Theorem unitinvinv 14540

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 2238	. . . . . . 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 14459	. . . . . 6 |- ( R e. Ring -> R e. SRing )
6	2, 4, 5	unitgrpbasd 14531	. . . . 5 |- ( R e. Ring -> U = ( Base ` ( (mulGrp ` R ) ↾s U ) ) )
7	6	eleq2d 2308	. . . 4 |- ( R e. Ring -> ( X e. U <-> X e. ( Base ` ( (mulGrp ` R ) ↾s U ) ) ) )
8	7	pm5.32i 458	. . 3 |- ( ( R e. Ring /\ X e. U ) <-> ( R e. Ring /\ X e. ( Base ` ( (mulGrp ` R ) ↾s U ) ) ) )
9	1, 3	unitgrp 14532	. . . 4 |- ( R e. Ring -> ( (mulGrp ` R ) ↾s U ) e. Grp )
10		eqid 2238	. . . . 5 |- ( Base ` ( (mulGrp ` R ) ↾s U ) ) = ( Base ` ( (mulGrp ` R ) ↾s U ) )
11		eqid 2238	. . . . 5 |- ( invg ` ( (mulGrp ` R ) ↾s U ) ) = ( invg ` ( (mulGrp ` R ) ↾s U ) )
12	10, 11	grpinvinv 13931	. . . 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 14538	. . . . 5 |- ( R e. Ring -> I = ( invg ` ( (mulGrp ` R ) ↾s U ) ) )
19	18	fveq1d 5700	. . . . 5 |- ( R e. Ring -> ( I ` X ) = ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) )
20	18, 19	fveq12d 5705	. . . 4 |- ( R e. Ring -> ( I ` ( I ` X ) ) = ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) ) )
21	20	eqeq1d 2247	. . 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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   ` cfv 5380  (class class class)co 6088   Basecbs 13410   ↾_s cress 13411   Grpcgrp 13864   invgcminusg 13865  mulGrpcmgp 14324   Ringcrg 14407  Unitcui 14502   invrcinvr 14536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-pre-ltirr 8297  ax-pre-lttrn 8299  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-tpos 6520  df-pnf 8368  df-mnf 8369  df-ltxr 8371  df-inn 9314  df-2 9372  df-3 9373  df-ndx 13413  df-slot 13414  df-base 13416  df-sets 13417  df-iress 13418  df-plusg 13503  df-mulr 13504  df-0g 13671  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-grp 13867  df-minusg 13868  df-cmn 14179  df-abl 14180  df-mgp 14325  df-ur 14370  df-srg 14375  df-ring 14409  df-oppr 14482  df-dvdsr 14504  df-unit 14505  df-invr 14537

This theorem is used by: (None)