Theorem aprcotr 14047

Description: The apartness relation given by df-apr 14043 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 2206	. . . 4 |- ( ( ph /\ X # Y ) -> (Unit ` R ) = (Unit ` R ) )
4		eqidd 2206	. . . 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 13956	. . . . . . . . 9 |- ( R e. LRing -> R e. Ring )
8	5, 7	syl 14	. . . . . . . 8 |- ( ph -> R e. Ring )
9	8	ringgrpd 13767	. . . . . . 7 |- ( ph -> R e. Grp )
10		aprcotr.x	. . . . . . . 8 |- ( ph -> X e. B )
11	10, 1	eleqtrd 2284	. . . . . . 7 |- ( ph -> X e. ( Base ` R ) )
12		aprcotr.z	. . . . . . . 8 |- ( ph -> Z e. B )
13	12, 1	eleqtrd 2284	. . . . . . 7 |- ( ph -> Z e. ( Base ` R ) )
14		aprcotr.y	. . . . . . . 8 |- ( ph -> Y e. B )
15	14, 1	eleqtrd 2284	. . . . . . 7 |- ( ph -> Y e. ( Base ` R ) )
16		eqid 2205	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
17		eqid 2205	. . . . . . . 8 |- ( +g ` R ) = ( +g ` R )
18		eqid 2205	. . . . . . . 8 |- ( -g ` R ) = ( -g ` R )
19	16, 17, 18	grpnpncan 13427	. . . . . . 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 1252	. . . . . 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 2206	. . . . . . 7 |- ( ph -> ( -g ` R ) = ( -g ` R ) )
24		eqidd 2206	. . . . . . 7 |- ( ph -> (Unit ` R ) = (Unit ` R ) )
25	1, 22, 23, 24, 8, 10, 14	aprval 14044	. . . . . 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 2282	. . . 4 |- ( ( ph /\ X # Y ) -> ( ( X ( -g ` R ) Z ) ( +g ` R ) ( Z ( -g ` R ) Y ) ) e. (Unit ` R ) )
28	16, 18	grpsubcl 13412	. . . . . . 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 1250	. . . . . 6 |- ( ph -> ( X ( -g ` R ) Z ) e. ( Base ` R ) )
30	29, 1	eleqtrrd 2285	. . . . 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 13412	. . . . . . 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 1250	. . . . . 6 |- ( ph -> ( Z ( -g ` R ) Y ) e. ( Base ` R ) )
34	33, 1	eleqtrrd 2285	. . . . 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 13958	. . 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 14044	. . . . . 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 14044	. . . . . 6 |- ( ph -> ( Z # Y <-> ( Z ( -g ` R ) Y ) e. (Unit ` R ) ) )
41	1, 22, 8, 12, 14	aprsym 14046	. . . . . 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 788	. . 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 710   = wceq 1373   e. wcel 2176   class class class wbr 4044   ` cfv 5271  (class class class)co 5944   Basecbs 12832   +g cplusg 12909   Grpcgrp 13332   -gcsg 13334   Ringcrg 13758  Unitcui 13849  LRingclring 13952  #_rcapr 14042

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-lttrn 8039  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-tpos 6331  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-3 9096  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-iress 12840  df-plusg 12922  df-mulr 12923  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-minusg 13336  df-sbg 13337  df-cmn 13622  df-abl 13623  df-mgp 13683  df-ur 13722  df-srg 13726  df-ring 13760  df-oppr 13830  df-dvdsr 13851  df-unit 13852  df-invr 13883  df-dvr 13894  df-nzr 13942  df-lring 13953  df-apr 14043

This theorem is referenced by:  aprap  14048