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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 12334 | . 2 ⊢ 7 = (6 + 1) | |
| 2 | 6cn 12357 | . . 3 ⊢ 6 ∈ ℂ | |
| 3 | ax-1cn 11213 | . . 3 ⊢ 1 ∈ ℂ | |
| 4 | 2, 3 | addcli 11267 | . 2 ⊢ (6 + 1) ∈ ℂ |
| 5 | 1, 4 | eqeltri 2837 | 1 ⊢ 7 ∈ ℂ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 1c1 11156 + caddc 11158 6c6 12325 7c7 12326 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-1cn 11213 ax-addcl 11215 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-clel 2816 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 |
| This theorem is referenced by: 8cn 12363 8m1e7 12399 7p2e9 12427 7p3e10 12808 7t2e14 12842 7t4e28 12844 7t7e49 12847 cos2bnd 16224 23prm 17156 139prm 17161 163prm 17162 317prm 17163 631prm 17164 1259lem1 17168 1259lem2 17169 1259lem3 17170 1259lem4 17171 1259lem5 17172 1259prm 17173 2503lem1 17174 2503lem2 17175 2503lem3 17176 4001lem1 17178 4001lem4 17181 4001prm 17182 log2ublem3 26991 log2ub 26992 bclbnd 27324 bposlem8 27335 2lgslem3d 27443 ex-prmo 30478 hgt750lem 34666 hgt750lem2 34667 60lcm7e420 42011 3exp7 42054 3lexlogpow5ineq1 42055 aks4d1p1 42077 sq7 42330 235t711 42339 ex-decpmul 42340 3cubeslem3r 42698 fmtno5lem4 47543 257prm 47548 fmtno4nprmfac193 47561 fmtno5fac 47569 m3prm 47579 139prmALT 47583 127prm 47586 m7prm 47587 2exp340mod341 47720 8exp8mod9 47723 |
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