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Mirrors > Home > ILE Home > Th. List > 2timesi | GIF version |
Description: Two times a number. (Contributed by NM, 1-Aug-1999.) |
Ref | Expression |
---|---|
2times.1 | ⊢ 𝐴 ∈ ℂ |
Ref | Expression |
---|---|
2timesi | ⊢ (2 · 𝐴) = (𝐴 + 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2times.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | 2times 9045 | . 2 ⊢ (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (2 · 𝐴) = (𝐴 + 𝐴) |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 (class class class)co 5874 ℂcc 7808 + caddc 7813 · cmul 7815 2c2 8968 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-resscn 7902 ax-1cn 7903 ax-icn 7905 ax-addcl 7906 ax-mulcl 7908 ax-mulcom 7911 ax-mulass 7913 ax-distr 7914 ax-1rid 7917 ax-cnre 7921 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-iota 5178 df-fv 5224 df-ov 5877 df-2 8976 |
This theorem is referenced by: 2t2e4 9071 nn0le2xi 9224 binom2i 10625 sinq34lt0t 14145 tangtx 14152 ex-dvds 14364 |
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