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Mirrors > Home > MPE Home > Th. List > drngdomn | Structured version Visualization version GIF version |
Description: A division ring is a domain. (Contributed by Mario Carneiro, 29-Mar-2015.) |
Ref | Expression |
---|---|
drngdomn | ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Domn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drngnzr 20728 | . 2 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ NzRing) | |
2 | eqid 2726 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | eqid 2726 | . . . . 5 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
4 | eqid 2726 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
5 | 2, 3, 4 | isdrng 20713 | . . . 4 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = ((Base‘𝑅) ∖ {(0g‘𝑅)}))) |
6 | 5 | simprbi 495 | . . 3 ⊢ (𝑅 ∈ DivRing → (Unit‘𝑅) = ((Base‘𝑅) ∖ {(0g‘𝑅)})) |
7 | drngring 20716 | . . . 4 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
8 | eqid 2726 | . . . . 5 ⊢ (RLReg‘𝑅) = (RLReg‘𝑅) | |
9 | 8, 3 | unitrrg 20683 | . . . 4 ⊢ (𝑅 ∈ Ring → (Unit‘𝑅) ⊆ (RLReg‘𝑅)) |
10 | 7, 9 | syl 17 | . . 3 ⊢ (𝑅 ∈ DivRing → (Unit‘𝑅) ⊆ (RLReg‘𝑅)) |
11 | 6, 10 | eqsstrrd 4019 | . 2 ⊢ (𝑅 ∈ DivRing → ((Base‘𝑅) ∖ {(0g‘𝑅)}) ⊆ (RLReg‘𝑅)) |
12 | 2, 8, 4 | isdomn2 20691 | . 2 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ((Base‘𝑅) ∖ {(0g‘𝑅)}) ⊆ (RLReg‘𝑅))) |
13 | 1, 11, 12 | sylanbrc 581 | 1 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Domn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∖ cdif 3944 ⊆ wss 3947 {csn 4633 ‘cfv 6556 Basecbs 17215 0gc0g 17456 Ringcrg 20218 Unitcui 20339 NzRingcnzr 20496 RLRegcrlreg 20671 Domncdomn 20672 DivRingcdr 20709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-2nd 8006 df-tpos 8243 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-er 8736 df-en 8977 df-dom 8978 df-sdom 8979 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-nn 12267 df-2 12329 df-3 12330 df-sets 17168 df-slot 17186 df-ndx 17198 df-base 17216 df-ress 17245 df-plusg 17281 df-mulr 17282 df-0g 17458 df-mgm 18635 df-sgrp 18714 df-mnd 18730 df-grp 18933 df-minusg 18934 df-cmn 19782 df-abl 19783 df-mgp 20120 df-rng 20138 df-ur 20167 df-ring 20220 df-oppr 20318 df-dvdsr 20341 df-unit 20342 df-invr 20372 df-nzr 20497 df-rlreg 20674 df-domn 20675 df-drng 20711 |
This theorem is referenced by: drngmcl 20730 drngmul0or 20740 fldidom 20751 fldidomOLD 20752 fidomndrng 20754 abvtriv 20815 ply1unit 33449 ply1dg1rt 33453 aks6d1c5lem3 41837 drngmullcan 42195 drngmulrcan 42196 |
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