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Mirrors > Home > MPE Home > Th. List > 2timesi | Structured version Visualization version GIF version |
Description: Two times a number. (Contributed by NM, 1-Aug-1999.) |
Ref | Expression |
---|---|
2timesi.1 | ⊢ 𝐴 ∈ ℂ |
Ref | Expression |
---|---|
2timesi | ⊢ (2 · 𝐴) = (𝐴 + 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2timesi.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | 2times 11145 | . 2 ⊢ (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (2 · 𝐴) = (𝐴 + 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 (class class class)co 6650 ℂcc 9934 + caddc 9939 · cmul 9941 2c2 11070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-mulcl 9998 ax-mulcom 10000 ax-mulass 10002 ax-distr 10003 ax-1rid 10006 ax-cnre 10009 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-2 11079 |
This theorem is referenced by: 2t2e4 11177 nn0le2xi 11347 binom2i 12974 rddif 14080 abs3lemi 14149 iseraltlem2 14413 prmreclem6 15625 mod2xi 15773 numexp2x 15783 prmlem2 15827 iihalf2 22732 pcoass 22824 ovolunlem1a 23264 tangtx 24257 sinq34lt0t 24261 eff1o 24295 ang180lem2 24540 dvatan 24662 basellem2 24808 basellem5 24811 chtub 24937 bposlem9 25017 ex-dvds 27313 norm3lem 28006 normpari 28011 polid2i 28014 ballotth 30599 heiborlem6 33615 rmspecsqrtnqOLD 37471 dirkertrigeqlem1 40315 fourierdlem94 40417 fourierdlem102 40425 fourierdlem111 40434 fourierdlem112 40435 fourierdlem113 40436 fourierdlem114 40437 sqwvfoura 40445 sqwvfourb 40446 fouriersw 40448 fmtnorec3 41460 2t6m3t4e0 42126 zlmodzxzequa 42285 |
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