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Mirrors > Home > MPE Home > Th. List > nrgdomn | Structured version Visualization version GIF version |
Description: A nonzero normed ring is a domain. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
nrgdomn | ⊢ (𝑅 ∈ NrmRing → (𝑅 ∈ Domn ↔ 𝑅 ∈ NzRing)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | domnnzr 19996 | . 2 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) | |
2 | simpr 485 | . . . 4 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) → 𝑅 ∈ NzRing) | |
3 | eqid 2818 | . . . . . . 7 ⊢ (norm‘𝑅) = (norm‘𝑅) | |
4 | eqid 2818 | . . . . . . 7 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅) | |
5 | 3, 4 | nrgabv 23197 | . . . . . 6 ⊢ (𝑅 ∈ NrmRing → (norm‘𝑅) ∈ (AbsVal‘𝑅)) |
6 | 5 | ne0d 4298 | . . . . 5 ⊢ (𝑅 ∈ NrmRing → (AbsVal‘𝑅) ≠ ∅) |
7 | 6 | adantr 481 | . . . 4 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) → (AbsVal‘𝑅) ≠ ∅) |
8 | 4 | abvn0b 20003 | . . . 4 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (AbsVal‘𝑅) ≠ ∅)) |
9 | 2, 7, 8 | sylanbrc 583 | . . 3 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) → 𝑅 ∈ Domn) |
10 | 9 | ex 413 | . 2 ⊢ (𝑅 ∈ NrmRing → (𝑅 ∈ NzRing → 𝑅 ∈ Domn)) |
11 | 1, 10 | impbid2 227 | 1 ⊢ (𝑅 ∈ NrmRing → (𝑅 ∈ Domn ↔ 𝑅 ∈ NzRing)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 ≠ wne 3013 ∅c0 4288 ‘cfv 6348 AbsValcabv 19516 NzRingcnzr 19958 Domncdomn 19981 normcnm 23113 NrmRingcnrg 23116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-ico 12732 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-mgp 19169 df-ring 19228 df-abv 19517 df-nzr 19959 df-domn 19985 df-nrg 23122 |
This theorem is referenced by: (None) |
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