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Mirrors > Home > ILE Home > Th. List > times2 | 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 8936 | . . 3 ⊢ 2 ∈ ℂ | |
2 | mulcom 7890 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 2 ∈ ℂ) → (𝐴 · 2) = (2 · 𝐴)) | |
3 | 1, 2 | mpan2 423 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (2 · 𝐴)) |
4 | 2times 8993 | . 2 ⊢ (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴)) | |
5 | 3, 4 | eqtrd 2203 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 (class class class)co 5850 ℂcc 7759 + caddc 7764 · cmul 7766 2c2 8916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-mulcom 7862 ax-mulass 7864 ax-distr 7865 ax-1rid 7868 ax-cnre 7872 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-iota 5158 df-fv 5204 df-ov 5853 df-2 8924 |
This theorem is referenced by: times2i 8996 avglt1 9103 times2d 9108 |
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