Theorem lringuplu 14341

Description: If the sum of two elements of a local ring is invertible, then at least one of the summands must be invertible. (Contributed by Jim Kingdon, 18-Feb-2025.) (Revised by SN, 23-Feb-2025.)

Hypotheses

Ref	Expression
lring.b	|- ( ph -> B = ( Base ` R ) )
lring.u	|- ( ph -> U = (Unit ` R ) )
lring.p	|- ( ph -> .+ = ( +g ` R ) )
lring.l	|- ( ph -> R e. LRing )
lring.s	|- ( ph -> ( X .+ Y ) e. U )
lring.x	|- ( ph -> X e. B )
lring.y	|- ( ph -> Y e. B )

Assertion

Ref	Expression
lringuplu	|- ( ph -> ( X e. U \/ Y e. U ) )

Proof of Theorem lringuplu

Dummy variables u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lring.l	. . . . . . . 8 |- ( ph -> R e. LRing )
2		lringring 14339	. . . . . . . 8 |- ( R e. LRing -> R e. Ring )
3	1, 2	syl 14	. . . . . . 7 |- ( ph -> R e. Ring )
4		lring.x	. . . . . . . 8 |- ( ph -> X e. B )
5		lring.b	. . . . . . . 8 |- ( ph -> B = ( Base ` R ) )
6	4, 5	eleqtrd 2311	. . . . . . 7 |- ( ph -> X e. ( Base ` R ) )
7		lring.s	. . . . . . . 8 |- ( ph -> ( X .+ Y ) e. U )
8		lring.u	. . . . . . . 8 |- ( ph -> U = (Unit ` R ) )
9	7, 8	eleqtrd 2311	. . . . . . 7 |- ( ph -> ( X .+ Y ) e. (Unit ` R ) )
10		eqid 2232	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
11		eqid 2232	. . . . . . . 8 |- (Unit ` R ) = (Unit ` R )
12		eqid 2232	. . . . . . . 8 |- (/r ` R ) = (/r ` R )
13		eqid 2232	. . . . . . . 8 |- ( .r ` R ) = ( .r ` R )
14	10, 11, 12, 13	dvrcan1 14285	. . . . . . 7 |- ( ( R e. Ring /\ X e. ( Base ` R ) /\ ( X .+ Y ) e. (Unit ` R ) ) -> ( ( X (/r ` R ) ( X .+ Y ) ) ( .r ` R ) ( X .+ Y ) ) = X )
15	3, 6, 9, 14	syl3anc 1274	. . . . . 6 |- ( ph -> ( ( X (/r ` R ) ( X .+ Y ) ) ( .r ` R ) ( X .+ Y ) ) = X )
16	15	adantr 276	. . . . 5 |- ( ( ph /\ ( X (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) -> ( ( X (/r ` R ) ( X .+ Y ) ) ( .r ` R ) ( X .+ Y ) ) = X )
17	3	adantr 276	. . . . . 6 |- ( ( ph /\ ( X (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) -> R e. Ring )
18		simpr 110	. . . . . 6 |- ( ( ph /\ ( X (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) -> ( X (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) )
19	9	adantr 276	. . . . . 6 |- ( ( ph /\ ( X (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) -> ( X .+ Y ) e. (Unit ` R ) )
20	11, 13	unitmulcl 14258	. . . . . 6 |- ( ( R e. Ring /\ ( X (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) /\ ( X .+ Y ) e. (Unit ` R ) ) -> ( ( X (/r ` R ) ( X .+ Y ) ) ( .r ` R ) ( X .+ Y ) ) e. (Unit ` R ) )
21	17, 18, 19, 20	syl3anc 1274	. . . . 5 |- ( ( ph /\ ( X (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) -> ( ( X (/r ` R ) ( X .+ Y ) ) ( .r ` R ) ( X .+ Y ) ) e. (Unit ` R ) )
22	16, 21	eqeltrrd 2310	. . . 4 |- ( ( ph /\ ( X (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) -> X e. (Unit ` R ) )
23	8	adantr 276	. . . 4 |- ( ( ph /\ ( X (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) -> U = (Unit ` R ) )
24	22, 23	eleqtrrd 2312	. . 3 |- ( ( ph /\ ( X (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) -> X e. U )
25	24	orcd 741	. 2 |- ( ( ph /\ ( X (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) -> ( X e. U \/ Y e. U ) )
26		lring.y	. . . . . . . 8 |- ( ph -> Y e. B )
27	26, 5	eleqtrd 2311	. . . . . . 7 |- ( ph -> Y e. ( Base ` R ) )
28	10, 11, 12, 13	dvrcan1 14285	. . . . . . 7 |- ( ( R e. Ring /\ Y e. ( Base ` R ) /\ ( X .+ Y ) e. (Unit ` R ) ) -> ( ( Y (/r ` R ) ( X .+ Y ) ) ( .r ` R ) ( X .+ Y ) ) = Y )
29	3, 27, 9, 28	syl3anc 1274	. . . . . 6 |- ( ph -> ( ( Y (/r ` R ) ( X .+ Y ) ) ( .r ` R ) ( X .+ Y ) ) = Y )
30	29	adantr 276	. . . . 5 |- ( ( ph /\ ( Y (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) -> ( ( Y (/r ` R ) ( X .+ Y ) ) ( .r ` R ) ( X .+ Y ) ) = Y )
31	3	adantr 276	. . . . . 6 |- ( ( ph /\ ( Y (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) -> R e. Ring )
32		simpr 110	. . . . . 6 |- ( ( ph /\ ( Y (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) -> ( Y (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) )
33	9	adantr 276	. . . . . 6 |- ( ( ph /\ ( Y (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) -> ( X .+ Y ) e. (Unit ` R ) )
34	11, 13	unitmulcl 14258	. . . . . 6 |- ( ( R e. Ring /\ ( Y (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) /\ ( X .+ Y ) e. (Unit ` R ) ) -> ( ( Y (/r ` R ) ( X .+ Y ) ) ( .r ` R ) ( X .+ Y ) ) e. (Unit ` R ) )
35	31, 32, 33, 34	syl3anc 1274	. . . . 5 |- ( ( ph /\ ( Y (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) -> ( ( Y (/r ` R ) ( X .+ Y ) ) ( .r ` R ) ( X .+ Y ) ) e. (Unit ` R ) )
36	30, 35	eqeltrrd 2310	. . . 4 |- ( ( ph /\ ( Y (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) -> Y e. (Unit ` R ) )
37	8	adantr 276	. . . 4 |- ( ( ph /\ ( Y (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) -> U = (Unit ` R ) )
38	36, 37	eleqtrrd 2312	. . 3 |- ( ( ph /\ ( Y (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) -> Y e. U )
39	38	olcd 742	. 2 |- ( ( ph /\ ( Y (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) -> ( X e. U \/ Y e. U ) )
40		eqid 2232	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
41	10, 11, 40, 12	dvrdir 14288	. . . . 5 |- ( ( R e. Ring /\ ( X e. ( Base ` R ) /\ Y e. ( Base ` R ) /\ ( X .+ Y ) e. (Unit ` R ) ) ) -> ( ( X ( +g ` R ) Y ) (/r ` R ) ( X .+ Y ) ) = ( ( X (/r ` R ) ( X .+ Y ) ) ( +g ` R ) ( Y (/r ` R ) ( X .+ Y ) ) ) )
42	3, 6, 27, 9, 41	syl13anc 1276	. . . 4 |- ( ph -> ( ( X ( +g ` R ) Y ) (/r ` R ) ( X .+ Y ) ) = ( ( X (/r ` R ) ( X .+ Y ) ) ( +g ` R ) ( Y (/r ` R ) ( X .+ Y ) ) ) )
43		lring.p	. . . . . . 7 |- ( ph -> .+ = ( +g ` R ) )
44	43	eqcomd 2238	. . . . . 6 |- ( ph -> ( +g ` R ) = .+ )
45	44	oveqd 6067	. . . . 5 |- ( ph -> ( X ( +g ` R ) Y ) = ( X .+ Y ) )
46	3	ringgrpd 14149	. . . . . . 7 |- ( ph -> R e. Grp )
47	10, 40, 46, 6, 27	grpcld 13727	. . . . . 6 |- ( ph -> ( X ( +g ` R ) Y ) e. ( Base ` R ) )
48		eqid 2232	. . . . . . 7 |- ( 1r ` R ) = ( 1r ` R )
49	10, 11, 12, 48	dvreq1 14287	. . . . . 6 |- ( ( R e. Ring /\ ( X ( +g ` R ) Y ) e. ( Base ` R ) /\ ( X .+ Y ) e. (Unit ` R ) ) -> ( ( ( X ( +g ` R ) Y ) (/r ` R ) ( X .+ Y ) ) = ( 1r ` R ) <-> ( X ( +g ` R ) Y ) = ( X .+ Y ) ) )
50	3, 47, 9, 49	syl3anc 1274	. . . . 5 |- ( ph -> ( ( ( X ( +g ` R ) Y ) (/r ` R ) ( X .+ Y ) ) = ( 1r ` R ) <-> ( X ( +g ` R ) Y ) = ( X .+ Y ) ) )
51	45, 50	mpbird 167	. . . 4 |- ( ph -> ( ( X ( +g ` R ) Y ) (/r ` R ) ( X .+ Y ) ) = ( 1r ` R ) )
52	42, 51	eqtr3d 2267	. . 3 |- ( ph -> ( ( X (/r ` R ) ( X .+ Y ) ) ( +g ` R ) ( Y (/r ` R ) ( X .+ Y ) ) ) = ( 1r ` R ) )
53		oveq2 6058	. . . . . 6 |- ( v = ( Y (/r ` R ) ( X .+ Y ) ) -> ( ( X (/r ` R ) ( X .+ Y ) ) ( +g ` R ) v ) = ( ( X (/r ` R ) ( X .+ Y ) ) ( +g ` R ) ( Y (/r ` R ) ( X .+ Y ) ) ) )
54	53	eqeq1d 2241	. . . . 5 |- ( v = ( Y (/r ` R ) ( X .+ Y ) ) -> ( ( ( X (/r ` R ) ( X .+ Y ) ) ( +g ` R ) v ) = ( 1r ` R ) <-> ( ( X (/r ` R ) ( X .+ Y ) ) ( +g ` R ) ( Y (/r ` R ) ( X .+ Y ) ) ) = ( 1r ` R ) ) )
55		eleq1 2295	. . . . . 6 |- ( v = ( Y (/r ` R ) ( X .+ Y ) ) -> ( v e. (Unit ` R ) <-> ( Y (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) )
56	55	orbi2d 798	. . . . 5 |- ( v = ( Y (/r ` R ) ( X .+ Y ) ) -> ( ( ( X (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) \/ v e. (Unit ` R ) ) <-> ( ( X (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) \/ ( Y (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) ) )
57	54, 56	imbi12d 234	. . . 4 |- ( v = ( Y (/r ` R ) ( X .+ Y ) ) -> ( ( ( ( X (/r ` R ) ( X .+ Y ) ) ( +g ` R ) v ) = ( 1r ` R ) -> ( ( X (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) \/ v e. (Unit ` R ) ) ) <-> ( ( ( X (/r ` R ) ( X .+ Y ) ) ( +g ` R ) ( Y (/r ` R ) ( X .+ Y ) ) ) = ( 1r ` R ) -> ( ( X (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) \/ ( Y (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) ) ) )
58		oveq1 6057	. . . . . . . 8 |- ( u = ( X (/r ` R ) ( X .+ Y ) ) -> ( u ( +g ` R ) v ) = ( ( X (/r ` R ) ( X .+ Y ) ) ( +g ` R ) v ) )
59	58	eqeq1d 2241	. . . . . . 7 |- ( u = ( X (/r ` R ) ( X .+ Y ) ) -> ( ( u ( +g ` R ) v ) = ( 1r ` R ) <-> ( ( X (/r ` R ) ( X .+ Y ) ) ( +g ` R ) v ) = ( 1r ` R ) ) )
60		eleq1 2295	. . . . . . . 8 |- ( u = ( X (/r ` R ) ( X .+ Y ) ) -> ( u e. (Unit ` R ) <-> ( X (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) )
61	60	orbi1d 799	. . . . . . 7 |- ( u = ( X (/r ` R ) ( X .+ Y ) ) -> ( ( u e. (Unit ` R ) \/ v e. (Unit ` R ) ) <-> ( ( X (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) \/ v e. (Unit ` R ) ) ) )
62	59, 61	imbi12d 234	. . . . . 6 |- ( u = ( X (/r ` R ) ( X .+ Y ) ) -> ( ( ( u ( +g ` R ) v ) = ( 1r ` R ) -> ( u e. (Unit ` R ) \/ v e. (Unit ` R ) ) ) <-> ( ( ( X (/r ` R ) ( X .+ Y ) ) ( +g ` R ) v ) = ( 1r ` R ) -> ( ( X (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) \/ v e. (Unit ` R ) ) ) ) )
63	62	ralbidv 2542	. . . . 5 |- ( u = ( X (/r ` R ) ( X .+ Y ) ) -> ( A. v e. ( Base ` R ) ( ( u ( +g ` R ) v ) = ( 1r ` R ) -> ( u e. (Unit ` R ) \/ v e. (Unit ` R ) ) ) <-> A. v e. ( Base ` R ) ( ( ( X (/r ` R ) ( X .+ Y ) ) ( +g ` R ) v ) = ( 1r ` R ) -> ( ( X (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) \/ v e. (Unit ` R ) ) ) ) )
64	10, 40, 48, 11	islring 14337	. . . . . . 7 |- ( R e. LRing <-> ( R e. NzRing /\ A. u e. ( Base ` R ) A. v e. ( Base ` R ) ( ( u ( +g ` R ) v ) = ( 1r ` R ) -> ( u e. (Unit ` R ) \/ v e. (Unit ` R ) ) ) ) )
65	1, 64	sylib 122	. . . . . 6 |- ( ph -> ( R e. NzRing /\ A. u e. ( Base ` R ) A. v e. ( Base ` R ) ( ( u ( +g ` R ) v ) = ( 1r ` R ) -> ( u e. (Unit ` R ) \/ v e. (Unit ` R ) ) ) ) )
66	65	simprd 114	. . . . 5 |- ( ph -> A. u e. ( Base ` R ) A. v e. ( Base ` R ) ( ( u ( +g ` R ) v ) = ( 1r ` R ) -> ( u e. (Unit ` R ) \/ v e. (Unit ` R ) ) ) )
67	10, 11, 12	dvrcl 14280	. . . . . 6 |- ( ( R e. Ring /\ X e. ( Base ` R ) /\ ( X .+ Y ) e. (Unit ` R ) ) -> ( X (/r ` R ) ( X .+ Y ) ) e. ( Base ` R ) )
68	3, 6, 9, 67	syl3anc 1274	. . . . 5 |- ( ph -> ( X (/r ` R ) ( X .+ Y ) ) e. ( Base ` R ) )
69	63, 66, 68	rspcdva 2926	. . . 4 |- ( ph -> A. v e. ( Base ` R ) ( ( ( X (/r ` R ) ( X .+ Y ) ) ( +g ` R ) v ) = ( 1r ` R ) -> ( ( X (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) \/ v e. (Unit ` R ) ) ) )
70	10, 11, 12	dvrcl 14280	. . . . 5 |- ( ( R e. Ring /\ Y e. ( Base ` R ) /\ ( X .+ Y ) e. (Unit ` R ) ) -> ( Y (/r ` R ) ( X .+ Y ) ) e. ( Base ` R ) )
71	3, 27, 9, 70	syl3anc 1274	. . . 4 |- ( ph -> ( Y (/r ` R ) ( X .+ Y ) ) e. ( Base ` R ) )
72	57, 69, 71	rspcdva 2926	. . 3 |- ( ph -> ( ( ( X (/r ` R ) ( X .+ Y ) ) ( +g ` R ) ( Y (/r ` R ) ( X .+ Y ) ) ) = ( 1r ` R ) -> ( ( X (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) \/ ( Y (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) ) )
73	52, 72	mpd 13	. 2 |- ( ph -> ( ( X (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) \/ ( Y (/r ` R ) ( X .+ Y ) ) e. (Unit ` R ) ) )
74	25, 39, 73	mpjaodan 806	1 |- ( ph -> ( X e. U \/ Y e. U ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2203   A.wral 2520   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   .rcmulr 13291   1rcur 14103   Ringcrg 14140  Unitcui 14231  /_rcdvr 14276  NzRingcnzr 14324  LRingclring 14335

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 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-lttrn 8241  ax-pre-ltadd 8243

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 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-tpos 6476  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220  df-plusg 13303  df-mulr 13304  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-minusg 13717  df-cmn 14003  df-abl 14004  df-mgp 14065  df-ur 14104  df-srg 14108  df-ring 14142  df-oppr 14212  df-dvdsr 14233  df-unit 14234  df-invr 14266  df-dvr 14277  df-nzr 14325  df-lring 14336

This theorem is referenced by:  aprcotr  14431