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Mirrors > Home > MPE Home > Th. List > unitinvcl | Structured version Visualization version GIF version |
Description: The inverse of a unit exists and is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
unitinvcl.1 | ⊢ 𝑈 = (Unit‘𝑅) |
unitinvcl.2 | ⊢ 𝐼 = (invr‘𝑅) |
Ref | Expression |
---|---|
unitinvcl | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unitinvcl.1 | . . 3 ⊢ 𝑈 = (Unit‘𝑅) | |
2 | eqid 2759 | . . 3 ⊢ ((mulGrp‘𝑅) ↾s 𝑈) = ((mulGrp‘𝑅) ↾s 𝑈) | |
3 | 1, 2 | unitgrp 19503 | . 2 ⊢ (𝑅 ∈ Ring → ((mulGrp‘𝑅) ↾s 𝑈) ∈ Grp) |
4 | 1, 2 | unitgrpbas 19502 | . . 3 ⊢ 𝑈 = (Base‘((mulGrp‘𝑅) ↾s 𝑈)) |
5 | unitinvcl.2 | . . . 4 ⊢ 𝐼 = (invr‘𝑅) | |
6 | 1, 2, 5 | invrfval 19509 | . . 3 ⊢ 𝐼 = (invg‘((mulGrp‘𝑅) ↾s 𝑈)) |
7 | 4, 6 | grpinvcl 18233 | . 2 ⊢ ((((mulGrp‘𝑅) ↾s 𝑈) ∈ Grp ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) ∈ 𝑈) |
8 | 3, 7 | sylan 583 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1539 ∈ wcel 2112 ‘cfv 6341 (class class class)co 7157 ↾s cress 16557 Grpcgrp 18184 mulGrpcmgp 19322 Ringcrg 19380 Unitcui 19475 invrcinvr 19507 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5161 ax-sep 5174 ax-nul 5181 ax-pow 5239 ax-pr 5303 ax-un 7466 ax-cnex 10645 ax-resscn 10646 ax-1cn 10647 ax-icn 10648 ax-addcl 10649 ax-addrcl 10650 ax-mulcl 10651 ax-mulrcl 10652 ax-mulcom 10653 ax-addass 10654 ax-mulass 10655 ax-distr 10656 ax-i2m1 10657 ax-1ne0 10658 ax-1rid 10659 ax-rnegex 10660 ax-rrecex 10661 ax-cnre 10662 ax-pre-lttri 10663 ax-pre-lttrn 10664 ax-pre-ltadd 10665 ax-pre-mulgt0 10666 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2902 df-ne 2953 df-nel 3057 df-ral 3076 df-rex 3077 df-reu 3078 df-rmo 3079 df-rab 3080 df-v 3412 df-sbc 3700 df-csb 3809 df-dif 3864 df-un 3866 df-in 3868 df-ss 3878 df-pss 3880 df-nul 4229 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4803 df-iun 4889 df-br 5038 df-opab 5100 df-mpt 5118 df-tr 5144 df-id 5435 df-eprel 5440 df-po 5448 df-so 5449 df-fr 5488 df-we 5490 df-xp 5535 df-rel 5536 df-cnv 5537 df-co 5538 df-dm 5539 df-rn 5540 df-res 5541 df-ima 5542 df-pred 6132 df-ord 6178 df-on 6179 df-lim 6180 df-suc 6181 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7115 df-ov 7160 df-oprab 7161 df-mpo 7162 df-om 7587 df-tpos 7909 df-wrecs 7964 df-recs 8025 df-rdg 8063 df-er 8306 df-en 8542 df-dom 8543 df-sdom 8544 df-pnf 10729 df-mnf 10730 df-xr 10731 df-ltxr 10732 df-le 10733 df-sub 10924 df-neg 10925 df-nn 11689 df-2 11751 df-3 11752 df-ndx 16559 df-slot 16560 df-base 16562 df-sets 16563 df-ress 16564 df-plusg 16651 df-mulr 16652 df-0g 16788 df-mgm 17933 df-sgrp 17982 df-mnd 17993 df-grp 18187 df-minusg 18188 df-mgp 19323 df-ur 19335 df-ring 19382 df-oppr 19459 df-dvdsr 19477 df-unit 19478 df-invr 19508 |
This theorem is referenced by: ringinvcl 19512 unitdvcl 19523 drnginvrn0 19603 subrgugrp 19637 gzrngunit 20247 uc1pmon1p 24866 ig1peu 24886 sum2dchr 25972 dvrdir 31027 rdivmuldivd 31028 dvrcan5 31030 rhmunitinv 31061 |
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