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Theorem 0cnALT 10863
Description: Alternate proof of 0cn 10622 which does not reference ax-1cn 10584. (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.
StepHypRef Expression
1 ax-icn 10585 . . 3 i ∈ ℂ
2 cnre 10627 . . 3 (i ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ i = (𝑥 + (i · 𝑦)))
3 ax-rnegex 10597 . . . . . 6 (𝑥 ∈ ℝ → ∃𝑧 ∈ ℝ (𝑥 + 𝑧) = 0)
4 readdcl 10609 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑥 + 𝑧) ∈ ℝ)
5 eleq1 2901 . . . . . . . 8 ((𝑥 + 𝑧) = 0 → ((𝑥 + 𝑧) ∈ ℝ ↔ 0 ∈ ℝ))
64, 5syl5ibcom 248 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑥 + 𝑧) = 0 → 0 ∈ ℝ))
76rexlimdva 3270 . . . . . 6 (𝑥 ∈ ℝ → (∃𝑧 ∈ ℝ (𝑥 + 𝑧) = 0 → 0 ∈ ℝ))
83, 7mpd 15 . . . . 5 (𝑥 ∈ ℝ → 0 ∈ ℝ)
98adantr 484 . . . 4 ((𝑥 ∈ ℝ ∧ ∃𝑦 ∈ ℝ i = (𝑥 + (i · 𝑦))) → 0 ∈ ℝ)
109rexlimiva 3267 . . 3 (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ i = (𝑥 + (i · 𝑦)) → 0 ∈ ℝ)
111, 2, 10mp2b 10 . 2 0 ∈ ℝ
1211recni 10644 1 0 ∈ ℂ
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wcel 2114  wrex 3131  (class class class)co 7140  cc 10524  cr 10525  0cc0 10526  ici 10528   + caddc 10529   · cmul 10531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794  ax-resscn 10583  ax-icn 10585  ax-addrcl 10587  ax-rnegex 10597  ax-cnre 10599
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-ral 3135  df-rex 3136  df-v 3471  df-in 3915  df-ss 3925
This theorem is referenced by: (None)
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