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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 12324 | . 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 1540 ∈ wcel 2109 (class class class)co 7389 ℂcc 11072 + caddc 11077 · cmul 11079 2c2 12242 |
| 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 2008 ax-8 2111 ax-9 2119 ax-ext 2702 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-mulcl 11136 ax-mulcom 11138 ax-mulass 11140 ax-distr 11141 ax-1rid 11144 ax-cnre 11147 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2066 df-clab 2709 df-cleq 2722 df-clel 2804 df-rex 3055 df-rab 3409 df-v 3452 df-dif 3919 df-un 3921 df-ss 3933 df-nul 4299 df-if 4491 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-br 5110 df-iota 6466 df-fv 6521 df-ov 7392 df-2 12250 |
| This theorem is referenced by: div4p1lem1div2 12443 climcndslem1 15821 climcndslem2 15822 sadcaddlem 16433 dvexp3 25888 chordthmlem 26748 chordthmlem2 26749 chordthmlem4 26751 logfaclbnd 27139 rplogsumlem1 27401 nexple 32775 aks4d1p1p5 42058 fltne 42625 |
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