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Mirrors > Home > MPE Home > Th. List > unitlinv | 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 |
---|---|
unitlinv | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → ((𝐼‘𝑋) · 𝑋) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unitinvcl.1 | . . . 4 ⊢ 𝑈 = (Unit‘𝑅) | |
2 | eqid 2651 | . . . 4 ⊢ ((mulGrp‘𝑅) ↾s 𝑈) = ((mulGrp‘𝑅) ↾s 𝑈) | |
3 | 1, 2 | unitgrp 18713 | . . 3 ⊢ (𝑅 ∈ Ring → ((mulGrp‘𝑅) ↾s 𝑈) ∈ Grp) |
4 | 1, 2 | unitgrpbas 18712 | . . . 4 ⊢ 𝑈 = (Base‘((mulGrp‘𝑅) ↾s 𝑈)) |
5 | fvex 6239 | . . . . . 6 ⊢ (Unit‘𝑅) ∈ V | |
6 | 1, 5 | eqeltri 2726 | . . . . 5 ⊢ 𝑈 ∈ V |
7 | eqid 2651 | . . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
8 | unitinvcl.3 | . . . . . . 7 ⊢ · = (.r‘𝑅) | |
9 | 7, 8 | mgpplusg 18539 | . . . . . 6 ⊢ · = (+g‘(mulGrp‘𝑅)) |
10 | 2, 9 | ressplusg 16040 | . . . . 5 ⊢ (𝑈 ∈ V → · = (+g‘((mulGrp‘𝑅) ↾s 𝑈))) |
11 | 6, 10 | ax-mp 5 | . . . 4 ⊢ · = (+g‘((mulGrp‘𝑅) ↾s 𝑈)) |
12 | eqid 2651 | . . . 4 ⊢ (0g‘((mulGrp‘𝑅) ↾s 𝑈)) = (0g‘((mulGrp‘𝑅) ↾s 𝑈)) | |
13 | unitinvcl.2 | . . . . 5 ⊢ 𝐼 = (invr‘𝑅) | |
14 | 1, 2, 13 | invrfval 18719 | . . . 4 ⊢ 𝐼 = (invg‘((mulGrp‘𝑅) ↾s 𝑈)) |
15 | 4, 11, 12, 14 | grplinv 17515 | . . 3 ⊢ ((((mulGrp‘𝑅) ↾s 𝑈) ∈ Grp ∧ 𝑋 ∈ 𝑈) → ((𝐼‘𝑋) · 𝑋) = (0g‘((mulGrp‘𝑅) ↾s 𝑈))) |
16 | 3, 15 | sylan 487 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → ((𝐼‘𝑋) · 𝑋) = (0g‘((mulGrp‘𝑅) ↾s 𝑈))) |
17 | unitinvcl.4 | . . . 4 ⊢ 1 = (1r‘𝑅) | |
18 | 1, 2, 17 | unitgrpid 18715 | . . 3 ⊢ (𝑅 ∈ Ring → 1 = (0g‘((mulGrp‘𝑅) ↾s 𝑈))) |
19 | 18 | adantr 480 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 1 = (0g‘((mulGrp‘𝑅) ↾s 𝑈))) |
20 | 16, 19 | eqtr4d 2688 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → ((𝐼‘𝑋) · 𝑋) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ‘cfv 5926 (class class class)co 6690 ↾s cress 15905 +gcplusg 15988 .rcmulr 15989 0gc0g 16147 Grpcgrp 17469 mulGrpcmgp 18535 1rcur 18547 Ringcrg 18593 Unitcui 18685 invrcinvr 18717 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-tpos 7397 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-minusg 17473 df-mgp 18536 df-ur 18548 df-ring 18595 df-oppr 18669 df-dvdsr 18687 df-unit 18688 df-invr 18718 |
This theorem is referenced by: dvrcan1 18737 drnginvrl 18814 subrginv 18844 subrgunit 18846 unitrrg 19341 matinv 20531 matunit 20532 slesolinv 20534 nrginvrcnlem 22542 uc1pmon1p 23956 ornglmullt 29935 rhmunitinv 29950 kerunit 29951 lincresunit3lem3 42588 |
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