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Theorem unitlinv 18500

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
unitlinv	⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → ((𝐼‘𝑋) · 𝑋) = 1 )

**Proof of Theorem unitlinv**
Step	Hyp	Ref	Expression
1		unitinvcl.1	. . . 4 ⊢ 𝑈 = (Unit‘𝑅)
2		eqid 2610	. . . 4 ⊢ ((mulGrp‘𝑅) ↾_s 𝑈) = ((mulGrp‘𝑅) ↾_s 𝑈)
3	1, 2	unitgrp 18490	. . 3 ⊢ (𝑅 ∈ Ring → ((mulGrp‘𝑅) ↾_s 𝑈) ∈ Grp)
4	1, 2	unitgrpbas 18489	. . . 4 ⊢ 𝑈 = (Base‘((mulGrp‘𝑅) ↾_s 𝑈))
5		fvex 6113	. . . . . 6 ⊢ (Unit‘𝑅) ∈ V
6	1, 5	eqeltri 2684	. . . . 5 ⊢ 𝑈 ∈ V
7		eqid 2610	. . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
8		unitinvcl.3	. . . . . . 7 ⊢ · = (._r‘𝑅)
9	7, 8	mgpplusg 18316	. . . . . 6 ⊢ · = (+_g‘(mulGrp‘𝑅))
10	2, 9	ressplusg 15818	. . . . 5 ⊢ (𝑈 ∈ V → · = (+_g‘((mulGrp‘𝑅) ↾_s 𝑈)))
11	6, 10	ax-mp 5	. . . 4 ⊢ · = (+_g‘((mulGrp‘𝑅) ↾_s 𝑈))
12		eqid 2610	. . . 4 ⊢ (0_g‘((mulGrp‘𝑅) ↾_s 𝑈)) = (0_g‘((mulGrp‘𝑅) ↾_s 𝑈))
13		unitinvcl.2	. . . . 5 ⊢ 𝐼 = (inv_r‘𝑅)
14	1, 2, 13	invrfval 18496	. . . 4 ⊢ 𝐼 = (inv_g‘((mulGrp‘𝑅) ↾_s 𝑈))
15	4, 11, 12, 14	grplinv 17291	. . 3 ⊢ ((((mulGrp‘𝑅) ↾_s 𝑈) ∈ Grp ∧ 𝑋 ∈ 𝑈) → ((𝐼‘𝑋) · 𝑋) = (0_g‘((mulGrp‘𝑅) ↾_s 𝑈)))
16	3, 15	sylan 487	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → ((𝐼‘𝑋) · 𝑋) = (0_g‘((mulGrp‘𝑅) ↾_s 𝑈)))
17		unitinvcl.4	. . . 4 ⊢ 1 = (1_r‘𝑅)
18	1, 2, 17	unitgrpid 18492	. . 3 ⊢ (𝑅 ∈ Ring → 1 = (0_g‘((mulGrp‘𝑅) ↾_s 𝑈)))
19	18	adantr 480	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 1 = (0_g‘((mulGrp‘𝑅) ↾_s 𝑈)))
20	16, 19	eqtr4d 2647	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → ((𝐼‘𝑋) · 𝑋) = 1 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ‘cfv 5804 (class class class)co 6549 ↾_s cress 15696 +_gcplusg 15768 ._rcmulr 15769 0_gc0g 15923 Grpcgrp 17245 mulGrpcmgp 18312 1_rcur 18324 Ringcrg 18370 Unitcui 18462 inv_rcinvr 18494

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495

This theorem is referenced by: dvrcan1 18514 drnginvrl 18589 subrginv 18619 subrgunit 18621 unitrrg 19114 matinv 20302 matunit 20303 slesolinv 20305 nrginvrcnlem 22305 uc1pmon1p 23715 ornglmullt 29138 rhmunitinv 29153 kerunit 29154 lincresunit3lem3 42057

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