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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 6977 | . . . 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 7052 | . 2 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = ((1 · 𝐴) + (1 · 𝐴))) |
5 | mulid2 7025 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
6 | 5, 5 | oveq12d 5530 | . 2 ⊢ (𝐴 ∈ ℂ → ((1 · 𝐴) + (1 · 𝐴)) = (𝐴 + 𝐴)) |
7 | 4, 6 | eqtrd 2072 | 1 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∈ wcel 1393 (class class class)co 5512 ℂcc 6887 1c1 6890 + caddc 6892 · cmul 6894 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-resscn 6976 ax-1cn 6977 ax-icn 6979 ax-addcl 6980 ax-mulcl 6982 ax-mulcom 6985 ax-mulass 6987 ax-distr 6988 ax-1rid 6991 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515 |
This theorem is referenced by: eqneg 7708 2times 8038 |
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