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| Mirrors > Home > MPE Home > Th. List > zsscn | Structured version Visualization version GIF version | ||
| Description: The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.) |
| Ref | Expression |
|---|---|
| zsscn | ⊢ ℤ ⊆ ℂ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | zcn 12618 | . 2 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
| 2 | 1 | ssriv 3987 | 1 ⊢ ℤ ⊆ ℂ |
| Colors of variables: wff setvar class |
| Syntax hints: ⊆ wss 3951 ℂcc 11153 ℤcz 12613 |
| 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-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-neg 11495 df-z 12614 |
| This theorem is referenced by: zex 12622 elq 12992 zexpcl 14117 fsumzcl 15771 fprodzcl 15990 zrisefaccl 16056 zfallfaccl 16057 4sqlem11 16993 cygabl 19909 zringbas 21464 zring0 21469 fermltlchr 21544 lmbrf 23268 lmres 23308 sszcld 24839 lmmbrf 25296 iscauf 25314 caucfil 25317 lmclimf 25338 elqaalem3 26363 iaa 26367 aareccl 26368 wilthlem2 27112 wilthlem3 27113 lgsfcl2 27347 2sqlem6 27467 gsumzrsum 33062 znfermltl 33394 zringnm 33957 fsum2dsub 34622 reprsuc 34630 caures 37767 mzpexpmpt 42756 uzmptshftfval 44365 fzsscn 45323 dvnprodlem2 45962 elaa2lem 46248 oddibas 48089 2zrngbas 48158 2zrng0 48160 |
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