Theorem unitrinv 13683

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 13603	. . . . . 6 |- ( R e. Ring -> R e. SRing )
5	2, 3, 4	unitgrpbasd 13671	. . . . 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 13672	. . . 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 13183	. . . 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 13480	. . . . 5 |- ( R e. Ring -> .x. = ( +g ` (mulGrp ` R ) ) )
20		basfn 12736	. . . . . . 7 |- Base Fn _V
21		elex 2774	. . . . . . 7 |- ( R e. Ring -> R e. _V )
22		funfvex 5575	. . . . . . . 8 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
23	22	funfni 5358	. . . . . . 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 13665	. . . . . 6 |- ( R e. Ring -> U C_ ( Base ` R ) )
27	24, 26	ssexd 4173	. . . . 5 |- ( R e. Ring -> U e. _V )
28	17	mgpex 13481	. . . . 5 |- ( R e. Ring -> (mulGrp ` R ) e. _V )
29	3, 19, 27, 28	ressplusgd 12806	. . . 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 13678	. . . . 5 |- ( R e. Ring -> I = ( invg ` ( (mulGrp ` R ) ↾s U ) ) )
35	34	fveq1d 5560	. . . 4 |- ( R e. Ring -> ( I ` X ) = ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) )
36	29, 30, 35	oveq123d 5943	. . 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 13674	. . 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 5253   ` cfv 5258  (class class class)co 5922   Basecbs 12678   ↾_s cress 12679   +g cplusg 12755   .rcmulr 12756   0gc0g 12927   Grpcgrp 13132   invgcminusg 13133  mulGrpcmgp 13476   1rcur 13515   Ringcrg 13552  Unitcui 13643   invrcinvr 13676

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 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

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 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-tpos 6303  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-cmn 13416  df-abl 13417  df-mgp 13477  df-ur 13516  df-srg 13520  df-ring 13554  df-oppr 13624  df-dvdsr 13645  df-unit 13646  df-invr 13677

This theorem is referenced by:  1rinv  13684  0unit  13685  dvrid  13693  subrguss  13792  subrginv  13793  subrgunit  13795