Theorem 0cnALT 11494

Description: Alternate proof of 0cn 11251 which does not reference ax-1cn 11211. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 7-Jan-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
0cnALT	⊢ 0 ∈ ℂ

Proof of Theorem 0cnALT

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

Step	Hyp	Ref	Expression
1		ax-icn 11212	. . 3 ⊢ i ∈ ℂ
2		cnre 11256	. . 3 ⊢ (i ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ i = (𝑥 + (i · 𝑦)))
3		ax-rnegex 11224	. . . . . 6 ⊢ (𝑥 ∈ ℝ → ∃𝑧 ∈ ℝ (𝑥 + 𝑧) = 0)
4		readdcl 11236	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑥 + 𝑧) ∈ ℝ)
5		eleq1 2827	. . . . . . . 8 ⊢ ((𝑥 + 𝑧) = 0 → ((𝑥 + 𝑧) ∈ ℝ ↔ 0 ∈ ℝ))
6	4, 5	syl5ibcom 245	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑥 + 𝑧) = 0 → 0 ∈ ℝ))
7	6	rexlimdva 3153	. . . . . 6 ⊢ (𝑥 ∈ ℝ → (∃𝑧 ∈ ℝ (𝑥 + 𝑧) = 0 → 0 ∈ ℝ))
8	3, 7	mpd 15	. . . . 5 ⊢ (𝑥 ∈ ℝ → 0 ∈ ℝ)
9	8	adantr 480	. . . 4 ⊢ ((𝑥 ∈ ℝ ∧ ∃𝑦 ∈ ℝ i = (𝑥 + (i · 𝑦))) → 0 ∈ ℝ)
10	9	rexlimiva 3145	. . 3 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ i = (𝑥 + (i · 𝑦)) → 0 ∈ ℝ)
11	1, 2, 10	mp2b 10	. 2 ⊢ 0 ∈ ℝ
12	11	recni 11273	1 ⊢ 0 ∈ ℂ

Syntax hints:   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∃wrex 3068  (class class class)co 7431  ℂcc 11151  ℝcr 11152  0cc0 11153  ici 11155   + caddc 11156   · cmul 11158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-resscn 11210  ax-icn 11212  ax-addrcl 11214  ax-rnegex 11224  ax-cnre 11226

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727  df-clel 2814  df-rex 3069  df-ss 3980

This theorem is referenced by: (None)