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| Description: A number times 2. (Contributed by Mario Carneiro, 27-May-2016.) | 
| Ref | Expression | 
|---|---|
| 2timesd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) | 
| Ref | Expression | 
|---|---|
| times2d | ⊢ (𝜑 → (𝐴 · 2) = (𝐴 + 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 2timesd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
| 2 | times2 12404 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴)) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐴 · 2) = (𝐴 + 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 (class class class)co 7432 ℂcc 11154 + caddc 11159 · cmul 11161 2c2 12322 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-mulcl 11218 ax-mulcom 11220 ax-mulass 11222 ax-distr 11223 ax-1rid 11226 ax-cnre 11229 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 df-2 12330 | 
| This theorem is referenced by: div4p1lem1div2 12523 climcndslem1 15886 climcndslem2 15887 sadcaddlem 16495 dvexp3 26017 chordthmlem 26876 chordthmlem2 26877 chordthmlem4 26879 logfaclbnd 27267 rplogsumlem1 27529 nexple 32834 aks4d1p1p5 42077 fltne 42659 | 
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