Theorem reldvdsrsrg 13648

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 13645	. . . . 5 ⊢ ∥r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)})
2		fveq2 5558	. . . . . . . 8 ⊢ (𝑤 = 𝑅 → (Base‘𝑤) = (Base‘𝑅))
3	2	eleq2d 2266	. . . . . . 7 ⊢ (𝑤 = 𝑅 → (𝑥 ∈ (Base‘𝑤) ↔ 𝑥 ∈ (Base‘𝑅)))
4		fveq2 5558	. . . . . . . . . 10 ⊢ (𝑤 = 𝑅 → (.r‘𝑤) = (.r‘𝑅))
5	4	oveqd 5939	. . . . . . . . 9 ⊢ (𝑤 = 𝑅 → (𝑧(.r‘𝑤)𝑥) = (𝑧(.r‘𝑅)𝑥))
6	5	eqeq1d 2205	. . . . . . . 8 ⊢ (𝑤 = 𝑅 → ((𝑧(.r‘𝑤)𝑥) = 𝑦 ↔ (𝑧(.r‘𝑅)𝑥) = 𝑦))
7	2, 6	rexeqbidv 2710	. . . . . . 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 4099	. . . . 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 2774	. . . . 5 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ V)
11		basfn 12736	. . . . . . . 8 ⊢ Base Fn V
12		funfvex 5575	. . . . . . . . 9 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
13	12	funfni 5358	. . . . . . . 8 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
14	11, 10, 13	sylancr 414	. . . . . . 7 ⊢ (𝑅 ∈ SRing → (Base‘𝑅) ∈ V)
15		xpexg 4777	. . . . . . 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 2196	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
22		eqid 2196	. . . . . . . . . . . . 13 ⊢ (.r‘𝑅) = (.r‘𝑅)
23	21, 22	srgcl 13526	. . . . . . . . . . . 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 2274	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦) → 𝑦 ∈ (Base‘𝑅))
26	25	rexlimdva2 2617	. . . . . . . . 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 4313	. . . . . . 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 4669	. . . . . . 7 ⊢ ((Base‘𝑅) × (Base‘𝑅)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))}
30	28, 29	sseqtrrdi 3232	. . . . . 6 ⊢ (𝑅 ∈ SRing → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)} ⊆ ((Base‘𝑅) × (Base‘𝑅)))
31	16, 30	ssexd 4173	. . . . 5 ⊢ (𝑅 ∈ SRing → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)} ∈ V)
32	1, 9, 10, 31	fvmptd3 5655	. . . 4 ⊢ (𝑅 ∈ SRing → (∥r‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)})
33	32, 30	eqsstrd 3219	. . 3 ⊢ (𝑅 ∈ SRing → (∥r‘𝑅) ⊆ ((Base‘𝑅) × (Base‘𝑅)))
34		xpss 4771	. . 3 ⊢ ((Base‘𝑅) × (Base‘𝑅)) ⊆ (V × V)
35	33, 34	sstrdi 3195	. 2 ⊢ (𝑅 ∈ SRing → (∥r‘𝑅) ⊆ (V × V))
36		df-rel 4670	. 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 2167  ∃wrex 2476  Vcvv 2763   ⊆ wss 3157  {copab 4093   × cxp 4661  Rel wrel 4668   Fn wfn 5253  ‘cfv 5258  (class class class)co 5922  Basecbs 12678  ._rcmulr 12756  SRingcsrg 13519  ∥_rcdsr 13642

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-mgp 13477  df-srg 13520  df-dvdsr 13645

This theorem is referenced by:  dvdsrd  13650  isunitd  13662  subrgdvds  13791