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Theorem unitrinv 19428

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 2821	. . . 4 ⊢ ((mulGrp‘𝑅) ↾_s 𝑈) = ((mulGrp‘𝑅) ↾_s 𝑈)
3	1, 2	unitgrp 19417	. . 3 ⊢ (𝑅 ∈ Ring → ((mulGrp‘𝑅) ↾_s 𝑈) ∈ Grp)
4	1, 2	unitgrpbas 19416	. . . 4 ⊢ 𝑈 = (Base‘((mulGrp‘𝑅) ↾_s 𝑈))
5	1	fvexi 6684	. . . . 5 ⊢ 𝑈 ∈ V
6		eqid 2821	. . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
7		unitinvcl.3	. . . . . . 7 ⊢ · = (._r‘𝑅)
8	6, 7	mgpplusg 19243	. . . . . 6 ⊢ · = (+_g‘(mulGrp‘𝑅))
9	2, 8	ressplusg 16612	. . . . 5 ⊢ (𝑈 ∈ V → · = (+_g‘((mulGrp‘𝑅) ↾_s 𝑈)))
10	5, 9	ax-mp 5	. . . 4 ⊢ · = (+_g‘((mulGrp‘𝑅) ↾_s 𝑈))
11		eqid 2821	. . . 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 19423	. . . 4 ⊢ 𝐼 = (inv_g‘((mulGrp‘𝑅) ↾_s 𝑈))
14	4, 10, 11, 13	grprinv 18153	. . 3 ⊢ ((((mulGrp‘𝑅) ↾_s 𝑈) ∈ Grp ∧ 𝑋 ∈ 𝑈) → (𝑋 · (𝐼‘𝑋)) = (0_g‘((mulGrp‘𝑅) ↾_s 𝑈)))
15	3, 14	sylan 582	. 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 19419	. . 3 ⊢ (𝑅 ∈ Ring → 1 = (0_g‘((mulGrp‘𝑅) ↾_s 𝑈)))
18	17	adantr 483	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 1 = (0_g‘((mulGrp‘𝑅) ↾_s 𝑈)))
19	15, 18	eqtr4d 2859	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑋 · (𝐼‘𝑋)) = 1 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ‘cfv 6355 (class class class)co 7156 ↾_s cress 16484 +_gcplusg 16565 ._rcmulr 16566 0_gc0g 16713 Grpcgrp 18103 mulGrpcmgp 19239 1_rcur 19251 Ringcrg 19297 Unitcui 19389 inv_rcinvr 19421

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422

This theorem is referenced by: 1rinv 19429 0unit 19430 dvrid 19438 drnginvrr 19522 subrguss 19550 subrginv 19551 subrgunit 19553 matunit 21287 slesolinvbi 21290 nminvr 23278 nrginvrcnlem 23300 ply1divalg 24731 dchrn0 25826 dvrcan5 30864 invginvrid 44435

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