Theorem 0cnALT 11473

Description: Alternate proof of 0cn 11226 which does not reference ax-1cn 11186. (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 11187	. . 3 ⊢ i ∈ ℂ
2		cnre 11233	. . 3 ⊢ (i ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ i = (𝑥 + (i · 𝑦)))
3		ax-rnegex 11199	. . . . . 6 ⊢ (𝑥 ∈ ℝ → ∃𝑧 ∈ ℝ (𝑥 + 𝑧) = 0)
4		readdcl 11211	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑥 + 𝑧) ∈ ℝ)
5		eleq1 2850	. . . . . . . 8 ⊢ ((𝑥 + 𝑧) = 0 → ((𝑥 + 𝑧) ∈ ℝ ↔ 0 ∈ ℝ))
6	4, 5	syl5ibcom 248	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑥 + 𝑧) = 0 → 0 ∈ ℝ))
7	6	rexlimdva 3165	. . . . . 6 ⊢ (𝑥 ∈ ℝ → (∃𝑧 ∈ ℝ (𝑥 + 𝑧) = 0 → 0 ∈ ℝ))
8	3, 7	mpd 16	. . . . 5 ⊢ (𝑥 ∈ ℝ → 0 ∈ ℝ)
9	8	adantr 486	. . . 4 ⊢ ((𝑥 ∈ ℝ ∧ ∃𝑦 ∈ ℝ i = (𝑥 + (i · 𝑦))) → 0 ∈ ℝ)
10	9	rexlimiva 3157	. . 3 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ i = (𝑥 + (i · 𝑦)) → 0 ∈ ℝ)
11	1, 2, 10	mp2b 10	. 2 ⊢ 0 ∈ ℝ
12	11	recni 11251	1 ⊢ 0 ∈ ℂ

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3088  (class class class)co 7417  ℂcc 11126  ℝcr 11127  0cc0 11128  ici 11130   + caddc 11131   · cmul 11133

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-resscn 11185  ax-icn 11187  ax-addrcl 11189  ax-rnegex 11199  ax-cnre 11201

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2754  df-clel 2837  df-rex 3089  df-ss 3919

This theorem is used by: (None)