Theorem reldvdsrsrg 13324

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 13321	. . . . 5 ⊢ ∥r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)})
2		fveq2 5527	. . . . . . . 8 ⊢ (𝑤 = 𝑅 → (Base‘𝑤) = (Base‘𝑅))
3	2	eleq2d 2257	. . . . . . 7 ⊢ (𝑤 = 𝑅 → (𝑥 ∈ (Base‘𝑤) ↔ 𝑥 ∈ (Base‘𝑅)))
4		fveq2 5527	. . . . . . . . . 10 ⊢ (𝑤 = 𝑅 → (.r‘𝑤) = (.r‘𝑅))
5	4	oveqd 5905	. . . . . . . . 9 ⊢ (𝑤 = 𝑅 → (𝑧(.r‘𝑤)𝑥) = (𝑧(.r‘𝑅)𝑥))
6	5	eqeq1d 2196	. . . . . . . 8 ⊢ (𝑤 = 𝑅 → ((𝑧(.r‘𝑤)𝑥) = 𝑦 ↔ (𝑧(.r‘𝑅)𝑥) = 𝑦))
7	2, 6	rexeqbidv 2696	. . . . . . 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 4081	. . . . 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 2760	. . . . 5 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ V)
11		basfn 12533	. . . . . . . 8 ⊢ Base Fn V
12		funfvex 5544	. . . . . . . . 9 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
13	12	funfni 5328	. . . . . . . 8 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
14	11, 10, 13	sylancr 414	. . . . . . 7 ⊢ (𝑅 ∈ SRing → (Base‘𝑅) ∈ V)
15		xpexg 4752	. . . . . . 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 2187	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
22		eqid 2187	. . . . . . . . . . . . 13 ⊢ (.r‘𝑅) = (.r‘𝑅)
23	21, 22	srgcl 13207	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ SRing ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑧(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
24	18, 19, 20, 23	syl3anc 1248	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦) → (𝑧(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
25	17, 24	eqeltrrd 2265	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦) → 𝑦 ∈ (Base‘𝑅))
26	25	rexlimdva2 2607	. . . . . . . . 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 4290	. . . . . . 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 4644	. . . . . . 7 ⊢ ((Base‘𝑅) × (Base‘𝑅)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))}
30	28, 29	sseqtrrdi 3216	. . . . . 6 ⊢ (𝑅 ∈ SRing → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)} ⊆ ((Base‘𝑅) × (Base‘𝑅)))
31	16, 30	ssexd 4155	. . . . 5 ⊢ (𝑅 ∈ SRing → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)} ∈ V)
32	1, 9, 10, 31	fvmptd3 5622	. . . 4 ⊢ (𝑅 ∈ SRing → (∥r‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)})
33	32, 30	eqsstrd 3203	. . 3 ⊢ (𝑅 ∈ SRing → (∥r‘𝑅) ⊆ ((Base‘𝑅) × (Base‘𝑅)))
34		xpss 4746	. . 3 ⊢ ((Base‘𝑅) × (Base‘𝑅)) ⊆ (V × V)
35	33, 34	sstrdi 3179	. 2 ⊢ (𝑅 ∈ SRing → (∥r‘𝑅) ⊆ (V × V))
36		df-rel 4645	. 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 1363   ∈ wcel 2158  ∃wrex 2466  Vcvv 2749   ⊆ wss 3141  {copab 4075   × cxp 4636  Rel wrel 4643   Fn wfn 5223  ‘cfv 5228  (class class class)co 5888  Basecbs 12475  ._rcmulr 12551  SRingcsrg 13200  ∥_rcdsr 13318

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-pre-ltirr 7936  ax-pre-ltadd 7940

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8007  df-mnf 8008  df-ltxr 8010  df-inn 8933  df-2 8991  df-3 8992  df-ndx 12478  df-slot 12479  df-base 12481  df-sets 12482  df-plusg 12563  df-mulr 12564  df-0g 12724  df-mgm 12793  df-sgrp 12826  df-mnd 12837  df-mgp 13163  df-srg 13201  df-dvdsr 13321

This theorem is referenced by:  dvdsrd  13326  isunitd  13338  subrgdvds  13419