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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 8803 | . . 3 ⊢ 2 = (1 + 1) | |
2 | 1 | oveq1i 5792 | . 2 ⊢ (2 · 𝐴) = ((1 + 1) · 𝐴) |
3 | 1p1times 7920 | . 2 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴)) | |
4 | 2, 3 | syl5eq 2185 | 1 ⊢ (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 (class class class)co 5782 ℂcc 7642 1c1 7645 + caddc 7647 · cmul 7649 2c2 8795 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-resscn 7736 ax-1cn 7737 ax-icn 7739 ax-addcl 7740 ax-mulcl 7742 ax-mulcom 7745 ax-mulass 7747 ax-distr 7748 ax-1rid 7751 ax-cnre 7755 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-iota 5096 df-fv 5139 df-ov 5785 df-2 8803 |
This theorem is referenced by: times2 8873 2timesi 8874 2halves 8973 halfaddsub 8978 avglt2 8983 2timesd 8986 expubnd 10381 subsq2 10431 sinmul 11487 sin2t 11492 cos2t 11493 |
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