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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 12335 | . 2 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
2 | zaddcl 12371 | . 2 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 + 𝑦) ∈ ℤ) | |
3 | znegcl 12366 | . 2 ⊢ (𝑥 ∈ ℤ → -𝑥 ∈ ℤ) | |
4 | 1z 12361 | . 2 ⊢ 1 ∈ ℤ | |
5 | zmulcl 12380 | . 2 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 · 𝑦) ∈ ℤ) | |
6 | 1, 2, 3, 4, 5 | cnsubrglem 20659 | 1 ⊢ ℤ ∈ (SubRing‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 ‘cfv 6432 ℤcz 12330 SubRingcsubrg 20031 ℂfldccnfld 20608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 ax-addf 10961 ax-mulf 10962 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-om 7708 df-1st 7825 df-2nd 7826 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-er 8490 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-nn 11985 df-2 12047 df-3 12048 df-4 12049 df-5 12050 df-6 12051 df-7 12052 df-8 12053 df-9 12054 df-n0 12245 df-z 12331 df-dec 12449 df-uz 12594 df-fz 13251 df-struct 16859 df-sets 16876 df-slot 16894 df-ndx 16906 df-base 16924 df-ress 16953 df-plusg 16986 df-mulr 16987 df-starv 16988 df-tset 16992 df-ple 16993 df-ds 16995 df-unif 16996 df-0g 17163 df-mgm 18337 df-sgrp 18386 df-mnd 18397 df-grp 18591 df-minusg 18592 df-subg 18763 df-cmn 19399 df-mgp 19732 df-ur 19749 df-ring 19796 df-cring 19797 df-subrg 20033 df-cnfld 20609 |
This theorem is referenced by: zringcrng 20683 zring1 20692 zringlpirlem1 20695 zringunit 20699 zringcyg 20702 zringsubgval 20703 prmirred 20707 zndvds 20768 zrhpsgnmhm 20800 rzgrp 20839 zringnrg 23962 zlmclm 24286 zclmncvs 24323 wilthlem2 26229 wilthlem3 26230 lgsqrlem1 26505 lgseisenlem4 26537 dchrisum0flblem1 26667 |
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