![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > times2 | Structured version Visualization version GIF version |
Description: A number times 2. (Contributed by NM, 16-Oct-2007.) |
Ref | Expression |
---|---|
times2 | ⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2cn 11426 | . . 3 ⊢ 2 ∈ ℂ | |
2 | mulcom 10338 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 2 ∈ ℂ) → (𝐴 · 2) = (2 · 𝐴)) | |
3 | 1, 2 | mpan2 682 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (2 · 𝐴)) |
4 | 2times 11494 | . 2 ⊢ (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴)) | |
5 | 3, 4 | eqtrd 2861 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 (class class class)co 6905 ℂcc 10250 + caddc 10255 · cmul 10257 2c2 11406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-mulcl 10314 ax-mulcom 10316 ax-mulass 10318 ax-distr 10319 ax-1rid 10322 ax-cnre 10325 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-iota 6086 df-fv 6131 df-ov 6908 df-2 11414 |
This theorem is referenced by: times2i 11497 avglt1 11596 times2d 11602 |
Copyright terms: Public domain | W3C validator |