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Mirrors > Home > MPE Home > Th. List > zsubrg | Structured version Visualization version GIF version |
Description: The integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.) |
Ref | Expression |
---|---|
zsubrg | ⊢ ℤ ∈ (SubRing‘ℂfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 11671 | . 2 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
2 | zaddcl 11707 | . 2 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 + 𝑦) ∈ ℤ) | |
3 | znegcl 11702 | . 2 ⊢ (𝑥 ∈ ℤ → -𝑥 ∈ ℤ) | |
4 | 1z 11697 | . 2 ⊢ 1 ∈ ℤ | |
5 | zmulcl 11716 | . 2 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 · 𝑦) ∈ ℤ) | |
6 | 1, 2, 3, 4, 5 | cnsubrglem 20118 | 1 ⊢ ℤ ∈ (SubRing‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2157 ‘cfv 6101 ℤcz 11666 SubRingcsubrg 19094 ℂfldccnfld 20068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-addf 10303 ax-mulf 10304 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-z 11667 df-dec 11784 df-uz 11931 df-fz 12581 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-ress 16192 df-plusg 16280 df-mulr 16281 df-starv 16282 df-tset 16286 df-ple 16287 df-ds 16289 df-unif 16290 df-0g 16417 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-grp 17741 df-minusg 17742 df-subg 17904 df-cmn 18510 df-mgp 18806 df-ur 18818 df-ring 18865 df-cring 18866 df-subrg 19096 df-cnfld 20069 |
This theorem is referenced by: zringcrng 20142 zring1 20151 zringlpirlem1 20154 zringunit 20158 zringcyg 20161 prmirred 20165 zndvds 20219 zrhpsgnmhm 20251 rzgrp 20292 zringnrg 22919 zlmclm 23239 zclmncvs 23275 wilthlem2 25147 wilthlem3 25148 lgsqrlem1 25423 lgseisenlem4 25455 dchrisum0flblem1 25549 zringsubgval 42982 |
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