Theorem unitinvinv 13620

Description: The inverse of the inverse of a unit is the same element. (Contributed by Mario Carneiro, 4-Dec-2014.)

Hypotheses

Ref	Expression
unitinvcl.1	|- U = (Unit ` R )
unitinvcl.2	|- I = ( invr ` R )

Assertion

Ref	Expression
unitinvinv	|- ( ( R e. Ring /\ X e. U ) -> ( I ` ( I ` X ) ) = X )

Proof of Theorem unitinvinv

Step	Hyp	Ref	Expression
1		unitinvcl.1	. . . . . . 7 |- U = (Unit ` R )
2	1	a1i 9	. . . . . 6 |- ( R e. Ring -> U = (Unit ` R ) )
3		eqid 2193	. . . . . . 7 |- ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s U )
4	3	a1i 9	. . . . . 6 |- ( R e. Ring -> ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s U ) )
5		ringsrg 13543	. . . . . 6 |- ( R e. Ring -> R e. SRing )
6	2, 4, 5	unitgrpbasd 13611	. . . . 5 |- ( R e. Ring -> U = ( Base ` ( (mulGrp ` R ) ↾s U ) ) )
7	6	eleq2d 2263	. . . 4 |- ( R e. Ring -> ( X e. U <-> X e. ( Base ` ( (mulGrp ` R ) ↾s U ) ) ) )
8	7	pm5.32i 454	. . 3 |- ( ( R e. Ring /\ X e. U ) <-> ( R e. Ring /\ X e. ( Base ` ( (mulGrp ` R ) ↾s U ) ) ) )
9	1, 3	unitgrp 13612	. . . 4 |- ( R e. Ring -> ( (mulGrp ` R ) ↾s U ) e. Grp )
10		eqid 2193	. . . . 5 |- ( Base ` ( (mulGrp ` R ) ↾s U ) ) = ( Base ` ( (mulGrp ` R ) ↾s U ) )
11		eqid 2193	. . . . 5 |- ( invg ` ( (mulGrp ` R ) ↾s U ) ) = ( invg ` ( (mulGrp ` R ) ↾s U ) )
12	10, 11	grpinvinv 13139	. . . 4 |- ( ( ( (mulGrp ` R ) ↾s U ) e. Grp /\ X e. ( Base ` ( (mulGrp ` R ) ↾s U ) ) ) -> ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) ) = X )
13	9, 12	sylan 283	. . 3 |- ( ( R e. Ring /\ X e. ( Base ` ( (mulGrp ` R ) ↾s U ) ) ) -> ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) ) = X )
14	8, 13	sylbi 121	. 2 |- ( ( R e. Ring /\ X e. U ) -> ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) ) = X )
15		unitinvcl.2	. . . . . . 7 |- I = ( invr ` R )
16	15	a1i 9	. . . . . 6 |- ( R e. Ring -> I = ( invr ` R ) )
17		id 19	. . . . . 6 |- ( R e. Ring -> R e. Ring )
18	2, 4, 16, 17	invrfvald 13618	. . . . 5 |- ( R e. Ring -> I = ( invg ` ( (mulGrp ` R ) ↾s U ) ) )
19	18	fveq1d 5556	. . . . 5 |- ( R e. Ring -> ( I ` X ) = ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) )
20	18, 19	fveq12d 5561	. . . 4 |- ( R e. Ring -> ( I ` ( I ` X ) ) = ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) ) )
21	20	eqeq1d 2202	. . 3 |- ( R e. Ring -> ( ( I ` ( I ` X ) ) = X <-> ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) ) = X ) )
22	21	adantr 276	. 2 |- ( ( R e. Ring /\ X e. U ) -> ( ( I ` ( I ` X ) ) = X <-> ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) ) = X ) )
23	14, 22	mpbird 167	1 |- ( ( R e. Ring /\ X e. U ) -> ( I ` ( I ` X ) ) = X )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   ` cfv 5254  (class class class)co 5918   Basecbs 12618   ↾_s cress 12619   Grpcgrp 13072   invgcminusg 13073  mulGrpcmgp 13416   Ringcrg 13492  Unitcui 13583   invrcinvr 13616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-tpos 6298  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-cmn 13356  df-abl 13357  df-mgp 13417  df-ur 13456  df-srg 13460  df-ring 13494  df-oppr 13564  df-dvdsr 13585  df-unit 13586  df-invr 13617

This theorem is referenced by: (None)