Theorem unitlinv 13358

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 2188	. . . . . 6 |- ( R e. Ring -> ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s U ) )
4		ringsrg 13282	. . . . . 6 |- ( R e. Ring -> R e. SRing )
5	2, 3, 4	unitgrpbasd 13347	. . . . 5 |- ( R e. Ring -> U = ( Base ` ( (mulGrp ` R ) ↾s U ) ) )
6	5	eleq2d 2257	. . . 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 2187	. . . . 5 |- ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s U )
9	1, 8	unitgrp 13348	. . . 4 |- ( R e. Ring -> ( (mulGrp ` R ) ↾s U ) e. Grp )
10		eqid 2187	. . . . 5 |- ( Base ` ( (mulGrp ` R ) ↾s U ) ) = ( Base ` ( (mulGrp ` R ) ↾s U ) )
11		eqid 2187	. . . . 5 |- ( +g ` ( (mulGrp ` R ) ↾s U ) ) = ( +g ` ( (mulGrp ` R ) ↾s U ) )
12		eqid 2187	. . . . 5 |- ( 0g ` ( (mulGrp ` R ) ↾s U ) ) = ( 0g ` ( (mulGrp ` R ) ↾s U ) )
13		eqid 2187	. . . . 5 |- ( invg ` ( (mulGrp ` R ) ↾s U ) ) = ( invg ` ( (mulGrp ` R ) ↾s U ) )
14	10, 11, 12, 13	grplinv 12944	. . . 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 2187	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
18		unitinvcl.3	. . . . . 6 |- .x. = ( .r ` R )
19	17, 18	mgpplusgg 13166	. . . . 5 |- ( R e. Ring -> .x. = ( +g ` (mulGrp ` R ) ) )
20		basfn 12533	. . . . . . 7 |- Base Fn _V
21		elex 2760	. . . . . . 7 |- ( R e. Ring -> R e. _V )
22		funfvex 5544	. . . . . . . 8 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
23	22	funfni 5328	. . . . . . 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 2188	. . . . . . 7 |- ( R e. Ring -> ( Base ` R ) = ( Base ` R ) )
26	25, 2, 4	unitssd 13341	. . . . . 6 |- ( R e. Ring -> U C_ ( Base ` R ) )
27	24, 26	ssexd 4155	. . . . 5 |- ( R e. Ring -> U e. _V )
28	17	mgpex 13167	. . . . 5 |- ( R e. Ring -> (mulGrp ` R ) e. _V )
29	3, 19, 27, 28	ressplusgd 12601	. . . 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 13354	. . . . 5 |- ( R e. Ring -> I = ( invg ` ( (mulGrp ` R ) ↾s U ) ) )
34	33	fveq1d 5529	. . . 4 |- ( R e. Ring -> ( I ` X ) = ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) )
35		eqidd 2188	. . . 4 |- ( R e. Ring -> X = X )
36	29, 34, 35	oveq123d 5909	. . 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 13350	. . 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 2230	1 |- ( ( R e. Ring /\ X e. U ) -> ( ( I ` X ) .x. X ) = .1. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1363   e. wcel 2158   _Vcvv 2749   Fn wfn 5223   ` cfv 5228  (class class class)co 5888   Basecbs 12475   ↾_s cress 12476   +g cplusg 12550   .rcmulr 12551   0gc0g 12722   Grpcgrp 12896   invgcminusg 12897  mulGrpcmgp 13162   1rcur 13196   Ringcrg 13233  Unitcui 13319   invrcinvr 13352

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-pre-ltirr 7936  ax-pre-lttrn 7938  ax-pre-ltadd 7940

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-tpos 6259  df-pnf 8007  df-mnf 8008  df-ltxr 8010  df-inn 8933  df-2 8991  df-3 8992  df-ndx 12478  df-slot 12479  df-base 12481  df-sets 12482  df-iress 12483  df-plusg 12563  df-mulr 12564  df-0g 12724  df-mgm 12793  df-sgrp 12826  df-mnd 12837  df-grp 12899  df-minusg 12900  df-cmn 13120  df-abl 13121  df-mgp 13163  df-ur 13197  df-srg 13201  df-ring 13235  df-oppr 13301  df-dvdsr 13321  df-unit 13322  df-invr 13353

This theorem is referenced by:  dvrcan1  13372  subrginv  13421  subrgunit  13423