Theorem 0cnALT 10874

Description: Alternate proof of 0cn 10633 which does not reference ax-1cn 10595. (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 10596	. . 3 ⊢ i ∈ ℂ
2		cnre 10638	. . 3 ⊢ (i ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ i = (𝑥 + (i · 𝑦)))
3		ax-rnegex 10608	. . . . . 6 ⊢ (𝑥 ∈ ℝ → ∃𝑧 ∈ ℝ (𝑥 + 𝑧) = 0)
4		readdcl 10620	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑥 + 𝑧) ∈ ℝ)
5		eleq1 2900	. . . . . . . 8 ⊢ ((𝑥 + 𝑧) = 0 → ((𝑥 + 𝑧) ∈ ℝ ↔ 0 ∈ ℝ))
6	4, 5	syl5ibcom 247	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑥 + 𝑧) = 0 → 0 ∈ ℝ))
7	6	rexlimdva 3284	. . . . . 6 ⊢ (𝑥 ∈ ℝ → (∃𝑧 ∈ ℝ (𝑥 + 𝑧) = 0 → 0 ∈ ℝ))
8	3, 7	mpd 15	. . . . 5 ⊢ (𝑥 ∈ ℝ → 0 ∈ ℝ)
9	8	adantr 483	. . . 4 ⊢ ((𝑥 ∈ ℝ ∧ ∃𝑦 ∈ ℝ i = (𝑥 + (i · 𝑦))) → 0 ∈ ℝ)
10	9	rexlimiva 3281	. . 3 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ i = (𝑥 + (i · 𝑦)) → 0 ∈ ℝ)
11	1, 2, 10	mp2b 10	. 2 ⊢ 0 ∈ ℝ
12	11	recni 10655	1 ⊢ 0 ∈ ℂ

Syntax hints:   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∃wrex 3139  (class class class)co 7156  ℂcc 10535  ℝcr 10536  0cc0 10537  ici 10539   + caddc 10540   · cmul 10542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-resscn 10594  ax-icn 10596  ax-addrcl 10598  ax-rnegex 10608  ax-cnre 10610

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-ral 3143  df-rex 3144  df-in 3943  df-ss 3952

This theorem is referenced by: (None)