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Mirrors > Home > MPE Home > Th. List > 2times | Structured version Visualization version 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 | โข (๐ด โ โ โ (2 ยท ๐ด) = (๐ด + ๐ด)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2 12275 | . . 3 โข 2 = (1 + 1) | |
2 | 1 | oveq1i 7419 | . 2 โข (2 ยท ๐ด) = ((1 + 1) ยท ๐ด) |
3 | 1p1times 11385 | . 2 โข (๐ด โ โ โ ((1 + 1) ยท ๐ด) = (๐ด + ๐ด)) | |
4 | 2, 3 | eqtrid 2785 | 1 โข (๐ด โ โ โ (2 ยท ๐ด) = (๐ด + ๐ด)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1542 โ wcel 2107 (class class class)co 7409 โcc 11108 1c1 11111 + caddc 11113 ยท cmul 11115 2c2 12267 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-mulcl 11172 ax-mulcom 11174 ax-mulass 11176 ax-distr 11177 ax-1rid 11180 ax-cnre 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-iota 6496 df-fv 6552 df-ov 7412 df-2 12275 |
This theorem is referenced by: times2 12349 2timesi 12350 2txmxeqx 12352 2halves 12440 halfaddsub 12445 avglt2 12451 2timesd 12455 expubnd 14142 absmax 15276 sinmul 16115 sin2t 16120 cos2t 16121 sadadd2lem2 16391 pythagtriplem4 16752 pythagtriplem14 16761 pythagtriplem16 16763 2sqreultlem 26950 2sqreunnltlem 26953 cncph 30072 pellexlem2 41568 acongrep 41719 sub2times 43982 2timesgt 43998 |
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