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Theorem reldvdsrsrg 13409
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 (∥r𝑅))

Proof of Theorem reldvdsrsrg
Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dvdsr 13406 . . . . 5 r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r𝑤)𝑥) = 𝑦)})
2 fveq2 5530 . . . . . . . 8 (𝑤 = 𝑅 → (Base‘𝑤) = (Base‘𝑅))
32eleq2d 2259 . . . . . . 7 (𝑤 = 𝑅 → (𝑥 ∈ (Base‘𝑤) ↔ 𝑥 ∈ (Base‘𝑅)))
4 fveq2 5530 . . . . . . . . . 10 (𝑤 = 𝑅 → (.r𝑤) = (.r𝑅))
54oveqd 5908 . . . . . . . . 9 (𝑤 = 𝑅 → (𝑧(.r𝑤)𝑥) = (𝑧(.r𝑅)𝑥))
65eqeq1d 2198 . . . . . . . 8 (𝑤 = 𝑅 → ((𝑧(.r𝑤)𝑥) = 𝑦 ↔ (𝑧(.r𝑅)𝑥) = 𝑦))
72, 6rexeqbidv 2699 . . . . . . 7 (𝑤 = 𝑅 → (∃𝑧 ∈ (Base‘𝑤)(𝑧(.r𝑤)𝑥) = 𝑦 ↔ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r𝑅)𝑥) = 𝑦))
83, 7anbi12d 473 . . . . . 6 (𝑤 = 𝑅 → ((𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r𝑤)𝑥) = 𝑦) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r𝑅)𝑥) = 𝑦)))
98opabbidv 4084 . . . . 5 (𝑤 = 𝑅 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r𝑤)𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r𝑅)𝑥) = 𝑦)})
10 elex 2763 . . . . 5 (𝑅 ∈ SRing → 𝑅 ∈ V)
11 basfn 12544 . . . . . . . 8 Base Fn V
12 funfvex 5547 . . . . . . . . 9 ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
1312funfni 5331 . . . . . . . 8 ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
1411, 10, 13sylancr 414 . . . . . . 7 (𝑅 ∈ SRing → (Base‘𝑅) ∈ V)
15 xpexg 4755 . . . . . . 7 (((Base‘𝑅) ∈ V ∧ (Base‘𝑅) ∈ V) → ((Base‘𝑅) × (Base‘𝑅)) ∈ V)
1614, 14, 15syl2anc 411 . . . . . 6 (𝑅 ∈ SRing → ((Base‘𝑅) × (Base‘𝑅)) ∈ V)
17 simpr 110 . . . . . . . . . . 11 ((((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝑧(.r𝑅)𝑥) = 𝑦) → (𝑧(.r𝑅)𝑥) = 𝑦)
18 simplll 533 . . . . . . . . . . . 12 ((((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝑧(.r𝑅)𝑥) = 𝑦) → 𝑅 ∈ SRing)
19 simplr 528 . . . . . . . . . . . 12 ((((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝑧(.r𝑅)𝑥) = 𝑦) → 𝑧 ∈ (Base‘𝑅))
20 simpllr 534 . . . . . . . . . . . 12 ((((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝑧(.r𝑅)𝑥) = 𝑦) → 𝑥 ∈ (Base‘𝑅))
21 eqid 2189 . . . . . . . . . . . . 13 (Base‘𝑅) = (Base‘𝑅)
22 eqid 2189 . . . . . . . . . . . . 13 (.r𝑅) = (.r𝑅)
2321, 22srgcl 13291 . . . . . . . . . . . 12 ((𝑅 ∈ SRing ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑧(.r𝑅)𝑥) ∈ (Base‘𝑅))
2418, 19, 20, 23syl3anc 1249 . . . . . . . . . . 11 ((((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝑧(.r𝑅)𝑥) = 𝑦) → (𝑧(.r𝑅)𝑥) ∈ (Base‘𝑅))
2517, 24eqeltrrd 2267 . . . . . . . . . 10 ((((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝑧(.r𝑅)𝑥) = 𝑦) → 𝑦 ∈ (Base‘𝑅))
2625rexlimdva2 2610 . . . . . . . . 9 ((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) → (∃𝑧 ∈ (Base‘𝑅)(𝑧(.r𝑅)𝑥) = 𝑦𝑦 ∈ (Base‘𝑅)))
2726imdistanda 448 . . . . . . . 8 (𝑅 ∈ SRing → ((𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r𝑅)𝑥) = 𝑦) → (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))))
2827ssopab2dv 4293 . . . . . . 7 (𝑅 ∈ SRing → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r𝑅)𝑥) = 𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))})
29 df-xp 4647 . . . . . . 7 ((Base‘𝑅) × (Base‘𝑅)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))}
3028, 29sseqtrrdi 3219 . . . . . 6 (𝑅 ∈ SRing → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r𝑅)𝑥) = 𝑦)} ⊆ ((Base‘𝑅) × (Base‘𝑅)))
3116, 30ssexd 4158 . . . . 5 (𝑅 ∈ SRing → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r𝑅)𝑥) = 𝑦)} ∈ V)
321, 9, 10, 31fvmptd3 5625 . . . 4 (𝑅 ∈ SRing → (∥r𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r𝑅)𝑥) = 𝑦)})
3332, 30eqsstrd 3206 . . 3 (𝑅 ∈ SRing → (∥r𝑅) ⊆ ((Base‘𝑅) × (Base‘𝑅)))
34 xpss 4749 . . 3 ((Base‘𝑅) × (Base‘𝑅)) ⊆ (V × V)
3533, 34sstrdi 3182 . 2 (𝑅 ∈ SRing → (∥r𝑅) ⊆ (V × V))
36 df-rel 4648 . 2 (Rel (∥r𝑅) ↔ (∥r𝑅) ⊆ (V × V))
3735, 36sylibr 134 1 (𝑅 ∈ SRing → Rel (∥r𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  wrex 2469  Vcvv 2752  wss 3144  {copab 4078   × cxp 4639  Rel wrel 4646   Fn wfn 5226  cfv 5231  (class class class)co 5891  Basecbs 12486  .rcmulr 12562  SRingcsrg 13284  rcdsr 13403
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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-addcom 7930  ax-addass 7932  ax-i2m1 7935  ax-0lt1 7936  ax-0id 7938  ax-rnegex 7939  ax-pre-ltirr 7942  ax-pre-ltadd 7946
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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-iota 5193  df-fun 5233  df-fn 5234  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-pnf 8013  df-mnf 8014  df-ltxr 8016  df-inn 8939  df-2 8997  df-3 8998  df-ndx 12489  df-slot 12490  df-base 12492  df-sets 12493  df-plusg 12574  df-mulr 12575  df-0g 12735  df-mgm 12804  df-sgrp 12837  df-mnd 12850  df-mgp 13242  df-srg 13285  df-dvdsr 13406
This theorem is referenced by:  dvdsrd  13411  isunitd  13423  subrgdvds  13549
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