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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 12277 | . 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 1541 ∈ wcel 2113 (class class class)co 7358 ℂcc 11024 + caddc 11029 · cmul 11031 2c2 12200 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-ext 2708 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-mulcl 11088 ax-mulcom 11090 ax-mulass 11092 ax-distr 11093 ax-1rid 11096 ax-cnre 11099 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-sb 2068 df-clab 2715 df-cleq 2728 df-clel 2811 df-rex 3061 df-rab 3400 df-v 3442 df-dif 3904 df-un 3906 df-ss 3918 df-nul 4286 df-if 4480 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-iota 6448 df-fv 6500 df-ov 7361 df-2 12208 |
| This theorem is referenced by: div4p1lem1div2 12396 climcndslem1 15772 climcndslem2 15773 sadcaddlem 16384 dvexp3 25938 chordthmlem 26798 chordthmlem2 26799 chordthmlem4 26801 logfaclbnd 27189 rplogsumlem1 27451 nexple 32925 aks4d1p1p5 42329 fltne 42887 |
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