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| Mirrors > Home > MPE Home > Th. List > 9cn | Structured version Visualization version GIF version | ||
| Description: The number 9 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 |
|---|---|
| 9cn | ⊢ 9 ∈ ℂ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-9 12336 | . 2 ⊢ 9 = (8 + 1) | |
| 2 | 8cn 12363 | . . 3 ⊢ 8 ∈ ℂ | |
| 3 | ax-1cn 11213 | . . 3 ⊢ 1 ∈ ℂ | |
| 4 | 2, 3 | addcli 11267 | . 2 ⊢ (8 + 1) ∈ ℂ |
| 5 | 1, 4 | eqeltri 2837 | 1 ⊢ 9 ∈ ℂ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 1c1 11156 + caddc 11158 8c8 12327 9c9 12328 |
| 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 df-9 12336 |
| This theorem is referenced by: 10m1e9 12829 9t2e18 12855 9t8e72 12861 9t9e81 12862 9t11e99 12863 0.999... 15917 cos2bnd 16224 3dvds 16368 3dvdsdec 16369 3dvds2dec 16370 2exp8 17126 139prm 17161 163prm 17162 317prm 17163 631prm 17164 1259lem1 17168 1259lem2 17169 1259lem3 17170 1259lem4 17171 1259lem5 17172 2503lem1 17174 2503lem2 17175 2503lem3 17176 2503prm 17177 4001lem1 17178 4001lem2 17179 4001lem3 17180 4001lem4 17181 sqrt2cxp2logb9e3 26842 mcubic 26890 cubic2 26891 cubic 26892 quartlem1 26900 log2tlbnd 26988 log2ublem3 26991 log2ub 26992 bposlem8 27335 ex-lcm 30477 9p10ne21 30489 1mhdrd 32898 hgt750lem2 34667 60gcd7e1 42006 3lexlogpow5ineq1 42055 3lexlogpow2ineq2 42060 3lexlogpow5ineq5 42061 sq9 42332 sum9cubes 42682 fmtno5lem4 47543 257prm 47548 fmtno4nprmfac193 47561 139prmALT 47583 127prm 47586 8exp8mod9 47723 nfermltl8rev 47729 evengpop3 47785 |
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