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Mirrors > Home > MPE Home > Th. List > unitinvinv | Structured version Visualization version GIF version |
Description: The inverse of the inverse of a unit is the same element. (Contributed by Mario Carneiro, 4-Dec-2014.) |
Ref | Expression |
---|---|
unitinvcl.1 | ⊢ 𝑈 = (Unit‘𝑅) |
unitinvcl.2 | ⊢ 𝐼 = (invr‘𝑅) |
Ref | Expression |
---|---|
unitinvinv | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝐼‘(𝐼‘𝑋)) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unitinvcl.1 | . . 3 ⊢ 𝑈 = (Unit‘𝑅) | |
2 | eqid 2771 | . . 3 ⊢ ((mulGrp‘𝑅) ↾s 𝑈) = ((mulGrp‘𝑅) ↾s 𝑈) | |
3 | 1, 2 | unitgrp 19152 | . 2 ⊢ (𝑅 ∈ Ring → ((mulGrp‘𝑅) ↾s 𝑈) ∈ Grp) |
4 | 1, 2 | unitgrpbas 19151 | . . 3 ⊢ 𝑈 = (Base‘((mulGrp‘𝑅) ↾s 𝑈)) |
5 | unitinvcl.2 | . . . 4 ⊢ 𝐼 = (invr‘𝑅) | |
6 | 1, 2, 5 | invrfval 19158 | . . 3 ⊢ 𝐼 = (invg‘((mulGrp‘𝑅) ↾s 𝑈)) |
7 | 4, 6 | grpinvinv 17965 | . 2 ⊢ ((((mulGrp‘𝑅) ↾s 𝑈) ∈ Grp ∧ 𝑋 ∈ 𝑈) → (𝐼‘(𝐼‘𝑋)) = 𝑋) |
8 | 3, 7 | sylan 572 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝐼‘(𝐼‘𝑋)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ‘cfv 6185 (class class class)co 6974 ↾s cress 16338 Grpcgrp 17903 mulGrpcmgp 18974 Ringcrg 19032 Unitcui 19124 invrcinvr 19156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-tpos 7693 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-0g 16569 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-grp 17906 df-minusg 17907 df-mgp 18975 df-ur 18987 df-ring 19034 df-oppr 19108 df-dvdsr 19126 df-unit 19127 df-invr 19157 |
This theorem is referenced by: extdg1id 30714 |
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