![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > drnginvrcl | Structured version Visualization version GIF version |
Description: Closure of the multiplicative inverse in a division ring. (reccl 11108 analog.) (Contributed by NM, 19-Apr-2014.) |
Ref | Expression |
---|---|
invrcl.b | ⊢ 𝐵 = (Base‘𝑅) |
invrcl.z | ⊢ 0 = (0g‘𝑅) |
invrcl.i | ⊢ 𝐼 = (invr‘𝑅) |
Ref | Expression |
---|---|
drnginvrcl | ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invrcl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
2 | eqid 2778 | . . . 4 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
3 | invrcl.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
4 | 1, 2, 3 | drngunit 19233 | . . 3 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ (Unit‘𝑅) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ))) |
5 | drngring 19235 | . . . 4 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
6 | invrcl.i | . . . . . 6 ⊢ 𝐼 = (invr‘𝑅) | |
7 | 2, 6, 1 | ringinvcl 19152 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Unit‘𝑅)) → (𝐼‘𝑋) ∈ 𝐵) |
8 | 7 | ex 405 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝑋 ∈ (Unit‘𝑅) → (𝐼‘𝑋) ∈ 𝐵)) |
9 | 5, 8 | syl 17 | . . 3 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ (Unit‘𝑅) → (𝐼‘𝑋) ∈ 𝐵)) |
10 | 4, 9 | sylbird 252 | . 2 ⊢ (𝑅 ∈ DivRing → ((𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ∈ 𝐵)) |
11 | 10 | 3impib 1096 | 1 ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ≠ wne 2967 ‘cfv 6190 Basecbs 16342 0gc0g 16572 Ringcrg 19023 Unitcui 19115 invrcinvr 19147 DivRingcdr 19228 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-tpos 7697 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-er 8091 df-en 8309 df-dom 8310 df-sdom 8311 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-nn 11442 df-2 11506 df-3 11507 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-ress 16350 df-plusg 16437 df-mulr 16438 df-0g 16574 df-mgm 17713 df-sgrp 17755 df-mnd 17766 df-grp 17897 df-minusg 17898 df-mgp 18966 df-ur 18978 df-ring 19025 df-oppr 19099 df-dvdsr 19117 df-unit 19118 df-invr 19148 df-drng 19230 |
This theorem is referenced by: drngmul0or 19249 sdrgacs 19305 cntzsdrg 19306 abvrec 19332 abvdiv 19333 lvecvs0or 19605 lssvs0or 19607 lvecinv 19610 lspsnvs 19611 lspfixed 19625 lspexch 19626 lspsolv 19640 drngnidl 19726 matunitlindflem1 34329 lfl1 35651 eqlkr3 35682 lkrlsp 35683 tendoinvcl 37685 dochkr1 38059 dochkr1OLDN 38060 lcfl7lem 38080 lclkrlem2m 38100 lclkrlem2o 38102 lclkrlem2p 38103 lcfrlem1 38123 lcfrlem2 38124 lcfrlem3 38125 lcfrlem29 38152 lcfrlem31 38154 lcfrlem33 38156 mapdpglem17N 38269 mapdpglem18 38270 mapdpglem19 38271 mapdpglem21 38273 mapdpglem22 38274 hdmapip1 38497 hgmapvvlem1 38504 hgmapvvlem2 38505 hgmapvvlem3 38506 prjspersym 38664 |
Copyright terms: Public domain | W3C validator |