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| Mirrors > Home > ILE Home > Th. List > zringbas | Unicode 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 |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-zring 14324 |
. . . 4
| |
| 2 | 1 | a1i 9 |
. . 3
|
| 3 | cnfldbas 14293 |
. . . 4
| |
| 4 | 3 | a1i 9 |
. . 3
|
| 5 | cnfldex 14292 |
. . . 4
| |
| 6 | 5 | a1i 9 |
. . 3
|
| 7 | zsscn 9379 |
. . . 4
| |
| 8 | 7 | a1i 9 |
. . 3
|
| 9 | 2, 4, 6, 8 | ressbas2d 12871 |
. 2
|
| 10 | 9 | mptru 1381 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-cnex 8015 ax-resscn 8016 ax-1cn 8017 ax-1re 8018 ax-icn 8019 ax-addcl 8020 ax-addrcl 8021 ax-mulcl 8022 ax-mulrcl 8023 ax-addcom 8024 ax-mulcom 8025 ax-addass 8026 ax-mulass 8027 ax-distr 8028 ax-i2m1 8029 ax-0lt1 8030 ax-1rid 8031 ax-0id 8032 ax-rnegex 8033 ax-precex 8034 ax-cnre 8035 ax-pre-ltirr 8036 ax-pre-ltwlin 8037 ax-pre-lttrn 8038 ax-pre-apti 8039 ax-pre-ltadd 8040 ax-pre-mulgt0 8041 |
| This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rmo 2491 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-pw 3617 df-sn 3638 df-pr 3639 df-tp 3640 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-id 4339 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-rn 4685 df-res 4686 df-ima 4687 df-iota 5231 df-fun 5272 df-fn 5273 df-f 5274 df-f1 5275 df-fo 5276 df-f1o 5277 df-fv 5278 df-riota 5898 df-ov 5946 df-oprab 5947 df-mpo 5948 df-1st 6225 df-2nd 6226 df-pnf 8108 df-mnf 8109 df-xr 8110 df-ltxr 8111 df-le 8112 df-sub 8244 df-neg 8245 df-reap 8647 df-inn 9036 df-2 9094 df-3 9095 df-4 9096 df-5 9097 df-6 9098 df-7 9099 df-8 9100 df-9 9101 df-n0 9295 df-z 9372 df-dec 9504 df-uz 9648 df-rp 9775 df-fz 10130 df-cj 11124 df-abs 11281 df-struct 12805 df-ndx 12806 df-slot 12807 df-base 12809 df-sets 12810 df-iress 12811 df-plusg 12893 df-mulr 12894 df-starv 12895 df-tset 12899 df-ple 12900 df-ds 12902 df-unif 12903 df-topgen 13063 df-bl 14279 df-mopn 14280 df-fg 14282 df-metu 14283 df-cnfld 14290 df-zring 14324 |
| This theorem is referenced by: dvdsrzring 14336 zringinvg 14337 expghmap 14340 mulgghm2 14341 mulgrhm 14342 mulgrhm2 14343 znlidl 14367 znbas 14377 znzrh2 14379 znzrhfo 14381 zndvds 14382 znf1o 14384 znidom 14390 znidomb 14391 znunit 14392 znrrg 14393 lgseisenlem3 15520 lgseisenlem4 15521 |
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