Theorem unitrinv 13761

Description: A unit times its inverse is the ring unity. (Contributed by Mario Carneiro, 2-Dec-2014.)

Hypotheses

Ref	Expression
unitinvcl.1	|- U = (Unit ` R )
unitinvcl.2	|- I = ( invr ` R )
unitinvcl.3	|- .x. = ( .r ` R )
unitinvcl.4	|- .1. = ( 1r ` R )

Assertion

Ref	Expression
unitrinv	|- ( ( R e. Ring /\ X e. U ) -> ( X .x. ( I ` X ) ) = .1. )

Proof of Theorem unitrinv

Step	Hyp	Ref	Expression
1		unitinvcl.1	. . . . . . 7 |- U = (Unit ` R )
2	1	a1i 9	. . . . . 6 |- ( R e. Ring -> U = (Unit ` R ) )
3		eqidd 2197	. . . . . 6 |- ( R e. Ring -> ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s U ) )
4		ringsrg 13681	. . . . . 6 |- ( R e. Ring -> R e. SRing )
5	2, 3, 4	unitgrpbasd 13749	. . . . 5 |- ( R e. Ring -> U = ( Base ` ( (mulGrp ` R ) ↾s U ) ) )
6	5	eleq2d 2266	. . . 4 |- ( R e. Ring -> ( X e. U <-> X e. ( Base ` ( (mulGrp ` R ) ↾s U ) ) ) )
7	6	pm5.32i 454	. . 3 |- ( ( R e. Ring /\ X e. U ) <-> ( R e. Ring /\ X e. ( Base ` ( (mulGrp ` R ) ↾s U ) ) ) )
8		eqid 2196	. . . . 5 |- ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s U )
9	1, 8	unitgrp 13750	. . . 4 |- ( R e. Ring -> ( (mulGrp ` R ) ↾s U ) e. Grp )
10		eqid 2196	. . . . 5 |- ( Base ` ( (mulGrp ` R ) ↾s U ) ) = ( Base ` ( (mulGrp ` R ) ↾s U ) )
11		eqid 2196	. . . . 5 |- ( +g ` ( (mulGrp ` R ) ↾s U ) ) = ( +g ` ( (mulGrp ` R ) ↾s U ) )
12		eqid 2196	. . . . 5 |- ( 0g ` ( (mulGrp ` R ) ↾s U ) ) = ( 0g ` ( (mulGrp ` R ) ↾s U ) )
13		eqid 2196	. . . . 5 |- ( invg ` ( (mulGrp ` R ) ↾s U ) ) = ( invg ` ( (mulGrp ` R ) ↾s U ) )
14	10, 11, 12, 13	grprinv 13255	. . . 4 |- ( ( ( (mulGrp ` R ) ↾s U ) e. Grp /\ X e. ( Base ` ( (mulGrp ` R ) ↾s U ) ) ) -> ( X ( +g ` ( (mulGrp ` R ) ↾s U ) ) ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) ) = ( 0g ` ( (mulGrp ` R ) ↾s U ) ) )
15	9, 14	sylan 283	. . 3 |- ( ( R e. Ring /\ X e. ( Base ` ( (mulGrp ` R ) ↾s U ) ) ) -> ( X ( +g ` ( (mulGrp ` R ) ↾s U ) ) ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) ) = ( 0g ` ( (mulGrp ` R ) ↾s U ) ) )
16	7, 15	sylbi 121	. 2 |- ( ( R e. Ring /\ X e. U ) -> ( X ( +g ` ( (mulGrp ` R ) ↾s U ) ) ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) ) = ( 0g ` ( (mulGrp ` R ) ↾s U ) ) )
17		eqid 2196	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
18		unitinvcl.3	. . . . . 6 |- .x. = ( .r ` R )
19	17, 18	mgpplusgg 13558	. . . . 5 |- ( R e. Ring -> .x. = ( +g ` (mulGrp ` R ) ) )
20		basfn 12763	. . . . . . 7 |- Base Fn _V
21		elex 2774	. . . . . . 7 |- ( R e. Ring -> R e. _V )
22		funfvex 5578	. . . . . . . 8 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
23	22	funfni 5361	. . . . . . 7 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
24	20, 21, 23	sylancr 414	. . . . . 6 |- ( R e. Ring -> ( Base ` R ) e. _V )
25		eqidd 2197	. . . . . . 7 |- ( R e. Ring -> ( Base ` R ) = ( Base ` R ) )
26	25, 2, 4	unitssd 13743	. . . . . 6 |- ( R e. Ring -> U C_ ( Base ` R ) )
27	24, 26	ssexd 4174	. . . . 5 |- ( R e. Ring -> U e. _V )
28	17	mgpex 13559	. . . . 5 |- ( R e. Ring -> (mulGrp ` R ) e. _V )
29	3, 19, 27, 28	ressplusgd 12833	. . . 4 |- ( R e. Ring -> .x. = ( +g ` ( (mulGrp ` R ) ↾s U ) ) )
30		eqidd 2197	. . . 4 |- ( R e. Ring -> X = X )
31		unitinvcl.2	. . . . . . 7 |- I = ( invr ` R )
32	31	a1i 9	. . . . . 6 |- ( R e. Ring -> I = ( invr ` R ) )
33		id 19	. . . . . 6 |- ( R e. Ring -> R e. Ring )
34	2, 3, 32, 33	invrfvald 13756	. . . . 5 |- ( R e. Ring -> I = ( invg ` ( (mulGrp ` R ) ↾s U ) ) )
35	34	fveq1d 5563	. . . 4 |- ( R e. Ring -> ( I ` X ) = ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) )
36	29, 30, 35	oveq123d 5946	. . 3 |- ( R e. Ring -> ( X .x. ( I ` X ) ) = ( X ( +g ` ( (mulGrp ` R ) ↾s U ) ) ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) ) )
37	36	adantr 276	. 2 |- ( ( R e. Ring /\ X e. U ) -> ( X .x. ( I ` X ) ) = ( X ( +g ` ( (mulGrp ` R ) ↾s U ) ) ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) ) )
38		unitinvcl.4	. . . 4 |- .1. = ( 1r ` R )
39	1, 8, 38	unitgrpid 13752	. . 3 |- ( R e. Ring -> .1. = ( 0g ` ( (mulGrp ` R ) ↾s U ) ) )
40	39	adantr 276	. 2 |- ( ( R e. Ring /\ X e. U ) -> .1. = ( 0g ` ( (mulGrp ` R ) ↾s U ) ) )
41	16, 37, 40	3eqtr4d 2239	1 |- ( ( R e. Ring /\ X e. U ) -> ( X .x. ( I ` X ) ) = .1. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   _Vcvv 2763   Fn wfn 5254   ` cfv 5259  (class class class)co 5925   Basecbs 12705   ↾_s cress 12706   +g cplusg 12782   .rcmulr 12783   0gc0g 12960   Grpcgrp 13204   invgcminusg 13205  mulGrpcmgp 13554   1rcur 13593   Ringcrg 13630  Unitcui 13721   invrcinvr 13754

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-addcom 7998  ax-addass 8000  ax-i2m1 8003  ax-0lt1 8004  ax-0id 8006  ax-rnegex 8007  ax-pre-ltirr 8010  ax-pre-lttrn 8012  ax-pre-ltadd 8014

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-tpos 6312  df-pnf 8082  df-mnf 8083  df-ltxr 8085  df-inn 9010  df-2 9068  df-3 9069  df-ndx 12708  df-slot 12709  df-base 12711  df-sets 12712  df-iress 12713  df-plusg 12795  df-mulr 12796  df-0g 12962  df-mgm 13060  df-sgrp 13106  df-mnd 13121  df-grp 13207  df-minusg 13208  df-cmn 13494  df-abl 13495  df-mgp 13555  df-ur 13594  df-srg 13598  df-ring 13632  df-oppr 13702  df-dvdsr 13723  df-unit 13724  df-invr 13755

This theorem is referenced by:  1rinv  13762  0unit  13763  dvrid  13771  subrguss  13870  subrginv  13871  subrgunit  13873