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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 12222 | . 2 ⊢ 7 = (6 + 1) | |
2 | 6cn 12245 | . . 3 ⊢ 6 ∈ ℂ | |
3 | ax-1cn 11110 | . . 3 ⊢ 1 ∈ ℂ | |
4 | 2, 3 | addcli 11162 | . 2 ⊢ (6 + 1) ∈ ℂ |
5 | 1, 4 | eqeltri 2834 | 1 ⊢ 7 ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 (class class class)co 7358 ℂcc 11050 1c1 11053 + caddc 11055 6c6 12213 7c7 12214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-1cn 11110 ax-addcl 11112 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-cleq 2729 df-clel 2815 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 |
This theorem is referenced by: 8cn 12251 8m1e7 12287 7p2e9 12315 7p3e10 12694 7t2e14 12728 7t4e28 12730 7t7e49 12733 cos2bnd 16071 23prm 16992 139prm 16997 163prm 16998 317prm 16999 631prm 17000 1259lem1 17004 1259lem2 17005 1259lem3 17006 1259lem4 17007 1259lem5 17008 1259prm 17009 2503lem1 17010 2503lem2 17011 2503lem3 17012 4001lem1 17014 4001lem4 17017 4001prm 17018 log2ublem3 26301 log2ub 26302 bclbnd 26631 bposlem8 26642 2lgslem3d 26750 ex-prmo 29406 hgt750lem 33267 hgt750lem2 33268 60lcm7e420 40470 3exp7 40513 3lexlogpow5ineq1 40514 aks4d1p1 40536 235t711 40808 ex-decpmul 40809 3cubeslem3r 41013 fmtno5lem4 45755 257prm 45760 fmtno4nprmfac193 45773 fmtno5fac 45781 m3prm 45791 139prmALT 45795 127prm 45798 m7prm 45799 2exp340mod341 45932 8exp8mod9 45935 |
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