Theorem 0cnALT 11444

Description: Alternate proof of 0cn 11197 which does not reference ax-1cn 11157. (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 11158	. . 3 ⊢ i ∈ ℂ
2		cnre 11204	. . 3 ⊢ (i ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ i = (𝑥 + (i · 𝑦)))
3		ax-rnegex 11170	. . . . . 6 ⊢ (𝑥 ∈ ℝ → ∃𝑧 ∈ ℝ (𝑥 + 𝑧) = 0)
4		readdcl 11182	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑥 + 𝑧) ∈ ℝ)
5		eleq1 2857	. . . . . . . 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 3172	. . . . . 6 ⊢ (𝑥 ∈ ℝ → (∃𝑧 ∈ ℝ (𝑥 + 𝑧) = 0 → 0 ∈ ℝ))
8	3, 7	mpd 16	. . . . 5 ⊢ (𝑥 ∈ ℝ → 0 ∈ ℝ)
9	8	adantr 485	. . . 4 ⊢ ((𝑥 ∈ ℝ ∧ ∃𝑦 ∈ ℝ i = (𝑥 + (i · 𝑦))) → 0 ∈ ℝ)
10	9	rexlimiva 3164	. . 3 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ i = (𝑥 + (i · 𝑦)) → 0 ∈ ℝ)
11	1, 2, 10	mp2b 10	. 2 ⊢ 0 ∈ ℝ
12	11	recni 11222	1 ⊢ 0 ∈ ℂ

Syntax hints:   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  (class class class)co 7411  ℂcc 11097  ℝcr 11098  0cc0 11099  ici 11101   + caddc 11102   · cmul 11104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-resscn 11156  ax-icn 11158  ax-addrcl 11160  ax-rnegex 11170  ax-cnre 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-cleq 2761  df-clel 2844  df-rex 3096  df-ss 3930

This theorem is referenced by: (None)