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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 11773 | . 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 2110 (class class class)co 7155 ℂcc 10534 + caddc 10539 · cmul 10541 2c2 11691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-mulcl 10598 ax-mulcom 10600 ax-mulass 10602 ax-distr 10603 ax-1rid 10606 ax-cnre 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-iota 6313 df-fv 6362 df-ov 7158 df-2 11699 |
This theorem is referenced by: div4p1lem1div2 11891 climcndslem1 15203 climcndslem2 15204 sadcaddlem 15805 dvexp3 24574 chordthmlem 25409 chordthmlem2 25410 chordthmlem4 25412 logfaclbnd 25797 rplogsumlem1 26059 nexple 31268 fltne 39270 |
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