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| Mirrors > Home > MPE Home > Th. List > 5cn | Structured version Visualization version GIF version | ||
| Description: The number 5 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 | 
|---|---|
| 5cn | ⊢ 5 ∈ ℂ | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-5 12333 | . 2 ⊢ 5 = (4 + 1) | |
| 2 | 4cn 12352 | . . 3 ⊢ 4 ∈ ℂ | |
| 3 | ax-1cn 11214 | . . 3 ⊢ 1 ∈ ℂ | |
| 4 | 2, 3 | addcli 11268 | . 2 ⊢ (4 + 1) ∈ ℂ | 
| 5 | 1, 4 | eqeltri 2836 | 1 ⊢ 5 ∈ ℂ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2107 (class class class)co 7432 ℂcc 11154 1c1 11157 + caddc 11159 4c4 12324 5c5 12325 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-1cn 11214 ax-addcl 11216 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-cleq 2728 df-clel 2815 df-2 12330 df-3 12331 df-4 12332 df-5 12333 | 
| This theorem is referenced by: 6cn 12358 6m1e5 12398 5p2e7 12423 5p3e8 12424 5p4e9 12425 5p5e10 12806 5t2e10 12835 5recm6rec 12878 bpoly4 16096 ef01bndlem 16221 5ndvds3 16451 5ndvds6 16452 dec5dvds 17103 dec5nprm 17105 2exp11 17128 2exp16 17129 prmlem1 17146 17prm 17155 139prm 17162 163prm 17163 317prm 17164 631prm 17165 1259lem1 17169 1259lem2 17170 1259lem3 17171 1259lem4 17172 2503lem1 17175 2503lem2 17176 2503lem3 17177 4001lem1 17179 4001lem2 17180 4001lem3 17181 4001lem4 17182 4001prm 17183 log2ublem3 26992 log2ub 26993 ppiub 27249 bclbnd 27325 bposlem4 27332 bposlem5 27333 bposlem6 27334 bposlem8 27336 bposlem9 27337 lgsdir2lem1 27370 2lgslem3c 27443 2lgsoddprmlem3d 27458 ex-fac 30471 fib6 34409 hgt750lem2 34668 12lcm5e60 42010 lcmineqlem23 42053 3lexlogpow5ineq1 42056 3lexlogpow5ineq5 42062 aks4d1p1p4 42073 aks4d1p1p6 42075 aks4d1p1p7 42076 sqn5i 42325 4t5e20 42331 sq5 42333 235t711 42344 ex-decpmul 42345 inductionexd 44173 ceil5half3 47347 fmtno5lem1 47545 fmtno5lem2 47546 257prm 47553 fmtno4prmfac193 47565 fmtno4nprmfac193 47566 flsqrt5 47586 139prmALT 47588 127prm 47591 5tcu2e40 47607 41prothprmlem2 47610 41prothprm 47611 2exp340mod341 47725 gbpart8 47760 gpg5order 48019 linevalexample 48317 ackval3012 48618 5m4e1 49371 | 
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