Theorem unitinvinv 13886

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 2205	. . . . . . 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 13809	. . . . . 6 |- ( R e. Ring -> R e. SRing )
6	2, 4, 5	unitgrpbasd 13877	. . . . 5 |- ( R e. Ring -> U = ( Base ` ( (mulGrp ` R ) ↾s U ) ) )
7	6	eleq2d 2275	. . . 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 13878	. . . 4 |- ( R e. Ring -> ( (mulGrp ` R ) ↾s U ) e. Grp )
10		eqid 2205	. . . . 5 |- ( Base ` ( (mulGrp ` R ) ↾s U ) ) = ( Base ` ( (mulGrp ` R ) ↾s U ) )
11		eqid 2205	. . . . 5 |- ( invg ` ( (mulGrp ` R ) ↾s U ) ) = ( invg ` ( (mulGrp ` R ) ↾s U ) )
12	10, 11	grpinvinv 13399	. . . 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 13884	. . . . 5 |- ( R e. Ring -> I = ( invg ` ( (mulGrp ` R ) ↾s U ) ) )
19	18	fveq1d 5578	. . . . 5 |- ( R e. Ring -> ( I ` X ) = ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) )
20	18, 19	fveq12d 5583	. . . 4 |- ( R e. Ring -> ( I ` ( I ` X ) ) = ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) ) )
21	20	eqeq1d 2214	. . 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 1373   e. wcel 2176   ` cfv 5271  (class class class)co 5944   Basecbs 12832   ↾_s cress 12833   Grpcgrp 13332   invgcminusg 13333  mulGrpcmgp 13682   Ringcrg 13758  Unitcui 13849   invrcinvr 13882

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-lttrn 8039  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-tpos 6331  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-3 9096  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-iress 12840  df-plusg 12922  df-mulr 12923  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-minusg 13336  df-cmn 13622  df-abl 13623  df-mgp 13683  df-ur 13722  df-srg 13726  df-ring 13760  df-oppr 13830  df-dvdsr 13851  df-unit 13852  df-invr 13883

This theorem is referenced by: (None)