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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 12308 | . . 3 ⊢ 2 = (1 + 1) | |
2 | 1 | oveq1i 7429 | . 2 ⊢ (2 · 𝐴) = ((1 + 1) · 𝐴) |
3 | 1p1times 11417 | . 2 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴)) | |
4 | 2, 3 | eqtrid 2777 | 1 ⊢ (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 (class class class)co 7419 ℂcc 11138 1c1 11141 + caddc 11143 · cmul 11145 2c2 12300 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-mulcl 11202 ax-mulcom 11204 ax-mulass 11206 ax-distr 11207 ax-1rid 11210 ax-cnre 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-iota 6501 df-fv 6557 df-ov 7422 df-2 12308 |
This theorem is referenced by: times2 12382 2timesi 12383 2txmxeqx 12385 2halves 12473 halfaddsub 12478 avglt2 12484 2timesd 12488 expubnd 14177 absmax 15312 sinmul 16152 sin2t 16157 cos2t 16158 sadadd2lem2 16428 pythagtriplem4 16791 pythagtriplem14 16800 pythagtriplem16 16802 2sqreultlem 27425 2sqreunnltlem 27428 cncph 30701 pellexlem2 42392 acongrep 42543 sub2times 44792 2timesgt 44808 |
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