Theorem aprap 14421

Description: The relation given by df-apr 14416 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 14416	. . . 4 ⊢ #r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ∧ (𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟))})
2		fveq2 5669	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3	2	eleq2d 2302	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↔ 𝑥 ∈ (Base‘𝑅)))
4	2	eleq2d 2302	. . . . . . 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 5669	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (-g‘𝑟) = (-g‘𝑅))
7	6	oveqd 6066	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑥(-g‘𝑟)𝑦) = (𝑥(-g‘𝑅)𝑦))
8		fveq2 5669	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
9	7, 8	eleq12d 2303	. . . . . 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 4175	. . . 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 2824	. . . 4 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ V)
13		basfn 13260	. . . . . . . 8 ⊢ Base Fn V
14	13	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ LRing → Base Fn V)
15		funfvex 5686	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
16	15	funfni 5457	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
17	14, 12, 16	syl2anc 411	. . . . . 6 ⊢ (𝑅 ∈ LRing → (Base‘𝑅) ∈ V)
18		xpexg 4863	. . . . . 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 4823	. . . . . 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 4249	. . . 4 ⊢ (𝑅 ∈ LRing → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))} ∈ V)
23	1, 11, 12, 22	fvmptd3 5770	. . 3 ⊢ (𝑅 ∈ LRing → (#r‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))})
24	23, 20	eqsstrdi 3289	. 2 ⊢ (𝑅 ∈ LRing → (#r‘𝑅) ⊆ ((Base‘𝑅) × (Base‘𝑅)))
25		eqidd 2233	. . . 4 ⊢ ((𝑅 ∈ LRing ∧ 𝑥 ∈ (Base‘𝑅)) → (Base‘𝑅) = (Base‘𝑅))
26		eqidd 2233	. . . 4 ⊢ ((𝑅 ∈ LRing ∧ 𝑥 ∈ (Base‘𝑅)) → (#r‘𝑅) = (#r‘𝑅))
27		lringring 14328	. . . . 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 2232	. . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅)
31		eqid 2232	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
32	30, 31	lringnz 14329	. . . . 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 14418	. . 3 ⊢ ((𝑅 ∈ LRing ∧ 𝑥 ∈ (Base‘𝑅)) → ¬ 𝑥(#r‘𝑅)𝑥)
35	34	ralrimiva 2615	. 2 ⊢ (𝑅 ∈ LRing → ∀𝑥 ∈ (Base‘𝑅) ¬ 𝑥(#r‘𝑅)𝑥)
36		eqidd 2233	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (Base‘𝑅) = (Base‘𝑅))
37		eqidd 2233	. . . . 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 14419	. . . 4 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(#r‘𝑅)𝑦 → 𝑦(#r‘𝑅)𝑥))
42	41	ralrimivva 2624	. . 3 ⊢ (𝑅 ∈ LRing → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝑥(#r‘𝑅)𝑦 → 𝑦(#r‘𝑅)𝑥))
43		eqidd 2233	. . . . 5 ⊢ ((𝑅 ∈ LRing ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (Base‘𝑅) = (Base‘𝑅))
44		eqidd 2233	. . . . 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 14420	. . . 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 2625	. . 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 7558	. 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 2203   ≠ wne 2412  ∀wral 2520  Vcvv 2812   ⊆ wss 3210   class class class wbr 4108  {copab 4169   × cxp 4746   Fn wfn 5346  ‘cfv 5351  (class class class)co 6049   Ap wap 7557  Basecbs 13201  0_gc0g 13458  -_gcsg 13704  1_rcur 14092  Ringcrg 14129  Unitcui 14220  LRingclring 14324  #_rcapr 14415

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-addass 8225  ax-i2m1 8228  ax-0lt1 8229  ax-0id 8231  ax-rnegex 8232  ax-pre-ltirr 8235  ax-pre-lttrn 8237  ax-pre-ltadd 8239

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-tpos 6475  df-pap 7558  df-pnf 8306  df-mnf 8307  df-ltxr 8309  df-inn 9234  df-2 9292  df-3 9293  df-ndx 13204  df-slot 13205  df-base 13207  df-sets 13208  df-iress 13209  df-plusg 13292  df-mulr 13293  df-0g 13460  df-mgm 13558  df-sgrp 13604  df-mnd 13619  df-grp 13705  df-minusg 13706  df-sbg 13707  df-cmn 13992  df-abl 13993  df-mgp 14054  df-ur 14093  df-srg 14097  df-ring 14131  df-oppr 14201  df-dvdsr 14222  df-unit 14223  df-invr 14255  df-dvr 14266  df-nzr 14314  df-lring 14325  df-apr 14416

This theorem is referenced by: (None)