Theorem opprlring 14487

Description: The opposite of a local ring is also a local ring. (Contributed by NM, 18-Oct-2014.)

Hypothesis

Ref	Expression
opprlring.1	|- O = (oppr ` R )

Assertion

Ref	Expression
opprlring	|- ( R e. LRing <-> O e. LRing )

Proof of Theorem opprlring

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

Step	Hyp	Ref	Expression
1		lringring 14484	. 2 |- ( R e. LRing -> R e. Ring )
2		lringring 14484	. . 3 |- ( O e. LRing -> O e. Ring )
3		opprlring.1	. . . 4 |- O = (oppr ` R )
4	3	opprringb 14369	. . 3 |- ( R e. Ring <-> O e. Ring )
5	2, 4	sylibr 134	. 2 |- ( O e. LRing -> R e. Ring )
6	3	opprnzrbg 14475	. . . 4 |- ( R e. Ring -> ( R e. NzRing <-> O e. NzRing ) )
7		eqid 2238	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
8	3, 7	opprbasg 14363	. . . . 5 |- ( R e. Ring -> ( Base ` R ) = ( Base ` O ) )
9		eqid 2238	. . . . . . . . . 10 |- ( +g ` R ) = ( +g ` R )
10	3, 9	oppraddg 14364	. . . . . . . . 9 |- ( R e. Ring -> ( +g ` R ) = ( +g ` O ) )
11	10	oveqd 6096	. . . . . . . 8 |- ( R e. Ring -> ( x ( +g ` R ) y ) = ( x ( +g ` O ) y ) )
12		eqid 2238	. . . . . . . . 9 |- ( 1r ` R ) = ( 1r ` R )
13	3, 12	oppr1g 14371	. . . . . . . 8 |- ( R e. Ring -> ( 1r ` R ) = ( 1r ` O ) )
14	11, 13	eqeq12d 2253	. . . . . . 7 |- ( R e. Ring -> ( ( x ( +g ` R ) y ) = ( 1r ` R ) <-> ( x ( +g ` O ) y ) = ( 1r ` O ) ) )
15		eqidd 2239	. . . . . . . . . 10 |- ( R e. Ring -> (Unit ` R ) = (Unit ` R ) )
16	3	a1i 9	. . . . . . . . . 10 |- ( R e. Ring -> O = (oppr ` R ) )
17		id 19	. . . . . . . . . 10 |- ( R e. Ring -> R e. Ring )
18	15, 16, 17	opprunitd 14400	. . . . . . . . 9 |- ( R e. Ring -> (Unit ` R ) = (Unit ` O ) )
19	18	eleq2d 2308	. . . . . . . 8 |- ( R e. Ring -> ( x e. (Unit ` R ) <-> x e. (Unit ` O ) ) )
20	18	eleq2d 2308	. . . . . . . 8 |- ( R e. Ring -> ( y e. (Unit ` R ) <-> y e. (Unit ` O ) ) )
21	19, 20	orbi12d 805	. . . . . . 7 |- ( R e. Ring -> ( ( x e. (Unit ` R ) \/ y e. (Unit ` R ) ) <-> ( x e. (Unit ` O ) \/ y e. (Unit ` O ) ) ) )
22	14, 21	imbi12d 234	. . . . . 6 |- ( R e. Ring -> ( ( ( x ( +g ` R ) y ) = ( 1r ` R ) -> ( x e. (Unit ` R ) \/ y e. (Unit ` R ) ) ) <-> ( ( x ( +g ` O ) y ) = ( 1r ` O ) -> ( x e. (Unit ` O ) \/ y e. (Unit ` O ) ) ) ) )
23	8, 22	raleqbidv 2765	. . . . 5 |- ( R e. Ring -> ( A. y e. ( Base ` R ) ( ( x ( +g ` R ) y ) = ( 1r ` R ) -> ( x e. (Unit ` R ) \/ y e. (Unit ` R ) ) ) <-> A. y e. ( Base ` O ) ( ( x ( +g ` O ) y ) = ( 1r ` O ) -> ( x e. (Unit ` O ) \/ y e. (Unit ` O ) ) ) ) )
24	8, 23	raleqbidv 2765	. . . 4 |- ( R e. Ring -> ( A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( +g ` R ) y ) = ( 1r ` R ) -> ( x e. (Unit ` R ) \/ y e. (Unit ` R ) ) ) <-> A. x e. ( Base ` O ) A. y e. ( Base ` O ) ( ( x ( +g ` O ) y ) = ( 1r ` O ) -> ( x e. (Unit ` O ) \/ y e. (Unit ` O ) ) ) ) )
25	6, 24	anbi12d 477	. . 3 |- ( R e. Ring -> ( ( R e. NzRing /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( +g ` R ) y ) = ( 1r ` R ) -> ( x e. (Unit ` R ) \/ y e. (Unit ` R ) ) ) ) <-> ( O e. NzRing /\ A. x e. ( Base ` O ) A. y e. ( Base ` O ) ( ( x ( +g ` O ) y ) = ( 1r ` O ) -> ( x e. (Unit ` O ) \/ y e. (Unit ` O ) ) ) ) ) )
26		eqid 2238	. . . 4 |- (Unit ` R ) = (Unit ` R )
27	7, 9, 12, 26	islring 14482	. . 3 |- ( R e. LRing <-> ( R e. NzRing /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( +g ` R ) y ) = ( 1r ` R ) -> ( x e. (Unit ` R ) \/ y e. (Unit ` R ) ) ) ) )
28		eqid 2238	. . . 4 |- ( Base ` O ) = ( Base ` O )
29		eqid 2238	. . . 4 |- ( +g ` O ) = ( +g ` O )
30		eqid 2238	. . . 4 |- ( 1r ` O ) = ( 1r ` O )
31		eqid 2238	. . . 4 |- (Unit ` O ) = (Unit ` O )
32	28, 29, 30, 31	islring 14482	. . 3 |- ( O e. LRing <-> ( O e. NzRing /\ A. x e. ( Base ` O ) A. y e. ( Base ` O ) ( ( x ( +g ` O ) y ) = ( 1r ` O ) -> ( x e. (Unit ` O ) \/ y e. (Unit ` O ) ) ) ) )
33	25, 27, 32	3bitr4g 223	. 2 |- ( R e. Ring -> ( R e. LRing <-> O e. LRing ) )
34	1, 5, 33	pm5.21nii 716	1 |- ( R e. LRing <-> O e. LRing )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   A.wral 2528   ` cfv 5375  (class class class)co 6079   Basecbs 13335   +g cplusg 13414   1rcur 14245   Ringcrg 14283  opp_rcoppr 14355  Unitcui 14376  NzRingcnzr 14469  LRingclring 14480

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-pre-ltirr 8285  ax-pre-lttrn 8287  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-tpos 6510  df-pnf 8356  df-mnf 8357  df-ltxr 8359  df-inn 9288  df-2 9346  df-3 9347  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-plusg 13427  df-mulr 13428  df-0g 13595  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-grp 13791  df-minusg 13792  df-cmn 14072  df-abl 14073  df-mgp 14201  df-ur 14246  df-srg 14251  df-ring 14285  df-oppr 14356  df-dvdsr 14378  df-unit 14379  df-nzr 14470  df-lring 14481

This theorem is referenced by:  opprdrng  14603