Theorem aprap 14458

Description: The relation given by df-apr 14450 for a local ring is an apartness relation. (Contributed by Jim Kingdon, 20-Feb-2025.)

Assertion

Ref	Expression
aprap	⊢ (𝑅 ∈ LRing → (#r‘𝑅) Ap (Base‘𝑅))

Proof of Theorem aprap

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

Step	Hyp	Ref	Expression
1		df-apr 14450	. . . 4 ⊢ #r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ∧ (𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟))})
2		fveq2 5672	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3	2	eleq2d 2304	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↔ 𝑥 ∈ (Base‘𝑅)))
4	2	eleq2d 2304	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑦 ∈ (Base‘𝑟) ↔ 𝑦 ∈ (Base‘𝑅)))
5	3, 4	anbi12d 473	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ↔ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))))
6		fveq2 5672	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (-g‘𝑟) = (-g‘𝑅))
7	6	oveqd 6069	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑥(-g‘𝑟)𝑦) = (𝑥(-g‘𝑅)𝑦))
8		fveq2 5672	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
9	7, 8	eleq12d 2305	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟) ↔ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅)))
10	5, 9	anbi12d 473	. . . . 5 ⊢ (𝑟 = 𝑅 → (((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ∧ (𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟)) ↔ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))))
11	10	opabbidv 4178	. . . 4 ⊢ (𝑟 = 𝑅 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ∧ (𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))})
12		elex 2827	. . . 4 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ V)
13		basfn 13292	. . . . . . . 8 ⊢ Base Fn V
14	13	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ LRing → Base Fn V)
15		funfvex 5689	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
16	15	funfni 5460	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
17	14, 12, 16	syl2anc 411	. . . . . 6 ⊢ (𝑅 ∈ LRing → (Base‘𝑅) ∈ V)
18		xpexg 4866	. . . . . 6 ⊢ (((Base‘𝑅) ∈ V ∧ (Base‘𝑅) ∈ V) → ((Base‘𝑅) × (Base‘𝑅)) ∈ V)
19	17, 17, 18	syl2anc 411	. . . . 5 ⊢ (𝑅 ∈ LRing → ((Base‘𝑅) × (Base‘𝑅)) ∈ V)
20		opabssxp 4826	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))} ⊆ ((Base‘𝑅) × (Base‘𝑅))
21	20	a1i 9	. . . . 5 ⊢ (𝑅 ∈ LRing → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))} ⊆ ((Base‘𝑅) × (Base‘𝑅)))
22	19, 21	ssexd 4252	. . . 4 ⊢ (𝑅 ∈ LRing → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))} ∈ V)
23	1, 11, 12, 22	fvmptd3 5773	. . 3 ⊢ (𝑅 ∈ LRing → (#r‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))})
24	23, 20	eqsstrdi 3292	. 2 ⊢ (𝑅 ∈ LRing → (#r‘𝑅) ⊆ ((Base‘𝑅) × (Base‘𝑅)))
25		eqidd 2235	. . . 4 ⊢ ((𝑅 ∈ LRing ∧ 𝑥 ∈ (Base‘𝑅)) → (Base‘𝑅) = (Base‘𝑅))
26		eqidd 2235	. . . 4 ⊢ ((𝑅 ∈ LRing ∧ 𝑥 ∈ (Base‘𝑅)) → (#r‘𝑅) = (#r‘𝑅))
27		lringring 14361	. . . . 5 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ Ring)
28	27	adantr 276	. . . 4 ⊢ ((𝑅 ∈ LRing ∧ 𝑥 ∈ (Base‘𝑅)) → 𝑅 ∈ Ring)
29		simpr 110	. . . 4 ⊢ ((𝑅 ∈ LRing ∧ 𝑥 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
30		eqid 2234	. . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅)
31		eqid 2234	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
32	30, 31	lringnz 14362	. . . . 5 ⊢ (𝑅 ∈ LRing → (1r‘𝑅) ≠ (0g‘𝑅))
33	32	adantr 276	. . . 4 ⊢ ((𝑅 ∈ LRing ∧ 𝑥 ∈ (Base‘𝑅)) → (1r‘𝑅) ≠ (0g‘𝑅))
34	25, 26, 28, 29, 33	aprirr 14455	. . 3 ⊢ ((𝑅 ∈ LRing ∧ 𝑥 ∈ (Base‘𝑅)) → ¬ 𝑥(#r‘𝑅)𝑥)
35	34	ralrimiva 2617	. 2 ⊢ (𝑅 ∈ LRing → ∀𝑥 ∈ (Base‘𝑅) ¬ 𝑥(#r‘𝑅)𝑥)
36		eqidd 2235	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (Base‘𝑅) = (Base‘𝑅))
37		eqidd 2235	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (#r‘𝑅) = (#r‘𝑅))
38	27	adantr 276	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑅 ∈ Ring)
39		simprl 531	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑥 ∈ (Base‘𝑅))
40		simprr 533	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑦 ∈ (Base‘𝑅))
41	36, 37, 38, 39, 40	aprsym 14456	. . . 4 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(#r‘𝑅)𝑦 → 𝑦(#r‘𝑅)𝑥))
42	41	ralrimivva 2626	. . 3 ⊢ (𝑅 ∈ LRing → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝑥(#r‘𝑅)𝑦 → 𝑦(#r‘𝑅)𝑥))
43		eqidd 2235	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (Base‘𝑅) = (Base‘𝑅))
44		eqidd 2235	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (#r‘𝑅) = (#r‘𝑅))
45		simpl 109	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑅 ∈ LRing)
46		simpr1 1030	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑥 ∈ (Base‘𝑅))
47		simpr2 1031	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑦 ∈ (Base‘𝑅))
48		simpr3 1032	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑧 ∈ (Base‘𝑅))
49	43, 44, 45, 46, 47, 48	aprcotr 14457	. . . 4 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑥(#r‘𝑅)𝑦 → (𝑥(#r‘𝑅)𝑧 ∨ 𝑦(#r‘𝑅)𝑧)))
50	49	ralrimivvva 2627	. . 3 ⊢ (𝑅 ∈ LRing → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥(#r‘𝑅)𝑦 → (𝑥(#r‘𝑅)𝑧 ∨ 𝑦(#r‘𝑅)𝑧)))
51	42, 50	jca 306	. 2 ⊢ (𝑅 ∈ LRing → (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝑥(#r‘𝑅)𝑦 → 𝑦(#r‘𝑅)𝑥) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥(#r‘𝑅)𝑦 → (𝑥(#r‘𝑅)𝑧 ∨ 𝑦(#r‘𝑅)𝑧))))
52		df-pap 7561	. 2 ⊢ ((#r‘𝑅) Ap (Base‘𝑅) ↔ (((#r‘𝑅) ⊆ ((Base‘𝑅) × (Base‘𝑅)) ∧ ∀𝑥 ∈ (Base‘𝑅) ¬ 𝑥(#r‘𝑅)𝑥) ∧ (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝑥(#r‘𝑅)𝑦 → 𝑦(#r‘𝑅)𝑥) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥(#r‘𝑅)𝑦 → (𝑥(#r‘𝑅)𝑧 ∨ 𝑦(#r‘𝑅)𝑧)))))
53	24, 35, 51, 52	syl21anbrc 1209	1 ⊢ (𝑅 ∈ LRing → (#r‘𝑅) Ap (Base‘𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 716   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205   ≠ wne 2414  ∀wral 2522  Vcvv 2815   ⊆ wss 3213   class class class wbr 4111  {copab 4172   × cxp 4749   Fn wfn 5349  ‘cfv 5354  (class class class)co 6052   Ap wap 7560  Basecbs 13233  0_gc0g 13490  -_gcsg 13736  1_rcur 14124  Ringcrg 14161  Unitcui 14253  LRingclring 14357  #_rcapr 14449

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 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-addcom 8232  ax-addass 8234  ax-i2m1 8237  ax-0lt1 8238  ax-0id 8240  ax-rnegex 8241  ax-pre-ltirr 8244  ax-pre-lttrn 8246  ax-pre-ltadd 8248

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 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-tpos 6478  df-pap 7561  df-pnf 8315  df-mnf 8316  df-ltxr 8318  df-inn 9243  df-2 9301  df-3 9302  df-ndx 13236  df-slot 13237  df-base 13239  df-sets 13240  df-iress 13241  df-plusg 13324  df-mulr 13325  df-0g 13492  df-mgm 13590  df-sgrp 13636  df-mnd 13651  df-grp 13737  df-minusg 13738  df-sbg 13739  df-cmn 14024  df-abl 14025  df-mgp 14086  df-ur 14125  df-srg 14129  df-ring 14163  df-oppr 14233  df-dvdsr 14255  df-unit 14256  df-invr 14288  df-dvr 14299  df-nzr 14347  df-lring 14358  df-apr 14450

This theorem is referenced by:  aprlring  14460