Theorem 0cnALT 11139

Description: Alternate proof of 0cn 10898 which does not reference ax-1cn 10860. (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 10861	. . 3 ⊢ i ∈ ℂ
2		cnre 10903	. . 3 ⊢ (i ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ i = (𝑥 + (i · 𝑦)))
3		ax-rnegex 10873	. . . . . 6 ⊢ (𝑥 ∈ ℝ → ∃𝑧 ∈ ℝ (𝑥 + 𝑧) = 0)
4		readdcl 10885	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑥 + 𝑧) ∈ ℝ)
5		eleq1 2826	. . . . . . . 8 ⊢ ((𝑥 + 𝑧) = 0 → ((𝑥 + 𝑧) ∈ ℝ ↔ 0 ∈ ℝ))
6	4, 5	syl5ibcom 244	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑥 + 𝑧) = 0 → 0 ∈ ℝ))
7	6	rexlimdva 3212	. . . . . 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 3209	. . 3 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ i = (𝑥 + (i · 𝑦)) → 0 ∈ ℝ)
11	1, 2, 10	mp2b 10	. 2 ⊢ 0 ∈ ℝ
12	11	recni 10920	1 ⊢ 0 ∈ ℂ

Syntax hints:   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∃wrex 3064  (class class class)co 7255  ℂcc 10800  ℝcr 10801  0cc0 10802  ici 10804   + caddc 10805   · cmul 10807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-resscn 10859  ax-icn 10861  ax-addrcl 10863  ax-rnegex 10873  ax-cnre 10875

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-v 3424  df-in 3890  df-ss 3900

This theorem is referenced by: (None)