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Mirrors > Home > MPE Home > Th. List > times2d | Structured version Visualization version GIF version |
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 12387 | . 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 1533 ∈ wcel 2098 (class class class)co 7419 ℂcc 11143 + caddc 11148 · cmul 11150 2c2 12305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-mulcl 11207 ax-mulcom 11209 ax-mulass 11211 ax-distr 11212 ax-1rid 11215 ax-cnre 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-iota 6501 df-fv 6557 df-ov 7422 df-2 12313 |
This theorem is referenced by: div4p1lem1div2 12505 climcndslem1 15836 climcndslem2 15837 sadcaddlem 16440 dvexp3 25959 chordthmlem 26814 chordthmlem2 26815 chordthmlem4 26817 logfaclbnd 27205 rplogsumlem1 27467 nexple 33761 aks4d1p1p5 41680 fltne 42205 |
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