Theorem aprcotr 14431

Description: The apartness relation given by df-apr 14427 for a local ring is cotransitive. (Contributed by Jim Kingdon, 17-Feb-2025.)

Hypotheses

Ref	Expression
aprcotr.b	|- ( ph -> B = ( Base ` R ) )
aprcotr.ap	|- ( ph -> # = (#r ` R ) )
aprcotr.r	|- ( ph -> R e. LRing )
aprcotr.x	|- ( ph -> X e. B )
aprcotr.y	|- ( ph -> Y e. B )
aprcotr.z	|- ( ph -> Z e. B )

Assertion

Ref	Expression
aprcotr	|- ( ph -> ( X # Y -> ( X # Z \/ Y # Z ) ) )

Proof of Theorem aprcotr

Step	Hyp	Ref	Expression
1		aprcotr.b	. . . . 5 |- ( ph -> B = ( Base ` R ) )
2	1	adantr 276	. . . 4 |- ( ( ph /\ X # Y ) -> B = ( Base ` R ) )
3		eqidd 2233	. . . 4 |- ( ( ph /\ X # Y ) -> (Unit ` R ) = (Unit ` R ) )
4		eqidd 2233	. . . 4 |- ( ( ph /\ X # Y ) -> ( +g ` R ) = ( +g ` R ) )
5		aprcotr.r	. . . . 5 |- ( ph -> R e. LRing )
6	5	adantr 276	. . . 4 |- ( ( ph /\ X # Y ) -> R e. LRing )
7		lringring 14339	. . . . . . . . 9 |- ( R e. LRing -> R e. Ring )
8	5, 7	syl 14	. . . . . . . 8 |- ( ph -> R e. Ring )
9	8	ringgrpd 14149	. . . . . . 7 |- ( ph -> R e. Grp )
10		aprcotr.x	. . . . . . . 8 |- ( ph -> X e. B )
11	10, 1	eleqtrd 2311	. . . . . . 7 |- ( ph -> X e. ( Base ` R ) )
12		aprcotr.z	. . . . . . . 8 |- ( ph -> Z e. B )
13	12, 1	eleqtrd 2311	. . . . . . 7 |- ( ph -> Z e. ( Base ` R ) )
14		aprcotr.y	. . . . . . . 8 |- ( ph -> Y e. B )
15	14, 1	eleqtrd 2311	. . . . . . 7 |- ( ph -> Y e. ( Base ` R ) )
16		eqid 2232	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
17		eqid 2232	. . . . . . . 8 |- ( +g ` R ) = ( +g ` R )
18		eqid 2232	. . . . . . . 8 |- ( -g ` R ) = ( -g ` R )
19	16, 17, 18	grpnpncan 13808	. . . . . . 7 |- ( ( R e. Grp /\ ( X e. ( Base ` R ) /\ Z e. ( Base ` R ) /\ Y e. ( Base ` R ) ) ) -> ( ( X ( -g ` R ) Z ) ( +g ` R ) ( Z ( -g ` R ) Y ) ) = ( X ( -g ` R ) Y ) )
20	9, 11, 13, 15, 19	syl13anc 1276	. . . . . 6 |- ( ph -> ( ( X ( -g ` R ) Z ) ( +g ` R ) ( Z ( -g ` R ) Y ) ) = ( X ( -g ` R ) Y ) )
21	20	adantr 276	. . . . 5 |- ( ( ph /\ X # Y ) -> ( ( X ( -g ` R ) Z ) ( +g ` R ) ( Z ( -g ` R ) Y ) ) = ( X ( -g ` R ) Y ) )
22		aprcotr.ap	. . . . . . 7 |- ( ph -> # = (#r ` R ) )
23		eqidd 2233	. . . . . . 7 |- ( ph -> ( -g ` R ) = ( -g ` R ) )
24		eqidd 2233	. . . . . . 7 |- ( ph -> (Unit ` R ) = (Unit ` R ) )
25	1, 22, 23, 24, 8, 10, 14	aprval 14428	. . . . . 6 |- ( ph -> ( X # Y <-> ( X ( -g ` R ) Y ) e. (Unit ` R ) ) )
26	25	biimpa 296	. . . . 5 |- ( ( ph /\ X # Y ) -> ( X ( -g ` R ) Y ) e. (Unit ` R ) )
27	21, 26	eqeltrd 2309	. . . 4 |- ( ( ph /\ X # Y ) -> ( ( X ( -g ` R ) Z ) ( +g ` R ) ( Z ( -g ` R ) Y ) ) e. (Unit ` R ) )
28	16, 18	grpsubcl 13793	. . . . . . 7 |- ( ( R e. Grp /\ X e. ( Base ` R ) /\ Z e. ( Base ` R ) ) -> ( X ( -g ` R ) Z ) e. ( Base ` R ) )
29	9, 11, 13, 28	syl3anc 1274	. . . . . 6 |- ( ph -> ( X ( -g ` R ) Z ) e. ( Base ` R ) )
30	29, 1	eleqtrrd 2312	. . . . 5 |- ( ph -> ( X ( -g ` R ) Z ) e. B )
31	30	adantr 276	. . . 4 |- ( ( ph /\ X # Y ) -> ( X ( -g ` R ) Z ) e. B )
32	16, 18	grpsubcl 13793	. . . . . . 7 |- ( ( R e. Grp /\ Z e. ( Base ` R ) /\ Y e. ( Base ` R ) ) -> ( Z ( -g ` R ) Y ) e. ( Base ` R ) )
33	9, 13, 15, 32	syl3anc 1274	. . . . . 6 |- ( ph -> ( Z ( -g ` R ) Y ) e. ( Base ` R ) )
34	33, 1	eleqtrrd 2312	. . . . 5 |- ( ph -> ( Z ( -g ` R ) Y ) e. B )
35	34	adantr 276	. . . 4 |- ( ( ph /\ X # Y ) -> ( Z ( -g ` R ) Y ) e. B )
36	2, 3, 4, 6, 27, 31, 35	lringuplu 14341	. . 3 |- ( ( ph /\ X # Y ) -> ( ( X ( -g ` R ) Z ) e. (Unit ` R ) \/ ( Z ( -g ` R ) Y ) e. (Unit ` R ) ) )
37	1, 22, 23, 24, 8, 10, 12	aprval 14428	. . . . . 6 |- ( ph -> ( X # Z <-> ( X ( -g ` R ) Z ) e. (Unit ` R ) ) )
38	37	biimprd 158	. . . . 5 |- ( ph -> ( ( X ( -g ` R ) Z ) e. (Unit ` R ) -> X # Z ) )
39	38	adantr 276	. . . 4 |- ( ( ph /\ X # Y ) -> ( ( X ( -g ` R ) Z ) e. (Unit ` R ) -> X # Z ) )
40	1, 22, 23, 24, 8, 12, 14	aprval 14428	. . . . . 6 |- ( ph -> ( Z # Y <-> ( Z ( -g ` R ) Y ) e. (Unit ` R ) ) )
41	1, 22, 8, 12, 14	aprsym 14430	. . . . . 6 |- ( ph -> ( Z # Y -> Y # Z ) )
42	40, 41	sylbird 170	. . . . 5 |- ( ph -> ( ( Z ( -g ` R ) Y ) e. (Unit ` R ) -> Y # Z ) )
43	42	adantr 276	. . . 4 |- ( ( ph /\ X # Y ) -> ( ( Z ( -g ` R ) Y ) e. (Unit ` R ) -> Y # Z ) )
44	39, 43	orim12d 794	. . 3 |- ( ( ph /\ X # Y ) -> ( ( ( X ( -g ` R ) Z ) e. (Unit ` R ) \/ ( Z ( -g ` R ) Y ) e. (Unit ` R ) ) -> ( X # Z \/ Y # Z ) ) )
45	36, 44	mpd 13	. 2 |- ( ( ph /\ X # Y ) -> ( X # Z \/ Y # Z ) )
46	45	ex 115	1 |- ( ph -> ( X # Y -> ( X # Z \/ Y # Z ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   = wceq 1398   e. wcel 2203   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   Grpcgrp 13713   -gcsg 13715   Ringcrg 14140  Unitcui 14231  LRingclring 14335  #_rcapr 14426

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-sbg 13718  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  df-apr 14427

This theorem is referenced by:  aprap  14432