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Mirrors > Home > MPE Home > Th. List > zringbas | Structured version Visualization version GIF version |
Description: The integers are the base of the ring of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.) (Revised by AV, 9-Jun-2019.) |
Ref | Expression |
---|---|
zringbas | ⊢ ℤ = (Base‘ℤring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zsscn 12563 | . 2 ⊢ ℤ ⊆ ℂ | |
2 | df-zring 21011 | . . 3 ⊢ ℤring = (ℂfld ↾s ℤ) | |
3 | cnfldbas 20941 | . . 3 ⊢ ℂ = (Base‘ℂfld) | |
4 | 2, 3 | ressbas2 17179 | . 2 ⊢ (ℤ ⊆ ℂ → ℤ = (Base‘ℤring)) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ ℤ = (Base‘ℤring) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ⊆ wss 3948 ‘cfv 6541 ℂcc 11105 ℤcz 12555 Basecbs 17141 ℂfldccnfld 20937 ℤringczring 21010 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-starv 17209 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-cnfld 20938 df-zring 21011 |
This theorem is referenced by: dvdsrzring 21023 zringlpirlem1 21024 zringlpirlem3 21026 zringinvg 21027 zringunit 21028 zringndrg 21030 zringcyg 21031 prmirredlem 21034 prmirred 21036 expghm 21037 mulgghm2 21038 mulgrhm 21039 mulgrhm2 21040 zlmlmod 21068 chrrhm 21075 domnchr 21076 znlidl 21077 znbas 21091 znzrh2 21093 znzrhfo 21095 zndvds 21097 znf1o 21099 zzngim 21100 znfld 21108 znidomb 21109 znunit 21111 znrrg 21113 cygznlem3 21117 frgpcyg 21121 zrhpsgnodpm 21137 zlmassa 21448 dchrzrhmul 26739 lgsqrlem1 26839 lgsqrlem2 26840 lgsqrlem3 26841 lgsdchr 26848 lgseisenlem3 26870 lgseisenlem4 26871 dchrisum0flblem1 27001 fermltlchr 32467 znfermltl 32468 elrspunidl 32535 ply1fermltlchr 32651 mdetpmtr1 32792 mdetpmtr12 32794 mdetlap 32801 nmmulg 32937 cnzh 32939 rezh 32940 zrhf1ker 32944 zrhunitpreima 32947 elzrhunit 32948 qqhval2lem 32950 qqhf 32955 qqhghm 32957 qqhrhm 32958 qqhnm 32959 mzpmfp 41471 2zlidl 46786 zlmodzxzel 46985 zlmodzxzscm 46987 linevalexample 47030 zlmodzxzldeplem3 47137 zlmodzxzldep 47139 ldepsnlinclem1 47140 ldepsnlinclem2 47141 |
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