Theorem unitlinv 14111

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

Proof of Theorem unitlinv

Step	Hyp	Ref	Expression
1		unitinvcl.1	. . . . . . 7 |- U = (Unit ` R )
2	1	a1i 9	. . . . . 6 |- ( R e. Ring -> U = (Unit ` R ) )
3		eqidd 2230	. . . . . 6 |- ( R e. Ring -> ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s U ) )
4		ringsrg 14031	. . . . . 6 |- ( R e. Ring -> R e. SRing )
5	2, 3, 4	unitgrpbasd 14100	. . . . 5 |- ( R e. Ring -> U = ( Base ` ( (mulGrp ` R ) ↾s U ) ) )
6	5	eleq2d 2299	. . . 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 2229	. . . . 5 |- ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s U )
9	1, 8	unitgrp 14101	. . . 4 |- ( R e. Ring -> ( (mulGrp ` R ) ↾s U ) e. Grp )
10		eqid 2229	. . . . 5 |- ( Base ` ( (mulGrp ` R ) ↾s U ) ) = ( Base ` ( (mulGrp ` R ) ↾s U ) )
11		eqid 2229	. . . . 5 |- ( +g ` ( (mulGrp ` R ) ↾s U ) ) = ( +g ` ( (mulGrp ` R ) ↾s U ) )
12		eqid 2229	. . . . 5 |- ( 0g ` ( (mulGrp ` R ) ↾s U ) ) = ( 0g ` ( (mulGrp ` R ) ↾s U ) )
13		eqid 2229	. . . . 5 |- ( invg ` ( (mulGrp ` R ) ↾s U ) ) = ( invg ` ( (mulGrp ` R ) ↾s U ) )
14	10, 11, 12, 13	grplinv 13604	. . . 4 |- ( ( ( (mulGrp ` R ) ↾s U ) e. Grp /\ X e. ( Base ` ( (mulGrp ` R ) ↾s U ) ) ) -> ( ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) ( +g ` ( (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 ) ) ) -> ( ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) ( +g ` ( (mulGrp ` R ) ↾s U ) ) X ) = ( 0g ` ( (mulGrp ` R ) ↾s U ) ) )
16	7, 15	sylbi 121	. 2 |- ( ( R e. Ring /\ X e. U ) -> ( ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) ( +g ` ( (mulGrp ` R ) ↾s U ) ) X ) = ( 0g ` ( (mulGrp ` R ) ↾s U ) ) )
17		eqid 2229	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
18		unitinvcl.3	. . . . . 6 |- .x. = ( .r ` R )
19	17, 18	mgpplusgg 13908	. . . . 5 |- ( R e. Ring -> .x. = ( +g ` (mulGrp ` R ) ) )
20		basfn 13112	. . . . . . 7 |- Base Fn _V
21		elex 2811	. . . . . . 7 |- ( R e. Ring -> R e. _V )
22		funfvex 5649	. . . . . . . 8 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
23	22	funfni 5426	. . . . . . 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 2230	. . . . . . 7 |- ( R e. Ring -> ( Base ` R ) = ( Base ` R ) )
26	25, 2, 4	unitssd 14094	. . . . . 6 |- ( R e. Ring -> U C_ ( Base ` R ) )
27	24, 26	ssexd 4224	. . . . 5 |- ( R e. Ring -> U e. _V )
28	17	mgpex 13909	. . . . 5 |- ( R e. Ring -> (mulGrp ` R ) e. _V )
29	3, 19, 27, 28	ressplusgd 13183	. . . 4 |- ( R e. Ring -> .x. = ( +g ` ( (mulGrp ` R ) ↾s U ) ) )
30		unitinvcl.2	. . . . . . 7 |- I = ( invr ` R )
31	30	a1i 9	. . . . . 6 |- ( R e. Ring -> I = ( invr ` R ) )
32		id 19	. . . . . 6 |- ( R e. Ring -> R e. Ring )
33	2, 3, 31, 32	invrfvald 14107	. . . . 5 |- ( R e. Ring -> I = ( invg ` ( (mulGrp ` R ) ↾s U ) ) )
34	33	fveq1d 5634	. . . 4 |- ( R e. Ring -> ( I ` X ) = ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) )
35		eqidd 2230	. . . 4 |- ( R e. Ring -> X = X )
36	29, 34, 35	oveq123d 6031	. . 3 |- ( R e. Ring -> ( ( I ` X ) .x. X ) = ( ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) ( +g ` ( (mulGrp ` R ) ↾s U ) ) X ) )
37	36	adantr 276	. 2 |- ( ( R e. Ring /\ X e. U ) -> ( ( I ` X ) .x. X ) = ( ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) ( +g ` ( (mulGrp ` R ) ↾s U ) ) X ) )
38		unitinvcl.4	. . . 4 |- .1. = ( 1r ` R )
39	1, 8, 38	unitgrpid 14103	. . 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 2272	1 |- ( ( R e. Ring /\ X e. U ) -> ( ( I ` X ) .x. X ) = .1. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2799   Fn wfn 5316   ` cfv 5321  (class class class)co 6010   Basecbs 13053   ↾_s cress 13054   +g cplusg 13131   .rcmulr 13132   0gc0g 13310   Grpcgrp 13554   invgcminusg 13555  mulGrpcmgp 13904   1rcur 13943   Ringcrg 13980  Unitcui 14071   invrcinvr 14105

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-addass 8117  ax-i2m1 8120  ax-0lt1 8121  ax-0id 8123  ax-rnegex 8124  ax-pre-ltirr 8127  ax-pre-lttrn 8129  ax-pre-ltadd 8131

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-tpos 6402  df-pnf 8199  df-mnf 8200  df-ltxr 8202  df-inn 9127  df-2 9185  df-3 9186  df-ndx 13056  df-slot 13057  df-base 13059  df-sets 13060  df-iress 13061  df-plusg 13144  df-mulr 13145  df-0g 13312  df-mgm 13410  df-sgrp 13456  df-mnd 13471  df-grp 13557  df-minusg 13558  df-cmn 13844  df-abl 13845  df-mgp 13905  df-ur 13944  df-srg 13948  df-ring 13982  df-oppr 14052  df-dvdsr 14073  df-unit 14074  df-invr 14106

This theorem is referenced by:  dvrcan1  14125  rhmunitinv  14163  subrginv  14222  subrgunit  14224  unitrrg  14252