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Theorem unitlinv 18446

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 2609	. . . 4 ⊢ ((mulGrp‘𝑅) ↾_s 𝑈) = ((mulGrp‘𝑅) ↾_s 𝑈)
3	1, 2	unitgrp 18436	. . 3 ⊢ (𝑅 ∈ Ring → ((mulGrp‘𝑅) ↾_s 𝑈) ∈ Grp)
4	1, 2	unitgrpbas 18435	. . . 4 ⊢ 𝑈 = (Base‘((mulGrp‘𝑅) ↾_s 𝑈))
5		fvex 6098	. . . . . 6 ⊢ (Unit‘𝑅) ∈ V
6	1, 5	eqeltri 2683	. . . . 5 ⊢ 𝑈 ∈ V
7		eqid 2609	. . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
8		unitinvcl.3	. . . . . . 7 ⊢ · = (._r‘𝑅)
9	7, 8	mgpplusg 18262	. . . . . 6 ⊢ · = (+_g‘(mulGrp‘𝑅))
10	2, 9	ressplusg 15764	. . . . 5 ⊢ (𝑈 ∈ V → · = (+_g‘((mulGrp‘𝑅) ↾_s 𝑈)))
11	6, 10	ax-mp 5	. . . 4 ⊢ · = (+_g‘((mulGrp‘𝑅) ↾_s 𝑈))
12		eqid 2609	. . . 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 18442	. . . 4 ⊢ 𝐼 = (inv_g‘((mulGrp‘𝑅) ↾_s 𝑈))
15	4, 11, 12, 14	grplinv 17237	. . 3 ⊢ ((((mulGrp‘𝑅) ↾_s 𝑈) ∈ Grp ∧ 𝑋 ∈ 𝑈) → ((𝐼‘𝑋) · 𝑋) = (0_g‘((mulGrp‘𝑅) ↾_s 𝑈)))
16	3, 15	sylan 486	. 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 18438	. . 3 ⊢ (𝑅 ∈ Ring → 1 = (0_g‘((mulGrp‘𝑅) ↾_s 𝑈)))
19	18	adantr 479	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 1 = (0_g‘((mulGrp‘𝑅) ↾_s 𝑈)))
20	16, 19	eqtr4d 2646	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → ((𝐼‘𝑋) · 𝑋) = 1 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 382 = wceq 1474 ∈ wcel 1976 Vcvv 3172 ‘cfv 5790 (class class class)co 6527 ↾_s cress 15642 +_gcplusg 15714 ._rcmulr 15715 0_gc0g 15869 Grpcgrp 17191 mulGrpcmgp 18258 1_rcur 18270 Ringcrg 18316 Unitcui 18408 inv_rcinvr 18440

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1712 ax-4 1727 ax-5 1826 ax-6 1874 ax-7 1921 ax-8 1978 ax-9 1985 ax-10 2005 ax-11 2020 ax-12 2033 ax-13 2233 ax-ext 2589 ax-rep 4693 ax-sep 4703 ax-nul 4712 ax-pow 4764 ax-pr 4828 ax-un 6824 ax-cnex 9848 ax-resscn 9849 ax-1cn 9850 ax-icn 9851 ax-addcl 9852 ax-addrcl 9853 ax-mulcl 9854 ax-mulrcl 9855 ax-mulcom 9856 ax-addass 9857 ax-mulass 9858 ax-distr 9859 ax-i2m1 9860 ax-1ne0 9861 ax-1rid 9862 ax-rnegex 9863 ax-rrecex 9864 ax-cnre 9865 ax-pre-lttri 9866 ax-pre-lttrn 9867 ax-pre-ltadd 9868 ax-pre-mulgt0 9869

This theorem depends on definitions: df-bi 195 df-or 383 df-an 384 df-3or 1031 df-3an 1032 df-tru 1477 df-ex 1695 df-nf 1700 df-sb 1867 df-eu 2461 df-mo 2462 df-clab 2596 df-cleq 2602 df-clel 2605 df-nfc 2739 df-ne 2781 df-nel 2782 df-ral 2900 df-rex 2901 df-reu 2902 df-rmo 2903 df-rab 2904 df-v 3174 df-sbc 3402 df-csb 3499 df-dif 3542 df-un 3544 df-in 3546 df-ss 3553 df-pss 3555 df-nul 3874 df-if 4036 df-pw 4109 df-sn 4125 df-pr 4127 df-tp 4129 df-op 4131 df-uni 4367 df-iun 4451 df-br 4578 df-opab 4638 df-mpt 4639 df-tr 4675 df-eprel 4939 df-id 4943 df-po 4949 df-so 4950 df-fr 4987 df-we 4989 df-xp 5034 df-rel 5035 df-cnv 5036 df-co 5037 df-dm 5038 df-rn 5039 df-res 5040 df-ima 5041 df-pred 5583 df-ord 5629 df-on 5630 df-lim 5631 df-suc 5632 df-iota 5754 df-fun 5792 df-fn 5793 df-f 5794 df-f1 5795 df-fo 5796 df-f1o 5797 df-fv 5798 df-riota 6489 df-ov 6530 df-oprab 6531 df-mpt2 6532 df-om 6935 df-tpos 7216 df-wrecs 7271 df-recs 7332 df-rdg 7370 df-er 7606 df-en 7819 df-dom 7820 df-sdom 7821 df-pnf 9932 df-mnf 9933 df-xr 9934 df-ltxr 9935 df-le 9936 df-sub 10119 df-neg 10120 df-nn 10868 df-2 10926 df-3 10927 df-ndx 15644 df-slot 15645 df-base 15646 df-sets 15647 df-ress 15648 df-plusg 15727 df-mulr 15728 df-0g 15871 df-mgm 17011 df-sgrp 17053 df-mnd 17064 df-grp 17194 df-minusg 17195 df-mgp 18259 df-ur 18271 df-ring 18318 df-oppr 18392 df-dvdsr 18410 df-unit 18411 df-invr 18441

This theorem is referenced by: dvrcan1 18460 drnginvrl 18535 subrginv 18565 subrgunit 18567 unitrrg 19060 matinv 20244 matunit 20245 slesolinv 20247 nrginvrcnlem 22238 uc1pmon1p 23632 ornglmullt 28944 rhmunitinv 28959 kerunit 28960 lincresunit3lem3 42052

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