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Mirrors > Home > MPE Home > Th. List > unitrinv | Structured version Visualization version GIF version |
Description: A unit times its inverse is the identity. (Contributed by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
unitinvcl.1 | ⊢ 𝑈 = (Unit‘𝑅) |
unitinvcl.2 | ⊢ 𝐼 = (invr‘𝑅) |
unitinvcl.3 | ⊢ · = (.r‘𝑅) |
unitinvcl.4 | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
unitrinv | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑋 · (𝐼‘𝑋)) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unitinvcl.1 | . . . 4 ⊢ 𝑈 = (Unit‘𝑅) | |
2 | eqid 2777 | . . . 4 ⊢ ((mulGrp‘𝑅) ↾s 𝑈) = ((mulGrp‘𝑅) ↾s 𝑈) | |
3 | 1, 2 | unitgrp 19054 | . . 3 ⊢ (𝑅 ∈ Ring → ((mulGrp‘𝑅) ↾s 𝑈) ∈ Grp) |
4 | 1, 2 | unitgrpbas 19053 | . . . 4 ⊢ 𝑈 = (Base‘((mulGrp‘𝑅) ↾s 𝑈)) |
5 | 1 | fvexi 6460 | . . . . 5 ⊢ 𝑈 ∈ V |
6 | eqid 2777 | . . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
7 | unitinvcl.3 | . . . . . . 7 ⊢ · = (.r‘𝑅) | |
8 | 6, 7 | mgpplusg 18880 | . . . . . 6 ⊢ · = (+g‘(mulGrp‘𝑅)) |
9 | 2, 8 | ressplusg 16385 | . . . . 5 ⊢ (𝑈 ∈ V → · = (+g‘((mulGrp‘𝑅) ↾s 𝑈))) |
10 | 5, 9 | ax-mp 5 | . . . 4 ⊢ · = (+g‘((mulGrp‘𝑅) ↾s 𝑈)) |
11 | eqid 2777 | . . . 4 ⊢ (0g‘((mulGrp‘𝑅) ↾s 𝑈)) = (0g‘((mulGrp‘𝑅) ↾s 𝑈)) | |
12 | unitinvcl.2 | . . . . 5 ⊢ 𝐼 = (invr‘𝑅) | |
13 | 1, 2, 12 | invrfval 19060 | . . . 4 ⊢ 𝐼 = (invg‘((mulGrp‘𝑅) ↾s 𝑈)) |
14 | 4, 10, 11, 13 | grprinv 17856 | . . 3 ⊢ ((((mulGrp‘𝑅) ↾s 𝑈) ∈ Grp ∧ 𝑋 ∈ 𝑈) → (𝑋 · (𝐼‘𝑋)) = (0g‘((mulGrp‘𝑅) ↾s 𝑈))) |
15 | 3, 14 | sylan 575 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑋 · (𝐼‘𝑋)) = (0g‘((mulGrp‘𝑅) ↾s 𝑈))) |
16 | unitinvcl.4 | . . . 4 ⊢ 1 = (1r‘𝑅) | |
17 | 1, 2, 16 | unitgrpid 19056 | . . 3 ⊢ (𝑅 ∈ Ring → 1 = (0g‘((mulGrp‘𝑅) ↾s 𝑈))) |
18 | 17 | adantr 474 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 1 = (0g‘((mulGrp‘𝑅) ↾s 𝑈))) |
19 | 15, 18 | eqtr4d 2816 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑋 · (𝐼‘𝑋)) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 Vcvv 3397 ‘cfv 6135 (class class class)co 6922 ↾s cress 16256 +gcplusg 16338 .rcmulr 16339 0gc0g 16486 Grpcgrp 17809 mulGrpcmgp 18876 1rcur 18888 Ringcrg 18934 Unitcui 19026 invrcinvr 19058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-tpos 7634 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-minusg 17813 df-mgp 18877 df-ur 18889 df-ring 18936 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 |
This theorem is referenced by: 1rinv 19066 0unit 19067 dvrid 19075 drnginvrr 19159 subrguss 19187 subrginv 19188 subrgunit 19190 matunit 20890 slesolinvbi 20893 nminvr 22881 nrginvrcnlem 22903 ply1divalg 24334 dchrn0 25427 dvrcan5 30355 invginvrid 43145 |
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