Theorem unitlinv 13437

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 2190	. . . . . 6 |- ( R e. Ring -> ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s U ) )
4		ringsrg 13360	. . . . . 6 |- ( R e. Ring -> R e. SRing )
5	2, 3, 4	unitgrpbasd 13426	. . . . 5 |- ( R e. Ring -> U = ( Base ` ( (mulGrp ` R ) ↾s U ) ) )
6	5	eleq2d 2259	. . . 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 2189	. . . . 5 |- ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s U )
9	1, 8	unitgrp 13427	. . . 4 |- ( R e. Ring -> ( (mulGrp ` R ) ↾s U ) e. Grp )
10		eqid 2189	. . . . 5 |- ( Base ` ( (mulGrp ` R ) ↾s U ) ) = ( Base ` ( (mulGrp ` R ) ↾s U ) )
11		eqid 2189	. . . . 5 |- ( +g ` ( (mulGrp ` R ) ↾s U ) ) = ( +g ` ( (mulGrp ` R ) ↾s U ) )
12		eqid 2189	. . . . 5 |- ( 0g ` ( (mulGrp ` R ) ↾s U ) ) = ( 0g ` ( (mulGrp ` R ) ↾s U ) )
13		eqid 2189	. . . . 5 |- ( invg ` ( (mulGrp ` R ) ↾s U ) ) = ( invg ` ( (mulGrp ` R ) ↾s U ) )
14	10, 11, 12, 13	grplinv 12960	. . . 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 2189	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
18		unitinvcl.3	. . . . . 6 |- .x. = ( .r ` R )
19	17, 18	mgpplusgg 13239	. . . . 5 |- ( R e. Ring -> .x. = ( +g ` (mulGrp ` R ) ) )
20		basfn 12538	. . . . . . 7 |- Base Fn _V
21		elex 2763	. . . . . . 7 |- ( R e. Ring -> R e. _V )
22		funfvex 5547	. . . . . . . 8 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
23	22	funfni 5331	. . . . . . 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 2190	. . . . . . 7 |- ( R e. Ring -> ( Base ` R ) = ( Base ` R ) )
26	25, 2, 4	unitssd 13420	. . . . . 6 |- ( R e. Ring -> U C_ ( Base ` R ) )
27	24, 26	ssexd 4158	. . . . 5 |- ( R e. Ring -> U e. _V )
28	17	mgpex 13240	. . . . 5 |- ( R e. Ring -> (mulGrp ` R ) e. _V )
29	3, 19, 27, 28	ressplusgd 12606	. . . 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 13433	. . . . 5 |- ( R e. Ring -> I = ( invg ` ( (mulGrp ` R ) ↾s U ) ) )
34	33	fveq1d 5532	. . . 4 |- ( R e. Ring -> ( I ` X ) = ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) )
35		eqidd 2190	. . . 4 |- ( R e. Ring -> X = X )
36	29, 34, 35	oveq123d 5912	. . 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 13429	. . 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 2232	1 |- ( ( R e. Ring /\ X e. U ) -> ( ( I ` X ) .x. X ) = .1. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   _Vcvv 2752   Fn wfn 5226   ` cfv 5231  (class class class)co 5891   Basecbs 12480   ↾_s cress 12481   +g cplusg 12555   .rcmulr 12556   0gc0g 12727   Grpcgrp 12911   invgcminusg 12912  mulGrpcmgp 13235   1rcur 13274   Ringcrg 13311  Unitcui 13398   invrcinvr 13431

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-addcom 7929  ax-addass 7931  ax-i2m1 7934  ax-0lt1 7935  ax-0id 7937  ax-rnegex 7938  ax-pre-ltirr 7941  ax-pre-lttrn 7943  ax-pre-ltadd 7945

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-tpos 6264  df-pnf 8012  df-mnf 8013  df-ltxr 8015  df-inn 8938  df-2 8996  df-3 8997  df-ndx 12483  df-slot 12484  df-base 12486  df-sets 12487  df-iress 12488  df-plusg 12568  df-mulr 12569  df-0g 12729  df-mgm 12798  df-sgrp 12831  df-mnd 12844  df-grp 12914  df-minusg 12915  df-cmn 13186  df-abl 13187  df-mgp 13236  df-ur 13275  df-srg 13279  df-ring 13313  df-oppr 13379  df-dvdsr 13400  df-unit 13401  df-invr 13432

This theorem is referenced by:  dvrcan1  13451  rhmunitinv  13489  subrginv  13545  subrgunit  13547