Theorem unitrrg 14216

Description: Units are regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)

Hypotheses

Ref	Expression
unitrrg.e	|- E = (RLReg ` R )
unitrrg.u	|- U = (Unit ` R )

Assertion

Ref	Expression
unitrrg	|- ( R e. Ring -> U C_ E )

Proof of Theorem unitrrg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2229	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
2	1	a1i 9	. . . . 5 |- ( ( R e. Ring /\ x e. U ) -> ( Base ` R ) = ( Base ` R ) )
3		unitrrg.u	. . . . . 6 |- U = (Unit ` R )
4	3	a1i 9	. . . . 5 |- ( ( R e. Ring /\ x e. U ) -> U = (Unit ` R ) )
5		ringsrg 13996	. . . . . 6 |- ( R e. Ring -> R e. SRing )
6	5	adantr 276	. . . . 5 |- ( ( R e. Ring /\ x e. U ) -> R e. SRing )
7		simpr 110	. . . . 5 |- ( ( R e. Ring /\ x e. U ) -> x e. U )
8	2, 4, 6, 7	unitcld 14057	. . . 4 |- ( ( R e. Ring /\ x e. U ) -> x e. ( Base ` R ) )
9		oveq2 6002	. . . . . 6 |- ( ( x ( .r ` R ) y ) = ( 0g ` R ) -> ( ( ( invr ` R ) ` x ) ( .r ` R ) ( x ( .r ` R ) y ) ) = ( ( ( invr ` R ) ` x ) ( .r ` R ) ( 0g ` R ) ) )
10		eqid 2229	. . . . . . . . . . 11 |- ( invr ` R ) = ( invr ` R )
11		eqid 2229	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
12		eqid 2229	. . . . . . . . . . 11 |- ( 1r ` R ) = ( 1r ` R )
13	3, 10, 11, 12	unitlinv 14075	. . . . . . . . . 10 |- ( ( R e. Ring /\ x e. U ) -> ( ( ( invr ` R ) ` x ) ( .r ` R ) x ) = ( 1r ` R ) )
14	13	adantr 276	. . . . . . . . 9 |- ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) -> ( ( ( invr ` R ) ` x ) ( .r ` R ) x ) = ( 1r ` R ) )
15	14	oveq1d 6009	. . . . . . . 8 |- ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) -> ( ( ( ( invr ` R ) ` x ) ( .r ` R ) x ) ( .r ` R ) y ) = ( ( 1r ` R ) ( .r ` R ) y ) )
16		simpll 527	. . . . . . . . 9 |- ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) -> R e. Ring )
17	3, 10, 1	ringinvcl 14074	. . . . . . . . . 10 |- ( ( R e. Ring /\ x e. U ) -> ( ( invr ` R ) ` x ) e. ( Base ` R ) )
18	17	adantr 276	. . . . . . . . 9 |- ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) -> ( ( invr ` R ) ` x ) e. ( Base ` R ) )
19	8	adantr 276	. . . . . . . . 9 |- ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) -> x e. ( Base ` R ) )
20		simpr 110	. . . . . . . . 9 |- ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) -> y e. ( Base ` R ) )
21	1, 11	ringass 13965	. . . . . . . . 9 |- ( ( R e. Ring /\ ( ( ( invr ` R ) ` x ) e. ( Base ` R ) /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( ( ( ( invr ` R ) ` x ) ( .r ` R ) x ) ( .r ` R ) y ) = ( ( ( invr ` R ) ` x ) ( .r ` R ) ( x ( .r ` R ) y ) ) )
22	16, 18, 19, 20, 21	syl13anc 1273	. . . . . . . 8 |- ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) -> ( ( ( ( invr ` R ) ` x ) ( .r ` R ) x ) ( .r ` R ) y ) = ( ( ( invr ` R ) ` x ) ( .r ` R ) ( x ( .r ` R ) y ) ) )
23	1, 11, 12	ringlidm 13972	. . . . . . . . 9 |- ( ( R e. Ring /\ y e. ( Base ` R ) ) -> ( ( 1r ` R ) ( .r ` R ) y ) = y )
24	23	adantlr 477	. . . . . . . 8 |- ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) -> ( ( 1r ` R ) ( .r ` R ) y ) = y )
25	15, 22, 24	3eqtr3d 2270	. . . . . . 7 |- ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) -> ( ( ( invr ` R ) ` x ) ( .r ` R ) ( x ( .r ` R ) y ) ) = y )
26		eqid 2229	. . . . . . . . 9 |- ( 0g ` R ) = ( 0g ` R )
27	1, 11, 26	ringrz 13993	. . . . . . . 8 |- ( ( R e. Ring /\ ( ( invr ` R ) ` x ) e. ( Base ` R ) ) -> ( ( ( invr ` R ) ` x ) ( .r ` R ) ( 0g ` R ) ) = ( 0g ` R ) )
28	16, 18, 27	syl2anc 411	. . . . . . 7 |- ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) -> ( ( ( invr ` R ) ` x ) ( .r ` R ) ( 0g ` R ) ) = ( 0g ` R ) )
29	25, 28	eqeq12d 2244	. . . . . 6 |- ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) -> ( ( ( ( invr ` R ) ` x ) ( .r ` R ) ( x ( .r ` R ) y ) ) = ( ( ( invr ` R ) ` x ) ( .r ` R ) ( 0g ` R ) ) <-> y = ( 0g ` R ) ) )
30	9, 29	imbitrid 154	. . . . 5 |- ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) -> ( ( x ( .r ` R ) y ) = ( 0g ` R ) -> y = ( 0g ` R ) ) )
31	30	ralrimiva 2603	. . . 4 |- ( ( R e. Ring /\ x e. U ) -> A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) = ( 0g ` R ) -> y = ( 0g ` R ) ) )
32		unitrrg.e	. . . . 5 |- E = (RLReg ` R )
33	32, 1, 11, 26	isrrg 14212	. . . 4 |- ( x e. E <-> ( x e. ( Base ` R ) /\ A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) = ( 0g ` R ) -> y = ( 0g ` R ) ) ) )
34	8, 31, 33	sylanbrc 417	. . 3 |- ( ( R e. Ring /\ x e. U ) -> x e. E )
35	34	ex 115	. 2 |- ( R e. Ring -> ( x e. U -> x e. E ) )
36	35	ssrdv 3230	1 |- ( R e. Ring -> U C_ E )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   C_ wss 3197   ` cfv 5314  (class class class)co 5994   Basecbs 13018   .rcmulr 13097   0gc0g 13275   1rcur 13908  SRingcsrg 13912   Ringcrg 13945  Unitcui 14036   invrcinvr 14069  RLRegcrlreg 14204

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 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-addcom 8087  ax-addass 8089  ax-i2m1 8092  ax-0lt1 8093  ax-0id 8095  ax-rnegex 8096  ax-pre-ltirr 8099  ax-pre-lttrn 8101  ax-pre-ltadd 8103

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 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-tpos 6381  df-pnf 8171  df-mnf 8172  df-ltxr 8174  df-inn 9099  df-2 9157  df-3 9158  df-ndx 13021  df-slot 13022  df-base 13024  df-sets 13025  df-iress 13026  df-plusg 13109  df-mulr 13110  df-0g 13277  df-mgm 13375  df-sgrp 13421  df-mnd 13436  df-grp 13522  df-minusg 13523  df-cmn 13809  df-abl 13810  df-mgp 13870  df-ur 13909  df-srg 13913  df-ring 13947  df-oppr 14017  df-dvdsr 14038  df-unit 14039  df-invr 14070  df-rlreg 14207

This theorem is referenced by:  znrrg  14609