Theorem reldvdsrsrg 14021

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 14018	. . . . 5 ⊢ ∥r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)})
2		fveq2 5603	. . . . . . . 8 ⊢ (𝑤 = 𝑅 → (Base‘𝑤) = (Base‘𝑅))
3	2	eleq2d 2279	. . . . . . 7 ⊢ (𝑤 = 𝑅 → (𝑥 ∈ (Base‘𝑤) ↔ 𝑥 ∈ (Base‘𝑅)))
4		fveq2 5603	. . . . . . . . . 10 ⊢ (𝑤 = 𝑅 → (.r‘𝑤) = (.r‘𝑅))
5	4	oveqd 5991	. . . . . . . . 9 ⊢ (𝑤 = 𝑅 → (𝑧(.r‘𝑤)𝑥) = (𝑧(.r‘𝑅)𝑥))
6	5	eqeq1d 2218	. . . . . . . 8 ⊢ (𝑤 = 𝑅 → ((𝑧(.r‘𝑤)𝑥) = 𝑦 ↔ (𝑧(.r‘𝑅)𝑥) = 𝑦))
7	2, 6	rexeqbidv 2725	. . . . . . 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 4129	. . . . 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 2791	. . . . 5 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ V)
11		basfn 13057	. . . . . . . 8 ⊢ Base Fn V
12		funfvex 5620	. . . . . . . . 9 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
13	12	funfni 5399	. . . . . . . 8 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
14	11, 10, 13	sylancr 414	. . . . . . 7 ⊢ (𝑅 ∈ SRing → (Base‘𝑅) ∈ V)
15		xpexg 4810	. . . . . . 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 2209	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
22		eqid 2209	. . . . . . . . . . . . 13 ⊢ (.r‘𝑅) = (.r‘𝑅)
23	21, 22	srgcl 13899	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ SRing ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑧(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
24	18, 19, 20, 23	syl3anc 1252	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦) → (𝑧(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
25	17, 24	eqeltrrd 2287	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦) → 𝑦 ∈ (Base‘𝑅))
26	25	rexlimdva2 2631	. . . . . . . . 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 4346	. . . . . . 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 4702	. . . . . . 7 ⊢ ((Base‘𝑅) × (Base‘𝑅)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))}
30	28, 29	sseqtrrdi 3253	. . . . . 6 ⊢ (𝑅 ∈ SRing → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)} ⊆ ((Base‘𝑅) × (Base‘𝑅)))
31	16, 30	ssexd 4203	. . . . 5 ⊢ (𝑅 ∈ SRing → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)} ∈ V)
32	1, 9, 10, 31	fvmptd3 5701	. . . 4 ⊢ (𝑅 ∈ SRing → (∥r‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)})
33	32, 30	eqsstrd 3240	. . 3 ⊢ (𝑅 ∈ SRing → (∥r‘𝑅) ⊆ ((Base‘𝑅) × (Base‘𝑅)))
34		xpss 4804	. . 3 ⊢ ((Base‘𝑅) × (Base‘𝑅)) ⊆ (V × V)
35	33, 34	sstrdi 3216	. 2 ⊢ (𝑅 ∈ SRing → (∥r‘𝑅) ⊆ (V × V))
36		df-rel 4703	. 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 1375   ∈ wcel 2180  ∃wrex 2489  Vcvv 2779   ⊆ wss 3177  {copab 4123   × cxp 4694  Rel wrel 4701   Fn wfn 5289  ‘cfv 5294  (class class class)co 5974  Basecbs 12998  ._rcmulr 13077  SRingcsrg 13892  ∥_rcdsr 14015

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-addcom 8067  ax-addass 8069  ax-i2m1 8072  ax-0lt1 8073  ax-0id 8075  ax-rnegex 8076  ax-pre-ltirr 8079  ax-pre-ltadd 8083

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-iota 5254  df-fun 5296  df-fn 5297  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-pnf 8151  df-mnf 8152  df-ltxr 8154  df-inn 9079  df-2 9137  df-3 9138  df-ndx 13001  df-slot 13002  df-base 13004  df-sets 13005  df-plusg 13089  df-mulr 13090  df-0g 13257  df-mgm 13355  df-sgrp 13401  df-mnd 13416  df-mgp 13850  df-srg 13893  df-dvdsr 14018

This theorem is referenced by:  dvdsrd  14023  isunitd  14035  subrgdvds  14164