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| Mirrors > Home > MPE Home > Th. List > 8cn | Structured version Visualization version GIF version | ||
| Description: The number 8 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 |
|---|---|
| 8cn | ⊢ 8 ∈ ℂ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-8 12335 | . 2 ⊢ 8 = (7 + 1) | |
| 2 | 7cn 12360 | . . 3 ⊢ 7 ∈ ℂ | |
| 3 | ax-1cn 11213 | . . 3 ⊢ 1 ∈ ℂ | |
| 4 | 2, 3 | addcli 11267 | . 2 ⊢ (7 + 1) ∈ ℂ |
| 5 | 1, 4 | eqeltri 2837 | 1 ⊢ 8 ∈ ℂ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 1c1 11156 + caddc 11158 7c7 12326 8c8 12327 |
| 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 df-8 12335 |
| This theorem is referenced by: 9cn 12366 9m1e8 12400 8p2e10 12813 8t2e16 12848 8t5e40 12851 cos2bnd 16224 2exp11 17127 2exp16 17128 139prm 17161 163prm 17162 317prm 17163 631prm 17164 1259lem2 17169 1259lem3 17170 1259lem4 17171 1259lem5 17172 2503lem2 17175 2503lem3 17176 2503prm 17177 4001lem1 17178 4001lem2 17179 4001prm 17182 quart1cl 26897 quart1lem 26898 quart1 26899 quartlem1 26900 log2tlbnd 26988 log2ublem3 26991 log2ub 26992 bposlem8 27335 lgsdir2lem1 27369 lgsdir2lem5 27373 2lgslem3a 27440 2lgslem3b 27441 2lgslem3c 27442 2lgslem3d 27443 2lgslem3a1 27444 2lgslem3b1 27445 2lgslem3c1 27446 2lgslem3d1 27447 2lgsoddprmlem1 27452 2lgsoddprmlem2 27453 2lgsoddprmlem3a 27454 2lgsoddprmlem3b 27455 2lgsoddprmlem3c 27456 2lgsoddprmlem3d 27457 ex-exp 30469 hgt750lem2 34667 420lcm8e840 42012 3exp7 42054 3lexlogpow5ineq1 42055 3lexlogpow5ineq5 42061 aks4d1p1 42077 sq8 42331 ex-decpmul 42340 resqrtvalex 43658 imsqrtvalex 43659 fmtno5lem4 47543 257prm 47548 fmtnoprmfac2lem1 47553 fmtno4prmfac 47559 fmtno4nprmfac193 47561 fmtno5faclem3 47568 m3prm 47579 139prmALT 47583 127prm 47586 m7prm 47587 5tcu2e40 47602 2exp340mod341 47720 8exp8mod9 47723 nfermltl8rev 47729 evengpop3 47785 tgoldbachlt 47803 |
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