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Theorem dvdsrpropd 19369
Description: The divisibility relation depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
Hypotheses
Ref Expression
rngidpropd.1 (𝜑𝐵 = (Base‘𝐾))
rngidpropd.2 (𝜑𝐵 = (Base‘𝐿))
rngidpropd.3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))
Assertion
Ref Expression
dvdsrpropd (𝜑 → (∥r𝐾) = (∥r𝐿))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦

Proof of Theorem dvdsrpropd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 rngidpropd.3 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))
21anassrs 468 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ 𝑦𝐵) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))
32eqeq1d 2827 . . . . . . 7 (((𝜑𝑥𝐵) ∧ 𝑦𝐵) → ((𝑥(.r𝐾)𝑦) = 𝑧 ↔ (𝑥(.r𝐿)𝑦) = 𝑧))
43an32s 648 . . . . . 6 (((𝜑𝑦𝐵) ∧ 𝑥𝐵) → ((𝑥(.r𝐾)𝑦) = 𝑧 ↔ (𝑥(.r𝐿)𝑦) = 𝑧))
54rexbidva 3300 . . . . 5 ((𝜑𝑦𝐵) → (∃𝑥𝐵 (𝑥(.r𝐾)𝑦) = 𝑧 ↔ ∃𝑥𝐵 (𝑥(.r𝐿)𝑦) = 𝑧))
65pm5.32da 579 . . . 4 (𝜑 → ((𝑦𝐵 ∧ ∃𝑥𝐵 (𝑥(.r𝐾)𝑦) = 𝑧) ↔ (𝑦𝐵 ∧ ∃𝑥𝐵 (𝑥(.r𝐿)𝑦) = 𝑧)))
7 rngidpropd.1 . . . . . 6 (𝜑𝐵 = (Base‘𝐾))
87eleq2d 2902 . . . . 5 (𝜑 → (𝑦𝐵𝑦 ∈ (Base‘𝐾)))
97rexeqdv 3421 . . . . 5 (𝜑 → (∃𝑥𝐵 (𝑥(.r𝐾)𝑦) = 𝑧 ↔ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r𝐾)𝑦) = 𝑧))
108, 9anbi12d 630 . . . 4 (𝜑 → ((𝑦𝐵 ∧ ∃𝑥𝐵 (𝑥(.r𝐾)𝑦) = 𝑧) ↔ (𝑦 ∈ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r𝐾)𝑦) = 𝑧)))
11 rngidpropd.2 . . . . . 6 (𝜑𝐵 = (Base‘𝐿))
1211eleq2d 2902 . . . . 5 (𝜑 → (𝑦𝐵𝑦 ∈ (Base‘𝐿)))
1311rexeqdv 3421 . . . . 5 (𝜑 → (∃𝑥𝐵 (𝑥(.r𝐿)𝑦) = 𝑧 ↔ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r𝐿)𝑦) = 𝑧))
1412, 13anbi12d 630 . . . 4 (𝜑 → ((𝑦𝐵 ∧ ∃𝑥𝐵 (𝑥(.r𝐿)𝑦) = 𝑧) ↔ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r𝐿)𝑦) = 𝑧)))
156, 10, 143bitr3d 310 . . 3 (𝜑 → ((𝑦 ∈ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r𝐾)𝑦) = 𝑧) ↔ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r𝐿)𝑦) = 𝑧)))
1615opabbidv 5128 . 2 (𝜑 → {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r𝐾)𝑦) = 𝑧)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r𝐿)𝑦) = 𝑧)})
17 eqid 2825 . . 3 (Base‘𝐾) = (Base‘𝐾)
18 eqid 2825 . . 3 (∥r𝐾) = (∥r𝐾)
19 eqid 2825 . . 3 (.r𝐾) = (.r𝐾)
2017, 18, 19dvdsrval 19318 . 2 (∥r𝐾) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r𝐾)𝑦) = 𝑧)}
21 eqid 2825 . . 3 (Base‘𝐿) = (Base‘𝐿)
22 eqid 2825 . . 3 (∥r𝐿) = (∥r𝐿)
23 eqid 2825 . . 3 (.r𝐿) = (.r𝐿)
2421, 22, 23dvdsrval 19318 . 2 (∥r𝐿) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r𝐿)𝑦) = 𝑧)}
2516, 20, 243eqtr4g 2885 1 (𝜑 → (∥r𝐾) = (∥r𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wrex 3143  {copab 5124  cfv 6351  (class class class)co 7151  Basecbs 16476  .rcmulr 16559  rcdsr 19311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-dvdsr 19314
This theorem is referenced by:  unitpropd  19370
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