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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 8937 | . . 3 ⊢ 2 = (1 + 1) | |
2 | 1 | oveq1i 5863 | . 2 ⊢ (2 · 𝐴) = ((1 + 1) · 𝐴) |
3 | 1p1times 8053 | . 2 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴)) | |
4 | 2, 3 | eqtrid 2215 | 1 ⊢ (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 (class class class)co 5853 ℂcc 7772 1c1 7775 + caddc 7777 · cmul 7779 2c2 8929 |
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 7866 ax-1cn 7867 ax-icn 7869 ax-addcl 7870 ax-mulcl 7872 ax-mulcom 7875 ax-mulass 7877 ax-distr 7878 ax-1rid 7881 ax-cnre 7885 |
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 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-iota 5160 df-fv 5206 df-ov 5856 df-2 8937 |
This theorem is referenced by: times2 9007 2timesi 9008 2halves 9107 halfaddsub 9112 avglt2 9117 2timesd 9120 expubnd 10533 subsq2 10583 sinmul 11707 sin2t 11712 cos2t 11713 pythagtriplem4 12222 pythagtriplem14 12231 pythagtriplem16 12233 |
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