Theorem aprap 13918

Description: The relation given by df-apr 13913 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 13913	. . . 4 ⊢ #r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ∧ (𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟))})
2		fveq2 5561	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3	2	eleq2d 2266	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↔ 𝑥 ∈ (Base‘𝑅)))
4	2	eleq2d 2266	. . . . . . 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 5561	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (-g‘𝑟) = (-g‘𝑅))
7	6	oveqd 5942	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑥(-g‘𝑟)𝑦) = (𝑥(-g‘𝑅)𝑦))
8		fveq2 5561	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
9	7, 8	eleq12d 2267	. . . . . 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 4100	. . . 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 2774	. . . 4 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ V)
13		basfn 12761	. . . . . . . 8 ⊢ Base Fn V
14	13	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ LRing → Base Fn V)
15		funfvex 5578	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
16	15	funfni 5361	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
17	14, 12, 16	syl2anc 411	. . . . . 6 ⊢ (𝑅 ∈ LRing → (Base‘𝑅) ∈ V)
18		xpexg 4778	. . . . . 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 4738	. . . . . 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 4174	. . . 4 ⊢ (𝑅 ∈ LRing → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))} ∈ V)
23	1, 11, 12, 22	fvmptd3 5658	. . 3 ⊢ (𝑅 ∈ LRing → (#r‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))})
24	23, 20	eqsstrdi 3236	. 2 ⊢ (𝑅 ∈ LRing → (#r‘𝑅) ⊆ ((Base‘𝑅) × (Base‘𝑅)))
25		eqidd 2197	. . . 4 ⊢ ((𝑅 ∈ LRing ∧ 𝑥 ∈ (Base‘𝑅)) → (Base‘𝑅) = (Base‘𝑅))
26		eqidd 2197	. . . 4 ⊢ ((𝑅 ∈ LRing ∧ 𝑥 ∈ (Base‘𝑅)) → (#r‘𝑅) = (#r‘𝑅))
27		lringring 13826	. . . . 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 2196	. . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅)
31		eqid 2196	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
32	30, 31	lringnz 13827	. . . . 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 13915	. . 3 ⊢ ((𝑅 ∈ LRing ∧ 𝑥 ∈ (Base‘𝑅)) → ¬ 𝑥(#r‘𝑅)𝑥)
35	34	ralrimiva 2570	. 2 ⊢ (𝑅 ∈ LRing → ∀𝑥 ∈ (Base‘𝑅) ¬ 𝑥(#r‘𝑅)𝑥)
36		eqidd 2197	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (Base‘𝑅) = (Base‘𝑅))
37		eqidd 2197	. . . . 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 529	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑥 ∈ (Base‘𝑅))
40		simprr 531	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑦 ∈ (Base‘𝑅))
41	36, 37, 38, 39, 40	aprsym 13916	. . . 4 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(#r‘𝑅)𝑦 → 𝑦(#r‘𝑅)𝑥))
42	41	ralrimivva 2579	. . 3 ⊢ (𝑅 ∈ LRing → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝑥(#r‘𝑅)𝑦 → 𝑦(#r‘𝑅)𝑥))
43		eqidd 2197	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (Base‘𝑅) = (Base‘𝑅))
44		eqidd 2197	. . . . 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 1005	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑥 ∈ (Base‘𝑅))
47		simpr2 1006	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑦 ∈ (Base‘𝑅))
48		simpr3 1007	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑧 ∈ (Base‘𝑅))
49	43, 44, 45, 46, 47, 48	aprcotr 13917	. . . 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 2580	. . 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 7331	. 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 1184	1 ⊢ (𝑅 ∈ LRing → (#r‘𝑅) Ap (Base‘𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709   ∧ w3a 980   = wceq 1364   ∈ wcel 2167   ≠ wne 2367  ∀wral 2475  Vcvv 2763   ⊆ wss 3157   class class class wbr 4034  {copab 4094   × cxp 4662   Fn wfn 5254  ‘cfv 5259  (class class class)co 5925   Ap wap 7330  Basecbs 12703  0_gc0g 12958  -_gcsg 13204  1_rcur 13591  Ringcrg 13628  Unitcui 13719  LRingclring 13822  #_rcapr 13912

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-lttrn 8010  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-tpos 6312  df-pap 7331  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-3 9067  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-plusg 12793  df-mulr 12794  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-minusg 13206  df-sbg 13207  df-cmn 13492  df-abl 13493  df-mgp 13553  df-ur 13592  df-srg 13596  df-ring 13630  df-oppr 13700  df-dvdsr 13721  df-unit 13722  df-invr 13753  df-dvr 13764  df-nzr 13812  df-lring 13823  df-apr 13913

This theorem is referenced by: (None)