570  df-1nqqs 7571  df-rq 7572  df-ltnqqs 7573  df-enq0 7644  df-nq0 7645  df-0nq0 7646  df-plq0 7647  df-mq0 7648  df-inp 7686  df-i1p 7687  df-iplp 7688  df-enr 7946  df-nr 7947  df-plr 7948  df-0r 7951  df-c 8038  df-0 8039  df-add 8043

This theorem is referenced by: (None)