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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2 8930 | . . 3 | |
2 | 1 | oveq1i 5861 | . 2 |
3 | 1p1times 8046 | . 2 | |
4 | 2, 3 | eqtrid 2215 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1348 wcel 2141 (class class class)co 5851 cc 7765 c1 7768 caddc 7770 cmul 7772 c2 8922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-resscn 7859 ax-1cn 7860 ax-icn 7862 ax-addcl 7863 ax-mulcl 7865 ax-mulcom 7868 ax-mulass 7870 ax-distr 7871 ax-1rid 7874 ax-cnre 7878 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-iota 5158 df-fv 5204 df-ov 5854 df-2 8930 |
This theorem is referenced by: times2 9000 2timesi 9001 2halves 9100 halfaddsub 9105 avglt2 9110 2timesd 9113 expubnd 10526 subsq2 10576 sinmul 11700 sin2t 11705 cos2t 11706 pythagtriplem4 12215 pythagtriplem14 12224 pythagtriplem16 12226 |
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