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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 9007 |
. . 3
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2 | 1 | oveq1i 5905 |
. 2
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3 | 1p1times 8120 |
. 2
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4 | 2, 3 | eqtrid 2234 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 7932 ax-1cn 7933 ax-icn 7935 ax-addcl 7936 ax-mulcl 7938 ax-mulcom 7941 ax-mulass 7943 ax-distr 7944 ax-1rid 7947 ax-cnre 7951 |
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 5898 df-2 9007 |
This theorem is referenced by: times2 9077 2timesi 9078 2halves 9177 halfaddsub 9182 avglt2 9187 2timesd 9190 expubnd 10607 subsq2 10658 sinmul 11783 sin2t 11788 cos2t 11789 pythagtriplem4 12299 pythagtriplem14 12308 pythagtriplem16 12310 |
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