Theorem unitrrg 14239

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 14018	. . . . . 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 14080	. . . 4 |- ( ( R e. Ring /\ x e. U ) -> x e. ( Base ` R ) )
9		oveq2 6015	. . . . . 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 14098	. . . . . . . . . 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 6022	. . . . . . . 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 14097	. . . . . . . . . 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 13987	. . . . . . . . 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 13994	. . . . . . . . 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 14015	. . . . . . . 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 14235	. . . 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 5318  (class class class)co 6007   Basecbs 13040   .rcmulr 13119   0gc0g 13297   1rcur 13930  SRingcsrg 13934   Ringcrg 13967  Unitcui 14058   invrcinvr 14092  RLRegcrlreg 14227

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 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-addcom 8107  ax-addass 8109  ax-i2m1 8112  ax-0lt1 8113  ax-0id 8115  ax-rnegex 8116  ax-pre-ltirr 8119  ax-pre-lttrn 8121  ax-pre-ltadd 8123

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 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-tpos 6397  df-pnf 8191  df-mnf 8192  df-ltxr 8194  df-inn 9119  df-2 9177  df-3 9178  df-ndx 13043  df-slot 13044  df-base 13046  df-sets 13047  df-iress 13048  df-plusg 13131  df-mulr 13132  df-0g 13299  df-mgm 13397  df-sgrp 13443  df-mnd 13458  df-grp 13544  df-minusg 13545  df-cmn 13831  df-abl 13832  df-mgp 13892  df-ur 13931  df-srg 13935  df-ring 13969  df-oppr 14039  df-dvdsr 14060  df-unit 14061  df-invr 14093  df-rlreg 14230

This theorem is referenced by:  znrrg  14632