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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 12223 | . . 3 โข 2 = (1 + 1) | |
2 | 1 | oveq1i 7372 | . 2 โข (2 ยท ๐ด) = ((1 + 1) ยท ๐ด) |
3 | 1p1times 11333 | . 2 โข (๐ด โ โ โ ((1 + 1) ยท ๐ด) = (๐ด + ๐ด)) | |
4 | 2, 3 | eqtrid 2789 | 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 7362 โcc 11056 1c1 11059 + caddc 11061 ยท cmul 11063 2c2 12215 |
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 2708 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-mulcl 11120 ax-mulcom 11122 ax-mulass 11124 ax-distr 11125 ax-1rid 11128 ax-cnre 11131 |
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 2715 df-cleq 2729 df-clel 2815 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-iota 6453 df-fv 6509 df-ov 7365 df-2 12223 |
This theorem is referenced by: times2 12297 2timesi 12298 2txmxeqx 12300 2halves 12388 halfaddsub 12393 avglt2 12399 2timesd 12403 expubnd 14089 absmax 15221 sinmul 16061 sin2t 16066 cos2t 16067 sadadd2lem2 16337 pythagtriplem4 16698 pythagtriplem14 16707 pythagtriplem16 16709 2sqreultlem 26811 2sqreunnltlem 26814 cncph 29803 pellexlem2 41182 acongrep 41333 sub2times 43580 2timesgt 43596 |
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