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Mirrors > Home > MPE Home > Th. List > 2times | Structured version Visualization version 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 10956 | . . 3 ⊢ 2 = (1 + 1) | |
2 | 1 | oveq1i 6559 | . 2 ⊢ (2 · 𝐴) = ((1 + 1) · 𝐴) |
3 | 1p1times 10086 | . 2 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴)) | |
4 | 2, 3 | syl5eq 2656 | 1 ⊢ (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 (class class class)co 6549 ℂcc 9813 1c1 9816 + caddc 9818 · cmul 9820 2c2 10947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-mulcl 9877 ax-mulcom 9879 ax-mulass 9881 ax-distr 9882 ax-1rid 9885 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-2 10956 |
This theorem is referenced by: times2 11023 2timesi 11024 2txmxeqx 11026 2halves 11137 halfaddsub 11142 avglt2 11148 2timesd 11152 expubnd 12783 subsq2 12835 absmax 13917 sinmul 14741 sin2t 14746 cos2t 14747 sadadd2lem2 15010 pythagtriplem4 15362 pythagtriplem14 15371 pythagtriplem16 15373 cncph 27058 pellexlem2 36412 acongrep 36565 sub2times 38426 2timesgt 38441 |
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