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Mirrors > Home > MPE Home > Th. List > drnginvrcl | Structured version Visualization version GIF version |
Description: Closure of the multiplicative inverse in a division ring. (reccl 11650 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 2738 | . . . 4 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
3 | invrcl.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
4 | 1, 2, 3 | drngunit 20006 | . . 3 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ (Unit‘𝑅) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ))) |
5 | drngring 20008 | . . . 4 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
6 | invrcl.i | . . . . . 6 ⊢ 𝐼 = (invr‘𝑅) | |
7 | 2, 6, 1 | ringinvcl 19928 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Unit‘𝑅)) → (𝐼‘𝑋) ∈ 𝐵) |
8 | 7 | ex 413 | . . . 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 259 | . 2 ⊢ (𝑅 ∈ DivRing → ((𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ∈ 𝐵)) |
11 | 10 | 3impib 1115 | 1 ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ‘cfv 6426 Basecbs 16922 0gc0g 17160 Ringcrg 19793 Unitcui 19891 invrcinvr 19923 DivRingcdr 20001 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-2nd 7821 df-tpos 8029 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-er 8485 df-en 8721 df-dom 8722 df-sdom 8723 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-2 12046 df-3 12047 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ress 16952 df-plusg 16985 df-mulr 16986 df-0g 17162 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-grp 18590 df-minusg 18591 df-mgp 19731 df-ur 19748 df-ring 19795 df-oppr 19872 df-dvdsr 19893 df-unit 19894 df-invr 19924 df-drng 20003 |
This theorem is referenced by: drngmul0or 20022 sdrgacs 20079 cntzsdrg 20080 abvrec 20106 abvdiv 20107 lvecvs0or 20380 lssvs0or 20382 lvecinv 20385 lspsnvs 20386 lspfixed 20400 lspexch 20401 lspsolv 20415 drngnidl 20510 matunitlindflem1 35781 lfl1 37092 eqlkr3 37123 lkrlsp 37124 tendoinvcl 39126 dochkr1 39500 dochkr1OLDN 39501 lcfl7lem 39521 lclkrlem2m 39541 lclkrlem2o 39543 lclkrlem2p 39544 lcfrlem1 39564 lcfrlem2 39565 lcfrlem3 39566 lcfrlem29 39593 lcfrlem31 39595 lcfrlem33 39597 mapdpglem17N 39710 mapdpglem18 39711 mapdpglem19 39712 mapdpglem21 39714 mapdpglem22 39715 hdmapip1 39938 hgmapvvlem1 39945 hgmapvvlem2 39946 hgmapvvlem3 39947 drnginvrcld 40256 prjspersym 40454 prjspnfv01 40469 prjspner01 40470 |
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