Theorem ax0id 7998

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 8040.

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 7938	. 2 ⊢ ℂ = (R × R)
2		oveq1 5958	. . 3 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (⟨𝑥, 𝑦⟩ + 0) = (𝐴 + 0))
3		id 19	. . 3 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → ⟨𝑥, 𝑦⟩ = 𝐴)
4	2, 3	eqeq12d 2221	. 2 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → ((⟨𝑥, 𝑦⟩ + 0) = ⟨𝑥, 𝑦⟩ ↔ (𝐴 + 0) = 𝐴))
5		0r 7870	. . . 4 ⊢ 0R ∈ R
6		addcnsr 7954	. . . 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 7939	. . . . . 6 ⊢ 0 = ⟨0R, 0R⟩
9	8	eqcomi 2210	. . . . 5 ⊢ ⟨0R, 0R⟩ = 0
10	9	a1i 9	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → ⟨0R, 0R⟩ = 0)
11	10	oveq2d 5967	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (⟨𝑥, 𝑦⟩ + ⟨0R, 0R⟩) = (⟨𝑥, 𝑦⟩ + 0))
12		0idsr 7887	. . . . 5 ⊢ (𝑥 ∈ R → (𝑥 +R 0R) = 𝑥)
13	12	adantr 276	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 +R 0R) = 𝑥)
14		0idsr 7887	. . . . 5 ⊢ (𝑦 ∈ R → (𝑦 +R 0R) = 𝑦)
15	14	adantl 277	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑦 +R 0R) = 𝑦)
16	13, 15	opeq12d 3829	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → ⟨(𝑥 +R 0R), (𝑦 +R 0R)⟩ = ⟨𝑥, 𝑦⟩)
17	7, 11, 16	3eqtr3d 2247	. 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (⟨𝑥, 𝑦⟩ + 0) = ⟨𝑥, 𝑦⟩)
18	1, 4, 17	optocl 4755	1 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177  ⟨cop 3637  (class class class)co 5951  Rcnr 7417  0_Rc0r 7418   +_R cplr 7421  ℂcc 7930  0cc0 7932   + caddc 7935

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-eprel 4340  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-irdg 6463  df-1o 6509  df-2o 6510  df-oadd 6513  df-omul 6514  df-er 6627  df-ec 6629  df-qs 6633  df-ni 7424  df-pli 7425  df-mi 7426  df-lti 7427  df-plpq 7464  df-mpq 7465  df-enq 7467  df-nqqs 7468  df-plqqs 7469  df-mqqs 7470  df-1nqqs 7471  df-rq 7472  df-ltnqqs 7473  df-enq0 7544  df-nq0 7545  df-0nq0 7546  df-plq0 7547  df-mq0 7548  df-inp 7586  df-i1p 7587  df-iplp 7588  df-enr 7846  df-nr 7847  df-plr 7848  df-0r 7851  df-c 7938  df-0 7939  df-add 7943

This theorem is referenced by: (None)