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| Description: Alternate proof of 0cn 11253 which does not reference ax-1cn 11213. (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.) | 
| Ref | Expression | 
|---|---|
| 0cnALT | ⊢ 0 ∈ ℂ | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ax-icn 11214 | . . 3 ⊢ i ∈ ℂ | |
| 2 | cnre 11258 | . . 3 ⊢ (i ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ i = (𝑥 + (i · 𝑦))) | |
| 3 | ax-rnegex 11226 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → ∃𝑧 ∈ ℝ (𝑥 + 𝑧) = 0) | |
| 4 | readdcl 11238 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑥 + 𝑧) ∈ ℝ) | |
| 5 | eleq1 2829 | . . . . . . . 8 ⊢ ((𝑥 + 𝑧) = 0 → ((𝑥 + 𝑧) ∈ ℝ ↔ 0 ∈ ℝ)) | |
| 6 | 4, 5 | syl5ibcom 245 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑥 + 𝑧) = 0 → 0 ∈ ℝ)) | 
| 7 | 6 | rexlimdva 3155 | . . . . . 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 3147 | . . 3 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ i = (𝑥 + (i · 𝑦)) → 0 ∈ ℝ) | 
| 11 | 1, 2, 10 | mp2b 10 | . 2 ⊢ 0 ∈ ℝ | 
| 12 | 11 | recni 11275 | 1 ⊢ 0 ∈ ℂ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 ici 11157 + caddc 11158 · cmul 11160 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-resscn 11212 ax-icn 11214 ax-addrcl 11216 ax-rnegex 11226 ax-cnre 11228 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-clel 2816 df-rex 3071 df-ss 3968 | 
| This theorem is referenced by: (None) | 
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