Theorem unitlinv 14433

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 2239	. . . . . 6 |- ( R e. Ring -> ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s U ) )
4		ringsrg 14352	. . . . . 6 |- ( R e. Ring -> R e. SRing )
5	2, 3, 4	unitgrpbasd 14422	. . . . 5 |- ( R e. Ring -> U = ( Base ` ( (mulGrp ` R ) ↾s U ) ) )
6	5	eleq2d 2308	. . . 4 |- ( R e. Ring -> ( X e. U <-> X e. ( Base ` ( (mulGrp ` R ) ↾s U ) ) ) )
7	6	pm5.32i 458	. . 3 |- ( ( R e. Ring /\ X e. U ) <-> ( R e. Ring /\ X e. ( Base ` ( (mulGrp ` R ) ↾s U ) ) ) )
8		eqid 2238	. . . . 5 |- ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s U )
9	1, 8	unitgrp 14423	. . . 4 |- ( R e. Ring -> ( (mulGrp ` R ) ↾s U ) e. Grp )
10		eqid 2238	. . . . 5 |- ( Base ` ( (mulGrp ` R ) ↾s U ) ) = ( Base ` ( (mulGrp ` R ) ↾s U ) )
11		eqid 2238	. . . . 5 |- ( +g ` ( (mulGrp ` R ) ↾s U ) ) = ( +g ` ( (mulGrp ` R ) ↾s U ) )
12		eqid 2238	. . . . 5 |- ( 0g ` ( (mulGrp ` R ) ↾s U ) ) = ( 0g ` ( (mulGrp ` R ) ↾s U ) )
13		eqid 2238	. . . . 5 |- ( invg ` ( (mulGrp ` R ) ↾s U ) ) = ( invg ` ( (mulGrp ` R ) ↾s U ) )
14	10, 11, 12, 13	grplinv 13855	. . . 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 2238	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
18		unitinvcl.3	. . . . . 6 |- .x. = ( .r ` R )
19	17, 18	mgpplusgg 14221	. . . . 5 |- ( R e. Ring -> .x. = ( +g ` (mulGrp ` R ) ) )
20		basfn 13411	. . . . . . 7 |- Base Fn _V
21		elex 2833	. . . . . . 7 |- ( R e. Ring -> R e. _V )
22		funfvex 5712	. . . . . . . 8 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
23	22	funfni 5483	. . . . . . 7 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
24	20, 21, 23	sylancr 418	. . . . . 6 |- ( R e. Ring -> ( Base ` R ) e. _V )
25		eqidd 2239	. . . . . . 7 |- ( R e. Ring -> ( Base ` R ) = ( Base ` R ) )
26	25, 2, 4	unitssd 14416	. . . . . 6 |- ( R e. Ring -> U C_ ( Base ` R ) )
27	24, 26	ssexd 4273	. . . . 5 |- ( R e. Ring -> U e. _V )
28	17	mgpex 14223	. . . . 5 |- ( R e. Ring -> (mulGrp ` R ) e. _V )
29	3, 19, 27, 28	ressplusgd 13483	. . . 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 14429	. . . . 5 |- ( R e. Ring -> I = ( invg ` ( (mulGrp ` R ) ↾s U ) ) )
34	33	fveq1d 5697	. . . 4 |- ( R e. Ring -> ( I ` X ) = ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` X ) )
35		eqidd 2239	. . . 4 |- ( R e. Ring -> X = X )
36	29, 34, 35	oveq123d 6106	. . 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 14425	. . 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 2281	1 |- ( ( R e. Ring /\ X e. U ) -> ( ( I ` X ) .x. X ) = .1. )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   Basecbs 13352   ↾_s cress 13353   +g cplusg 13431   .rcmulr 13432   0gc0g 13610   Grpcgrp 13805   invgcminusg 13806  mulGrpcmgp 14217   1rcur 14262   Ringcrg 14300  Unitcui 14393   invrcinvr 14427

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-tpos 6516  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-plusg 13444  df-mulr 13445  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-minusg 13809  df-cmn 14089  df-abl 14090  df-mgp 14218  df-ur 14263  df-srg 14268  df-ring 14302  df-oppr 14373  df-dvdsr 14395  df-unit 14396  df-invr 14428

This theorem is used by:  dvrcan1  14447  rhmunitinv  14485  subrginv  14545  subrgunit  14547  unitrrg  14576