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Mirrors > Home > ILE Home > Th. List > 2timesd | GIF version |
Description: Two times a number. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
2timesd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
2timesd | ⊢ (𝜑 → (2 · 𝐴) = (𝐴 + 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2timesd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | 2times 8841 | . 2 ⊢ (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (2 · 𝐴) = (𝐴 + 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 (class class class)co 5767 ℂcc 7611 + caddc 7616 · cmul 7618 2c2 8764 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-resscn 7705 ax-1cn 7706 ax-icn 7708 ax-addcl 7709 ax-mulcl 7711 ax-mulcom 7714 ax-mulass 7716 ax-distr 7717 ax-1rid 7720 ax-cnre 7724 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-iota 5083 df-fv 5126 df-ov 5770 df-2 8772 |
This theorem is referenced by: xleaddadd 9663 fzctr 9903 flhalf 10068 q2submod 10151 modaddmodup 10153 m1expeven 10333 binom2 10396 nn0opthlem2d 10460 crre 10622 imval2 10659 resqrexlemdec 10776 amgm2 10883 maxabsle 10969 maxabslemab 10971 maxltsup 10983 max0addsup 10984 arisum2 11261 efival 11428 sinadd 11432 cosadd 11433 addsin 11438 subsin 11439 cosmul 11441 addcos 11442 subcos 11443 sin2t 11445 cos2t 11446 eirraplem 11472 bl2in 12561 cosordlem 12919 |
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