Theorem unitrrg 14343

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 2231	. . . . . 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 14122	. . . . . 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 14184	. . . 4 |- ( ( R e. Ring /\ x e. U ) -> x e. ( Base ` R ) )
9		oveq2 6036	. . . . . 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 2231	. . . . . . . . . . 11 |- ( invr ` R ) = ( invr ` R )
11		eqid 2231	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
12		eqid 2231	. . . . . . . . . . 11 |- ( 1r ` R ) = ( 1r ` R )
13	3, 10, 11, 12	unitlinv 14202	. . . . . . . . . 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 6043	. . . . . . . 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 14201	. . . . . . . . . 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 14091	. . . . . . . . 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 14098	. . . . . . . . 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 2272	. . . . . . 7 |- ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) -> ( ( ( invr ` R ) ` x ) ( .r ` R ) ( x ( .r ` R ) y ) ) = y )
26		eqid 2231	. . . . . . . . 9 |- ( 0g ` R ) = ( 0g ` R )
27	1, 11, 26	ringrz 14119	. . . . . . . 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 2246	. . . . . 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 2606	. . . 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 14339	. . . 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 3234	1 |- ( R e. Ring -> U C_ E )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   A.wral 2511   C_ wss 3201   ` cfv 5333  (class class class)co 6028   Basecbs 13143   .rcmulr 13222   0gc0g 13400   1rcur 14034  SRingcsrg 14038   Ringcrg 14071  Unitcui 14162   invrcinvr 14196  RLRegcrlreg 14331

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-lttrn 8189  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-tpos 6454  df-pnf 8259  df-mnf 8260  df-ltxr 8262  df-inn 9187  df-2 9245  df-3 9246  df-ndx 13146  df-slot 13147  df-base 13149  df-sets 13150  df-iress 13151  df-plusg 13234  df-mulr 13235  df-0g 13402  df-mgm 13500  df-sgrp 13546  df-mnd 13561  df-grp 13647  df-minusg 13648  df-cmn 13934  df-abl 13935  df-mgp 13996  df-ur 14035  df-srg 14039  df-ring 14073  df-oppr 14143  df-dvdsr 14164  df-unit 14165  df-invr 14197  df-rlreg 14334

This theorem is referenced by:  znrrg  14736