Theorem aprap 14293

Description: The relation given by df-apr 14288 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 14288	. . . 4 ⊢ #r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ∧ (𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟))})
2		fveq2 5635	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3	2	eleq2d 2299	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↔ 𝑥 ∈ (Base‘𝑅)))
4	2	eleq2d 2299	. . . . . . 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 5635	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (-g‘𝑟) = (-g‘𝑅))
7	6	oveqd 6030	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑥(-g‘𝑟)𝑦) = (𝑥(-g‘𝑅)𝑦))
8		fveq2 5635	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
9	7, 8	eleq12d 2300	. . . . . 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 4153	. . . 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 2812	. . . 4 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ V)
13		basfn 13134	. . . . . . . 8 ⊢ Base Fn V
14	13	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ LRing → Base Fn V)
15		funfvex 5652	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
16	15	funfni 5429	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
17	14, 12, 16	syl2anc 411	. . . . . 6 ⊢ (𝑅 ∈ LRing → (Base‘𝑅) ∈ V)
18		xpexg 4838	. . . . . 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 4798	. . . . . 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 4227	. . . 4 ⊢ (𝑅 ∈ LRing → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))} ∈ V)
23	1, 11, 12, 22	fvmptd3 5736	. . 3 ⊢ (𝑅 ∈ LRing → (#r‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))})
24	23, 20	eqsstrdi 3277	. 2 ⊢ (𝑅 ∈ LRing → (#r‘𝑅) ⊆ ((Base‘𝑅) × (Base‘𝑅)))
25		eqidd 2230	. . . 4 ⊢ ((𝑅 ∈ LRing ∧ 𝑥 ∈ (Base‘𝑅)) → (Base‘𝑅) = (Base‘𝑅))
26		eqidd 2230	. . . 4 ⊢ ((𝑅 ∈ LRing ∧ 𝑥 ∈ (Base‘𝑅)) → (#r‘𝑅) = (#r‘𝑅))
27		lringring 14201	. . . . 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 2229	. . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅)
31		eqid 2229	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
32	30, 31	lringnz 14202	. . . . 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 14290	. . 3 ⊢ ((𝑅 ∈ LRing ∧ 𝑥 ∈ (Base‘𝑅)) → ¬ 𝑥(#r‘𝑅)𝑥)
35	34	ralrimiva 2603	. 2 ⊢ (𝑅 ∈ LRing → ∀𝑥 ∈ (Base‘𝑅) ¬ 𝑥(#r‘𝑅)𝑥)
36		eqidd 2230	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (Base‘𝑅) = (Base‘𝑅))
37		eqidd 2230	. . . . 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 14291	. . . 4 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(#r‘𝑅)𝑦 → 𝑦(#r‘𝑅)𝑥))
42	41	ralrimivva 2612	. . 3 ⊢ (𝑅 ∈ LRing → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝑥(#r‘𝑅)𝑦 → 𝑦(#r‘𝑅)𝑥))
43		eqidd 2230	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (Base‘𝑅) = (Base‘𝑅))
44		eqidd 2230	. . . . 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 1027	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑥 ∈ (Base‘𝑅))
47		simpr2 1028	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑦 ∈ (Base‘𝑅))
48		simpr3 1029	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑧 ∈ (Base‘𝑅))
49	43, 44, 45, 46, 47, 48	aprcotr 14292	. . . 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 2613	. . 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 7460	. 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 1206	1 ⊢ (𝑅 ∈ LRing → (#r‘𝑅) Ap (Base‘𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 713   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200   ≠ wne 2400  ∀wral 2508  Vcvv 2800   ⊆ wss 3198   class class class wbr 4086  {copab 4147   × cxp 4721   Fn wfn 5319  ‘cfv 5324  (class class class)co 6013   Ap wap 7459  Basecbs 13075  0_gc0g 13332  -_gcsg 13578  1_rcur 13965  Ringcrg 14002  Unitcui 14093  LRingclring 14197  #_rcapr 14287

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-addass 8127  ax-i2m1 8130  ax-0lt1 8131  ax-0id 8133  ax-rnegex 8134  ax-pre-ltirr 8137  ax-pre-lttrn 8139  ax-pre-ltadd 8141

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-tpos 6406  df-pap 7460  df-pnf 8209  df-mnf 8210  df-ltxr 8212  df-inn 9137  df-2 9195  df-3 9196  df-ndx 13078  df-slot 13079  df-base 13081  df-sets 13082  df-iress 13083  df-plusg 13166  df-mulr 13167  df-0g 13334  df-mgm 13432  df-sgrp 13478  df-mnd 13493  df-grp 13579  df-minusg 13580  df-sbg 13581  df-cmn 13866  df-abl 13867  df-mgp 13927  df-ur 13966  df-srg 13970  df-ring 14004  df-oppr 14074  df-dvdsr 14095  df-unit 14096  df-invr 14128  df-dvr 14139  df-nzr 14187  df-lring 14198  df-apr 14288

This theorem is referenced by: (None)