Theorem unitrrg 14405

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 2232	. . . . . 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 14183	. . . . . 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 14245	. . . 4 |- ( ( R e. Ring /\ x e. U ) -> x e. ( Base ` R ) )
9		oveq2 6057	. . . . . 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 2232	. . . . . . . . . . 11 |- ( invr ` R ) = ( invr ` R )
11		eqid 2232	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
12		eqid 2232	. . . . . . . . . . 11 |- ( 1r ` R ) = ( 1r ` R )
13	3, 10, 11, 12	unitlinv 14263	. . . . . . . . . 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 6064	. . . . . . . 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 14262	. . . . . . . . . 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 14152	. . . . . . . . 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 1276	. . . . . . . 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 14159	. . . . . . . . 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 2273	. . . . . . 7 |- ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) -> ( ( ( invr ` R ) ` x ) ( .r ` R ) ( x ( .r ` R ) y ) ) = y )
26		eqid 2232	. . . . . . . . 9 |- ( 0g ` R ) = ( 0g ` R )
27	1, 11, 26	ringrz 14180	. . . . . . . 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 2247	. . . . . 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 2615	. . . 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 14400	. . . 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 3243	1 |- ( R e. Ring -> U C_ E )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   A.wral 2520   C_ wss 3210   ` cfv 5351  (class class class)co 6049   Basecbs 13204   .rcmulr 13283   0gc0g 13461   1rcur 14095  SRingcsrg 14099   Ringcrg 14132  Unitcui 14223   invrcinvr 14257  RLRegcrlreg 14392

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-addass 8228  ax-i2m1 8231  ax-0lt1 8232  ax-0id 8234  ax-rnegex 8235  ax-pre-ltirr 8238  ax-pre-lttrn 8240  ax-pre-ltadd 8242

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-tpos 6475  df-pnf 8309  df-mnf 8310  df-ltxr 8312  df-inn 9237  df-2 9295  df-3 9296  df-ndx 13207  df-slot 13208  df-base 13210  df-sets 13211  df-iress 13212  df-plusg 13295  df-mulr 13296  df-0g 13463  df-mgm 13561  df-sgrp 13607  df-mnd 13622  df-grp 13708  df-minusg 13709  df-cmn 13995  df-abl 13996  df-mgp 14057  df-ur 14096  df-srg 14100  df-ring 14134  df-oppr 14204  df-dvdsr 14225  df-unit 14226  df-invr 14258  df-rlreg 14395

This theorem is referenced by:  znrrg  14800