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Mirrors > Home > ILE Home > Th. List > 1p1times | GIF version |
Description: Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
1p1times | ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 7854 | . . . 4 ⊢ 1 ∈ ℂ | |
2 | 1 | a1i 9 | . . 3 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ) |
3 | id 19 | . . 3 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
4 | 2, 2, 3 | adddird 7932 | . 2 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = ((1 · 𝐴) + (1 · 𝐴))) |
5 | mulid2 7905 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
6 | 5, 5 | oveq12d 5868 | . 2 ⊢ (𝐴 ∈ ℂ → ((1 · 𝐴) + (1 · 𝐴)) = (𝐴 + 𝐴)) |
7 | 4, 6 | eqtrd 2203 | 1 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 (class class class)co 5850 ℂcc 7759 1c1 7762 + caddc 7764 · cmul 7766 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-resscn 7853 ax-1cn 7854 ax-icn 7856 ax-addcl 7857 ax-mulcl 7859 ax-mulcom 7862 ax-mulass 7864 ax-distr 7865 ax-1rid 7868 ax-cnre 7872 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-iota 5158 df-fv 5204 df-ov 5853 |
This theorem is referenced by: eqneg 8636 2times 8993 |
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