Theorem unitrrg 14499

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 2234	. . . . . 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 14275	. . . . . 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 14338	. . . 4 |- ( ( R e. Ring /\ x e. U ) -> x e. ( Base ` R ) )
9		oveq2 6066	. . . . . 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 2234	. . . . . . . . . . 11 |- ( invr ` R ) = ( invr ` R )
11		eqid 2234	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
12		eqid 2234	. . . . . . . . . . 11 |- ( 1r ` R ) = ( 1r ` R )
13	3, 10, 11, 12	unitlinv 14356	. . . . . . . . . 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 6073	. . . . . . . 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 14355	. . . . . . . . . 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 14244	. . . . . . . . 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 14251	. . . . . . . . 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 2275	. . . . . . 7 |- ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) -> ( ( ( invr ` R ) ` x ) ( .r ` R ) ( x ( .r ` R ) y ) ) = y )
26		eqid 2234	. . . . . . . . 9 |- ( 0g ` R ) = ( 0g ` R )
27	1, 11, 26	ringrz 14272	. . . . . . . 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 2249	. . . . . 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 2617	. . . 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 14494	. . . 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 3248	1 |- ( R e. Ring -> U C_ E )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   A.wral 2522   C_ wss 3214   ` cfv 5357  (class class class)co 6058   Basecbs 13296   .rcmulr 13375   0gc0g 13553   1rcur 14187  SRingcsrg 14191   Ringcrg 14224  Unitcui 14316   invrcinvr 14350  RLRegcrlreg 14486

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-tpos 6489  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13653  df-sgrp 13699  df-mnd 13714  df-grp 13800  df-minusg 13801  df-cmn 14087  df-abl 14088  df-mgp 14149  df-ur 14188  df-srg 14192  df-ring 14226  df-oppr 14296  df-dvdsr 14318  df-unit 14319  df-invr 14351  df-rlreg 14489

This theorem is referenced by:  znrrg  14920