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Mirrors > Home > ILE Home > Th. List > 2times | GIF version |
Description: Two times a number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.) (Proof shortened by AV, 26-Feb-2020.) |
Ref | Expression |
---|---|
2times | ⊢ (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2 9009 | . . 3 ⊢ 2 = (1 + 1) | |
2 | 1 | oveq1i 5907 | . 2 ⊢ (2 · 𝐴) = ((1 + 1) · 𝐴) |
3 | 1p1times 8122 | . 2 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴)) | |
4 | 2, 3 | eqtrid 2234 | 1 ⊢ (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 (class class class)co 5897 ℂcc 7840 1c1 7843 + caddc 7845 · cmul 7847 2c2 9001 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-resscn 7934 ax-1cn 7935 ax-icn 7937 ax-addcl 7938 ax-mulcl 7940 ax-mulcom 7943 ax-mulass 7945 ax-distr 7946 ax-1rid 7949 ax-cnre 7953 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-iota 5196 df-fv 5243 df-ov 5900 df-2 9009 |
This theorem is referenced by: times2 9079 2timesi 9080 2halves 9179 halfaddsub 9184 avglt2 9189 2timesd 9192 expubnd 10611 subsq2 10662 sinmul 11787 sin2t 11792 cos2t 11793 pythagtriplem4 12303 pythagtriplem14 12312 pythagtriplem16 12314 |
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