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Mirrors > Home > ILE Home > Th. List > 2times | Unicode 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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2 8689 |
. . 3
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2 | 1 | oveq1i 5738 |
. 2
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3 | 1p1times 7819 |
. 2
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4 | 2, 3 | syl5eq 2159 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-resscn 7637 ax-1cn 7638 ax-icn 7640 ax-addcl 7641 ax-mulcl 7643 ax-mulcom 7646 ax-mulass 7648 ax-distr 7649 ax-1rid 7652 ax-cnre 7656 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-iota 5046 df-fv 5089 df-ov 5731 df-2 8689 |
This theorem is referenced by: times2 8753 2timesi 8754 2halves 8853 halfaddsub 8858 avglt2 8863 2timesd 8866 expubnd 10243 subsq2 10293 sinmul 11302 sin2t 11307 cos2t 11308 |
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