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Mirrors > Home > MPE Home > Th. List > 7cn | Structured version Visualization version GIF version |
Description: The number 7 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.) |
Ref | Expression |
---|---|
7cn | ⊢ 7 ∈ ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-7 12276 | . 2 ⊢ 7 = (6 + 1) | |
2 | 6cn 12299 | . . 3 ⊢ 6 ∈ ℂ | |
3 | ax-1cn 11164 | . . 3 ⊢ 1 ∈ ℂ | |
4 | 2, 3 | addcli 11216 | . 2 ⊢ (6 + 1) ∈ ℂ |
5 | 1, 4 | eqeltri 2829 | 1 ⊢ 7 ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 (class class class)co 7405 ℂcc 11104 1c1 11107 + caddc 11109 6c6 12267 7c7 12268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-1cn 11164 ax-addcl 11166 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-cleq 2724 df-clel 2810 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 |
This theorem is referenced by: 8cn 12305 8m1e7 12341 7p2e9 12369 7p3e10 12748 7t2e14 12782 7t4e28 12784 7t7e49 12787 cos2bnd 16127 23prm 17048 139prm 17053 163prm 17054 317prm 17055 631prm 17056 1259lem1 17060 1259lem2 17061 1259lem3 17062 1259lem4 17063 1259lem5 17064 1259prm 17065 2503lem1 17066 2503lem2 17067 2503lem3 17068 4001lem1 17070 4001lem4 17073 4001prm 17074 log2ublem3 26442 log2ub 26443 bclbnd 26772 bposlem8 26783 2lgslem3d 26891 ex-prmo 29701 hgt750lem 33651 hgt750lem2 33652 60lcm7e420 40863 3exp7 40906 3lexlogpow5ineq1 40907 aks4d1p1 40929 235t711 41200 ex-decpmul 41201 3cubeslem3r 41410 fmtno5lem4 46210 257prm 46215 fmtno4nprmfac193 46228 fmtno5fac 46236 m3prm 46246 139prmALT 46250 127prm 46253 m7prm 46254 2exp340mod341 46387 8exp8mod9 46390 |
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