Theorem ax0id 8091

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 8133.

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 8031	. 2 ⊢ ℂ = (R × R)
2		oveq1 6020	. . 3 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (⟨𝑥, 𝑦⟩ + 0) = (𝐴 + 0))
3		id 19	. . 3 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → ⟨𝑥, 𝑦⟩ = 𝐴)
4	2, 3	eqeq12d 2244	. 2 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → ((⟨𝑥, 𝑦⟩ + 0) = ⟨𝑥, 𝑦⟩ ↔ (𝐴 + 0) = 𝐴))
5		0r 7963	. . . 4 ⊢ 0R ∈ R
6		addcnsr 8047	. . . 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 8032	. . . . . 6 ⊢ 0 = ⟨0R, 0R⟩
9	8	eqcomi 2233	. . . . 5 ⊢ ⟨0R, 0R⟩ = 0
10	9	a1i 9	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → ⟨0R, 0R⟩ = 0)
11	10	oveq2d 6029	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (⟨𝑥, 𝑦⟩ + ⟨0R, 0R⟩) = (⟨𝑥, 𝑦⟩ + 0))
12		0idsr 7980	. . . . 5 ⊢ (𝑥 ∈ R → (𝑥 +R 0R) = 𝑥)
13	12	adantr 276	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 +R 0R) = 𝑥)
14		0idsr 7980	. . . . 5 ⊢ (𝑦 ∈ R → (𝑦 +R 0R) = 𝑦)
15	14	adantl 277	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑦 +R 0R) = 𝑦)
16	13, 15	opeq12d 3868	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → ⟨(𝑥 +R 0R), (𝑦 +R 0R)⟩ = ⟨𝑥, 𝑦⟩)
17	7, 11, 16	3eqtr3d 2270	. 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (⟨𝑥, 𝑦⟩ + 0) = ⟨𝑥, 𝑦⟩)
18	1, 4, 17	optocl 4800	1 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  ⟨cop 3670  (class class class)co 6013  Rcnr 7510  0_Rc0r 7511   +_R cplr 7514  ℂcc 8023  0cc0 8025   + caddc 8028

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-2o 6578  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7517  df-pli 7518  df-mi 7519  df-lti 7520  df-plpq 7557  df-mpq 7558  df-enq 7560  df-nqqs 7561  df-plqqs 7562  df-mqqs 7563  df-1nqqs 7564  df-rq 7565  df-ltnqqs 7566  df-enq0 7637  df-nq0 7638  df-0nq0 7639  df-plq0 7640  df-mq0 7641  df-inp 7679  df-i1p 7680  df-iplp 7681  df-enr 7939  df-nr 7940  df-plr 7941  df-0r 7944  df-c 8031  df-0 8032  df-add 8036

This theorem is referenced by: (None)