Theorem reldvdsrsrg 13898

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 13895	. . . . 5 ⊢ ∥r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)})
2		fveq2 5583	. . . . . . . 8 ⊢ (𝑤 = 𝑅 → (Base‘𝑤) = (Base‘𝑅))
3	2	eleq2d 2276	. . . . . . 7 ⊢ (𝑤 = 𝑅 → (𝑥 ∈ (Base‘𝑤) ↔ 𝑥 ∈ (Base‘𝑅)))
4		fveq2 5583	. . . . . . . . . 10 ⊢ (𝑤 = 𝑅 → (.r‘𝑤) = (.r‘𝑅))
5	4	oveqd 5968	. . . . . . . . 9 ⊢ (𝑤 = 𝑅 → (𝑧(.r‘𝑤)𝑥) = (𝑧(.r‘𝑅)𝑥))
6	5	eqeq1d 2215	. . . . . . . 8 ⊢ (𝑤 = 𝑅 → ((𝑧(.r‘𝑤)𝑥) = 𝑦 ↔ (𝑧(.r‘𝑅)𝑥) = 𝑦))
7	2, 6	rexeqbidv 2720	. . . . . . 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 4114	. . . . 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 2784	. . . . 5 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ V)
11		basfn 12934	. . . . . . . 8 ⊢ Base Fn V
12		funfvex 5600	. . . . . . . . 9 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
13	12	funfni 5381	. . . . . . . 8 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
14	11, 10, 13	sylancr 414	. . . . . . 7 ⊢ (𝑅 ∈ SRing → (Base‘𝑅) ∈ V)
15		xpexg 4793	. . . . . . 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 2206	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
22		eqid 2206	. . . . . . . . . . . . 13 ⊢ (.r‘𝑅) = (.r‘𝑅)
23	21, 22	srgcl 13776	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ SRing ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑧(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
24	18, 19, 20, 23	syl3anc 1250	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦) → (𝑧(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
25	17, 24	eqeltrrd 2284	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦) → 𝑦 ∈ (Base‘𝑅))
26	25	rexlimdva2 2627	. . . . . . . . 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 4329	. . . . . . 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 4685	. . . . . . 7 ⊢ ((Base‘𝑅) × (Base‘𝑅)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))}
30	28, 29	sseqtrrdi 3243	. . . . . 6 ⊢ (𝑅 ∈ SRing → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)} ⊆ ((Base‘𝑅) × (Base‘𝑅)))
31	16, 30	ssexd 4188	. . . . 5 ⊢ (𝑅 ∈ SRing → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)} ∈ V)
32	1, 9, 10, 31	fvmptd3 5680	. . . 4 ⊢ (𝑅 ∈ SRing → (∥r‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)})
33	32, 30	eqsstrd 3230	. . 3 ⊢ (𝑅 ∈ SRing → (∥r‘𝑅) ⊆ ((Base‘𝑅) × (Base‘𝑅)))
34		xpss 4787	. . 3 ⊢ ((Base‘𝑅) × (Base‘𝑅)) ⊆ (V × V)
35	33, 34	sstrdi 3206	. 2 ⊢ (𝑅 ∈ SRing → (∥r‘𝑅) ⊆ (V × V))
36		df-rel 4686	. 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 1373   ∈ wcel 2177  ∃wrex 2486  Vcvv 2773   ⊆ wss 3167  {copab 4108   × cxp 4677  Rel wrel 4684   Fn wfn 5271  ‘cfv 5276  (class class class)co 5951  Basecbs 12876  ._rcmulr 12954  SRingcsrg 13769  ∥_rcdsr 13892

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-addcom 8032  ax-addass 8034  ax-i2m1 8037  ax-0lt1 8038  ax-0id 8040  ax-rnegex 8041  ax-pre-ltirr 8044  ax-pre-ltadd 8048

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-iota 5237  df-fun 5278  df-fn 5279  df-fv 5284  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-pnf 8116  df-mnf 8117  df-ltxr 8119  df-inn 9044  df-2 9102  df-3 9103  df-ndx 12879  df-slot 12880  df-base 12882  df-sets 12883  df-plusg 12966  df-mulr 12967  df-0g 13134  df-mgm 13232  df-sgrp 13278  df-mnd 13293  df-mgp 13727  df-srg 13770  df-dvdsr 13895

This theorem is referenced by:  dvdsrd  13900  isunitd  13912  subrgdvds  14041