| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 12329 | . . 3 ⊢ 2 = (1 + 1) | |
| 2 | 1 | oveq1i 7441 | . 2 ⊢ (2 · 𝐴) = ((1 + 1) · 𝐴) |
| 3 | 1p1times 11432 | . 2 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴)) | |
| 4 | 2, 3 | eqtrid 2789 | 1 ⊢ (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 1c1 11156 + caddc 11158 · cmul 11160 2c2 12321 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-mulcl 11217 ax-mulcom 11219 ax-mulass 11221 ax-distr 11222 ax-1rid 11225 ax-cnre 11228 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-2 12329 |
| This theorem is referenced by: times2 12403 2timesi 12404 2txmxeqx 12406 2halves 12494 halfaddsub 12499 avglt2 12505 2timesd 12509 expubnd 14217 absmax 15368 sinmul 16208 sin2t 16213 cos2t 16214 sadadd2lem2 16487 pythagtriplem4 16857 pythagtriplem14 16866 pythagtriplem16 16868 2sqreultlem 27491 2sqreunnltlem 27494 cncph 30838 pellexlem2 42841 acongrep 42992 sub2times 45284 2timesgt 45300 gpg3nbgrvtxlem 48023 |
| Copyright terms: Public domain | W3C validator |