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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 9049 | . 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 1353 ∈ wcel 2148 (class class class)co 5877 ℂcc 7811 + caddc 7816 · cmul 7818 2c2 8972 |
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 7905 ax-1cn 7906 ax-icn 7908 ax-addcl 7909 ax-mulcl 7911 ax-mulcom 7914 ax-mulass 7916 ax-distr 7917 ax-1rid 7920 ax-cnre 7924 |
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 2741 df-un 3135 df-in 3137 df-ss 3144 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-iota 5180 df-fv 5226 df-ov 5880 df-2 8980 |
This theorem is referenced by: xleaddadd 9889 fzctr 10135 flhalf 10304 q2submod 10387 modaddmodup 10389 m1expeven 10569 binom2 10634 nn0opthlem2d 10703 crre 10868 imval2 10905 resqrexlemdec 11022 amgm2 11129 maxabsle 11215 maxabslemab 11217 maxltsup 11229 max0addsup 11230 arisum2 11509 efival 11742 sinadd 11746 cosadd 11747 addsin 11752 subsin 11753 cosmul 11755 addcos 11756 subcos 11757 sin2t 11759 cos2t 11760 eirraplem 11786 pythagtriplem12 12277 pythagtriplem15 12280 pythagtriplem17 12282 difsqpwdvds 12339 bl2in 13988 cosordlem 14355 apdifflemf 14879 apdifflemr 14880 |
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