Theorem aprap 14324

Description: The relation given by df-apr 14319 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 14319	. . . 4 ⊢ #r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ∧ (𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟))})
2		fveq2 5642	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3	2	eleq2d 2300	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↔ 𝑥 ∈ (Base‘𝑅)))
4	2	eleq2d 2300	. . . . . . 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 5642	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (-g‘𝑟) = (-g‘𝑅))
7	6	oveqd 6040	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑥(-g‘𝑟)𝑦) = (𝑥(-g‘𝑅)𝑦))
8		fveq2 5642	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
9	7, 8	eleq12d 2301	. . . . . 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 4156	. . . 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 2813	. . . 4 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ V)
13		basfn 13164	. . . . . . . 8 ⊢ Base Fn V
14	13	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ LRing → Base Fn V)
15		funfvex 5659	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
16	15	funfni 5434	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
17	14, 12, 16	syl2anc 411	. . . . . 6 ⊢ (𝑅 ∈ LRing → (Base‘𝑅) ∈ V)
18		xpexg 4842	. . . . . 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 4802	. . . . . 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 4230	. . . 4 ⊢ (𝑅 ∈ LRing → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))} ∈ V)
23	1, 11, 12, 22	fvmptd3 5743	. . 3 ⊢ (𝑅 ∈ LRing → (#r‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))})
24	23, 20	eqsstrdi 3278	. 2 ⊢ (𝑅 ∈ LRing → (#r‘𝑅) ⊆ ((Base‘𝑅) × (Base‘𝑅)))
25		eqidd 2231	. . . 4 ⊢ ((𝑅 ∈ LRing ∧ 𝑥 ∈ (Base‘𝑅)) → (Base‘𝑅) = (Base‘𝑅))
26		eqidd 2231	. . . 4 ⊢ ((𝑅 ∈ LRing ∧ 𝑥 ∈ (Base‘𝑅)) → (#r‘𝑅) = (#r‘𝑅))
27		lringring 14232	. . . . 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 2230	. . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅)
31		eqid 2230	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
32	30, 31	lringnz 14233	. . . . 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 14321	. . 3 ⊢ ((𝑅 ∈ LRing ∧ 𝑥 ∈ (Base‘𝑅)) → ¬ 𝑥(#r‘𝑅)𝑥)
35	34	ralrimiva 2604	. 2 ⊢ (𝑅 ∈ LRing → ∀𝑥 ∈ (Base‘𝑅) ¬ 𝑥(#r‘𝑅)𝑥)
36		eqidd 2231	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (Base‘𝑅) = (Base‘𝑅))
37		eqidd 2231	. . . . 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 14322	. . . 4 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(#r‘𝑅)𝑦 → 𝑦(#r‘𝑅)𝑥))
42	41	ralrimivva 2613	. . 3 ⊢ (𝑅 ∈ LRing → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝑥(#r‘𝑅)𝑦 → 𝑦(#r‘𝑅)𝑥))
43		eqidd 2231	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (Base‘𝑅) = (Base‘𝑅))
44		eqidd 2231	. . . . 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 1029	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑥 ∈ (Base‘𝑅))
47		simpr2 1030	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑦 ∈ (Base‘𝑅))
48		simpr3 1031	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑧 ∈ (Base‘𝑅))
49	43, 44, 45, 46, 47, 48	aprcotr 14323	. . . 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 2614	. . 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 7472	. 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 1208	1 ⊢ (𝑅 ∈ LRing → (#r‘𝑅) Ap (Base‘𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 715   ∧ w3a 1004   = wceq 1397   ∈ wcel 2201   ≠ wne 2401  ∀wral 2509  Vcvv 2801   ⊆ wss 3199   class class class wbr 4089  {copab 4150   × cxp 4725   Fn wfn 5323  ‘cfv 5328  (class class class)co 6023   Ap wap 7471  Basecbs 13105  0_gc0g 13362  -_gcsg 13608  1_rcur 13996  Ringcrg 14033  Unitcui 14124  LRingclring 14228  #_rcapr 14318

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-addass 8139  ax-i2m1 8142  ax-0lt1 8143  ax-0id 8145  ax-rnegex 8146  ax-pre-ltirr 8149  ax-pre-lttrn 8151  ax-pre-ltadd 8153

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-tpos 6416  df-pap 7472  df-pnf 8221  df-mnf 8222  df-ltxr 8224  df-inn 9149  df-2 9207  df-3 9208  df-ndx 13108  df-slot 13109  df-base 13111  df-sets 13112  df-iress 13113  df-plusg 13196  df-mulr 13197  df-0g 13364  df-mgm 13462  df-sgrp 13508  df-mnd 13523  df-grp 13609  df-minusg 13610  df-sbg 13611  df-cmn 13896  df-abl 13897  df-mgp 13958  df-ur 13997  df-srg 14001  df-ring 14035  df-oppr 14105  df-dvdsr 14126  df-unit 14127  df-invr 14159  df-dvr 14170  df-nzr 14218  df-lring 14229  df-apr 14319

This theorem is referenced by: (None)