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Theorem 0cnALT 10612
 Description: Alternate proof of 0cn 10370 which does not reference ax-1cn 10332. (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 10333 . . 3 i ∈ ℂ
2 cnre 10375 . . 3 (i ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ i = (𝑥 + (i · 𝑦)))
3 ax-rnegex 10345 . . . . . 6 (𝑥 ∈ ℝ → ∃𝑧 ∈ ℝ (𝑥 + 𝑧) = 0)
4 readdcl 10357 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑥 + 𝑧) ∈ ℝ)
5 eleq1 2847 . . . . . . . 8 ((𝑥 + 𝑧) = 0 → ((𝑥 + 𝑧) ∈ ℝ ↔ 0 ∈ ℝ))
64, 5syl5ibcom 237 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑥 + 𝑧) = 0 → 0 ∈ ℝ))
76rexlimdva 3213 . . . . . 6 (𝑥 ∈ ℝ → (∃𝑧 ∈ ℝ (𝑥 + 𝑧) = 0 → 0 ∈ ℝ))
83, 7mpd 15 . . . . 5 (𝑥 ∈ ℝ → 0 ∈ ℝ)
98adantr 474 . . . 4 ((𝑥 ∈ ℝ ∧ ∃𝑦 ∈ ℝ i = (𝑥 + (i · 𝑦))) → 0 ∈ ℝ)
109rexlimiva 3210 . . 3 (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ i = (𝑥 + (i · 𝑦)) → 0 ∈ ℝ)
111, 2, 10mp2b 10 . 2 0 ∈ ℝ
1211recni 10393 1 0 ∈ ℂ
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∃wrex 3091  (class class class)co 6924  ℂcc 10272  ℝcr 10273  0cc0 10274  ici 10276   + caddc 10277   · cmul 10279 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754  ax-resscn 10331  ax-icn 10333  ax-addrcl 10335  ax-rnegex 10345  ax-cnre 10347 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-ral 3095  df-rex 3096  df-in 3799  df-ss 3806 This theorem is referenced by: (None)
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