Theorem reldvdsrsrg 13591

Description: The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2025.)

Assertion

Ref	Expression
reldvdsrsrg	⊢ (𝑅 ∈ SRing → Rel (∥r‘𝑅))

Proof of Theorem reldvdsrsrg

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

Step	Hyp	Ref	Expression
1		df-dvdsr 13588	. . . . 5 ⊢ ∥r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)})
2		fveq2 5555	. . . . . . . 8 ⊢ (𝑤 = 𝑅 → (Base‘𝑤) = (Base‘𝑅))
3	2	eleq2d 2263	. . . . . . 7 ⊢ (𝑤 = 𝑅 → (𝑥 ∈ (Base‘𝑤) ↔ 𝑥 ∈ (Base‘𝑅)))
4		fveq2 5555	. . . . . . . . . 10 ⊢ (𝑤 = 𝑅 → (.r‘𝑤) = (.r‘𝑅))
5	4	oveqd 5936	. . . . . . . . 9 ⊢ (𝑤 = 𝑅 → (𝑧(.r‘𝑤)𝑥) = (𝑧(.r‘𝑅)𝑥))
6	5	eqeq1d 2202	. . . . . . . 8 ⊢ (𝑤 = 𝑅 → ((𝑧(.r‘𝑤)𝑥) = 𝑦 ↔ (𝑧(.r‘𝑅)𝑥) = 𝑦))
7	2, 6	rexeqbidv 2707	. . . . . . 7 ⊢ (𝑤 = 𝑅 → (∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦 ↔ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦))
8	3, 7	anbi12d 473	. . . . . 6 ⊢ (𝑤 = 𝑅 → ((𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)))
9	8	opabbidv 4096	. . . . 5 ⊢ (𝑤 = 𝑅 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)})
10		elex 2771	. . . . 5 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ V)
11		basfn 12679	. . . . . . . 8 ⊢ Base Fn V
12		funfvex 5572	. . . . . . . . 9 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
13	12	funfni 5355	. . . . . . . 8 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
14	11, 10, 13	sylancr 414	. . . . . . 7 ⊢ (𝑅 ∈ SRing → (Base‘𝑅) ∈ V)
15		xpexg 4774	. . . . . . 7 ⊢ (((Base‘𝑅) ∈ V ∧ (Base‘𝑅) ∈ V) → ((Base‘𝑅) × (Base‘𝑅)) ∈ V)
16	14, 14, 15	syl2anc 411	. . . . . 6 ⊢ (𝑅 ∈ SRing → ((Base‘𝑅) × (Base‘𝑅)) ∈ V)
17		simpr 110	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦) → (𝑧(.r‘𝑅)𝑥) = 𝑦)
18		simplll 533	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦) → 𝑅 ∈ SRing)
19		simplr 528	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦) → 𝑧 ∈ (Base‘𝑅))
20		simpllr 534	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦) → 𝑥 ∈ (Base‘𝑅))
21		eqid 2193	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
22		eqid 2193	. . . . . . . . . . . . 13 ⊢ (.r‘𝑅) = (.r‘𝑅)
23	21, 22	srgcl 13469	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ SRing ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑧(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
24	18, 19, 20, 23	syl3anc 1249	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦) → (𝑧(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
25	17, 24	eqeltrrd 2271	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦) → 𝑦 ∈ (Base‘𝑅))
26	25	rexlimdva2 2614	. . . . . . . . 9 ⊢ ((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) → (∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦 → 𝑦 ∈ (Base‘𝑅)))
27	26	imdistanda 448	. . . . . . . 8 ⊢ (𝑅 ∈ SRing → ((𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦) → (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))))
28	27	ssopab2dv 4310	. . . . . . 7 ⊢ (𝑅 ∈ SRing → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))})
29		df-xp 4666	. . . . . . 7 ⊢ ((Base‘𝑅) × (Base‘𝑅)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))}
30	28, 29	sseqtrrdi 3229	. . . . . 6 ⊢ (𝑅 ∈ SRing → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)} ⊆ ((Base‘𝑅) × (Base‘𝑅)))
31	16, 30	ssexd 4170	. . . . 5 ⊢ (𝑅 ∈ SRing → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)} ∈ V)
32	1, 9, 10, 31	fvmptd3 5652	. . . 4 ⊢ (𝑅 ∈ SRing → (∥r‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)})
33	32, 30	eqsstrd 3216	. . 3 ⊢ (𝑅 ∈ SRing → (∥r‘𝑅) ⊆ ((Base‘𝑅) × (Base‘𝑅)))
34		xpss 4768	. . 3 ⊢ ((Base‘𝑅) × (Base‘𝑅)) ⊆ (V × V)
35	33, 34	sstrdi 3192	. 2 ⊢ (𝑅 ∈ SRing → (∥r‘𝑅) ⊆ (V × V))
36		df-rel 4667	. 2 ⊢ (Rel (∥r‘𝑅) ↔ (∥r‘𝑅) ⊆ (V × V))
37	35, 36	sylibr 134	1 ⊢ (𝑅 ∈ SRing → Rel (∥r‘𝑅))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  ∃wrex 2473  Vcvv 2760   ⊆ wss 3154  {copab 4090   × cxp 4658  Rel wrel 4665   Fn wfn 5250  ‘cfv 5255  (class class class)co 5919  Basecbs 12621  ._rcmulr 12699  SRingcsrg 13462  ∥_rcdsr 13585

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-plusg 12711  df-mulr 12712  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-mgp 13420  df-srg 13463  df-dvdsr 13588

This theorem is referenced by:  dvdsrd  13593  isunitd  13605  subrgdvds  13734