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Mirrors > Home > ILE Home > Th. List > 2times | GIF version |
Description: Two times a number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.) (Proof shortened by AV, 26-Feb-2020.) |
Ref | Expression |
---|---|
2times | ⊢ (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2 8779 | . . 3 ⊢ 2 = (1 + 1) | |
2 | 1 | oveq1i 5784 | . 2 ⊢ (2 · 𝐴) = ((1 + 1) · 𝐴) |
3 | 1p1times 7896 | . 2 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴)) | |
4 | 2, 3 | syl5eq 2184 | 1 ⊢ (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 (class class class)co 5774 ℂcc 7618 1c1 7621 + caddc 7623 · cmul 7625 2c2 8771 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-resscn 7712 ax-1cn 7713 ax-icn 7715 ax-addcl 7716 ax-mulcl 7718 ax-mulcom 7721 ax-mulass 7723 ax-distr 7724 ax-1rid 7727 ax-cnre 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-iota 5088 df-fv 5131 df-ov 5777 df-2 8779 |
This theorem is referenced by: times2 8849 2timesi 8850 2halves 8949 halfaddsub 8954 avglt2 8959 2timesd 8962 expubnd 10350 subsq2 10400 sinmul 11451 sin2t 11456 cos2t 11457 |
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