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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 12326 | . . 3 ⊢ 2 = (1 + 1) | |
2 | 1 | oveq1i 7440 | . 2 ⊢ (2 · 𝐴) = ((1 + 1) · 𝐴) |
3 | 1p1times 11429 | . 2 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴)) | |
4 | 2, 3 | eqtrid 2786 | 1 ⊢ (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 (class class class)co 7430 ℂcc 11150 1c1 11153 + caddc 11155 · cmul 11157 2c2 12318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-mulcl 11214 ax-mulcom 11216 ax-mulass 11218 ax-distr 11219 ax-1rid 11222 ax-cnre 11225 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-iota 6515 df-fv 6570 df-ov 7433 df-2 12326 |
This theorem is referenced by: times2 12400 2timesi 12401 2txmxeqx 12403 2halves 12491 halfaddsub 12496 avglt2 12502 2timesd 12506 expubnd 14213 absmax 15364 sinmul 16204 sin2t 16209 cos2t 16210 sadadd2lem2 16483 pythagtriplem4 16852 pythagtriplem14 16861 pythagtriplem16 16863 2sqreultlem 27505 2sqreunnltlem 27508 cncph 30847 pellexlem2 42817 acongrep 42968 sub2times 45222 2timesgt 45238 gpg3nbgrvtxlem 47957 |
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