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Theorem unitrinv 19835

Description: A unit times its inverse is the identity. (Contributed by Mario Carneiro, 2-Dec-2014.)

**Hypotheses**
Ref	Expression
unitinvcl.1	⊢ 𝑈 = (Unit‘𝑅)
unitinvcl.2	⊢ 𝐼 = (inv_r‘𝑅)
unitinvcl.3	⊢ · = (._r‘𝑅)
unitinvcl.4	⊢ 1 = (1_r‘𝑅)

**Assertion**
Ref	Expression
unitrinv	⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑋 · (𝐼‘𝑋)) = 1 )

**Proof of Theorem unitrinv**
Step	Hyp	Ref	Expression
1		unitinvcl.1	. . . 4 ⊢ 𝑈 = (Unit‘𝑅)
2		eqid 2738	. . . 4 ⊢ ((mulGrp‘𝑅) ↾_s 𝑈) = ((mulGrp‘𝑅) ↾_s 𝑈)
3	1, 2	unitgrp 19824	. . 3 ⊢ (𝑅 ∈ Ring → ((mulGrp‘𝑅) ↾_s 𝑈) ∈ Grp)
4	1, 2	unitgrpbas 19823	. . . 4 ⊢ 𝑈 = (Base‘((mulGrp‘𝑅) ↾_s 𝑈))
5	1	fvexi 6770	. . . . 5 ⊢ 𝑈 ∈ V
6		eqid 2738	. . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
7		unitinvcl.3	. . . . . . 7 ⊢ · = (._r‘𝑅)
8	6, 7	mgpplusg 19639	. . . . . 6 ⊢ · = (+_g‘(mulGrp‘𝑅))
9	2, 8	ressplusg 16926	. . . . 5 ⊢ (𝑈 ∈ V → · = (+_g‘((mulGrp‘𝑅) ↾_s 𝑈)))
10	5, 9	ax-mp 5	. . . 4 ⊢ · = (+_g‘((mulGrp‘𝑅) ↾_s 𝑈))
11		eqid 2738	. . . 4 ⊢ (0_g‘((mulGrp‘𝑅) ↾_s 𝑈)) = (0_g‘((mulGrp‘𝑅) ↾_s 𝑈))
12		unitinvcl.2	. . . . 5 ⊢ 𝐼 = (inv_r‘𝑅)
13	1, 2, 12	invrfval 19830	. . . 4 ⊢ 𝐼 = (inv_g‘((mulGrp‘𝑅) ↾_s 𝑈))
14	4, 10, 11, 13	grprinv 18544	. . 3 ⊢ ((((mulGrp‘𝑅) ↾_s 𝑈) ∈ Grp ∧ 𝑋 ∈ 𝑈) → (𝑋 · (𝐼‘𝑋)) = (0_g‘((mulGrp‘𝑅) ↾_s 𝑈)))
15	3, 14	sylan 579	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑋 · (𝐼‘𝑋)) = (0_g‘((mulGrp‘𝑅) ↾_s 𝑈)))
16		unitinvcl.4	. . . 4 ⊢ 1 = (1_r‘𝑅)
17	1, 2, 16	unitgrpid 19826	. . 3 ⊢ (𝑅 ∈ Ring → 1 = (0_g‘((mulGrp‘𝑅) ↾_s 𝑈)))
18	17	adantr 480	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 1 = (0_g‘((mulGrp‘𝑅) ↾_s 𝑈)))
19	15, 18	eqtr4d 2781	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑋 · (𝐼‘𝑋)) = 1 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 Vcvv 3422 ‘cfv 6418 (class class class)co 7255 ↾_s cress 16867 +_gcplusg 16888 ._rcmulr 16889 0_gc0g 17067 Grpcgrp 18492 mulGrpcmgp 19635 1_rcur 19652 Ringcrg 19698 Unitcui 19796 inv_rcinvr 19828

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-2nd 7805 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-0g 17069 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-grp 18495 df-minusg 18496 df-mgp 19636 df-ur 19653 df-ring 19700 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829

This theorem is referenced by: 1rinv 19836 0unit 19837 dvrid 19845 drnginvrr 19926 subrguss 19954 subrginv 19955 subrgunit 19957 matunit 21735 slesolinvbi 21738 nminvr 23739 nrginvrcnlem 23761 ply1divalg 25207 dchrn0 26303 dvrcan5 31392 invginvrid 45591

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