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Theorem dvdsrpropd 20433
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 467 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ 𝑦𝐵) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))
32eqeq1d 2737 . . . . . . 7 (((𝜑𝑥𝐵) ∧ 𝑦𝐵) → ((𝑥(.r𝐾)𝑦) = 𝑧 ↔ (𝑥(.r𝐿)𝑦) = 𝑧))
43an32s 652 . . . . . 6 (((𝜑𝑦𝐵) ∧ 𝑥𝐵) → ((𝑥(.r𝐾)𝑦) = 𝑧 ↔ (𝑥(.r𝐿)𝑦) = 𝑧))
54rexbidva 3175 . . . . 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 2825 . . . . 5 (𝜑 → (𝑦𝐵𝑦 ∈ (Base‘𝐾)))
97rexeqdv 3325 . . . . 5 (𝜑 → (∃𝑥𝐵 (𝑥(.r𝐾)𝑦) = 𝑧 ↔ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r𝐾)𝑦) = 𝑧))
108, 9anbi12d 632 . . . 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 2825 . . . . 5 (𝜑 → (𝑦𝐵𝑦 ∈ (Base‘𝐿)))
1311rexeqdv 3325 . . . . 5 (𝜑 → (∃𝑥𝐵 (𝑥(.r𝐿)𝑦) = 𝑧 ↔ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r𝐿)𝑦) = 𝑧))
1412, 13anbi12d 632 . . . 4 (𝜑 → ((𝑦𝐵 ∧ ∃𝑥𝐵 (𝑥(.r𝐿)𝑦) = 𝑧) ↔ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r𝐿)𝑦) = 𝑧)))
156, 10, 143bitr3d 309 . . 3 (𝜑 → ((𝑦 ∈ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r𝐾)𝑦) = 𝑧) ↔ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r𝐿)𝑦) = 𝑧)))
1615opabbidv 5214 . 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 2735 . . 3 (Base‘𝐾) = (Base‘𝐾)
18 eqid 2735 . . 3 (∥r𝐾) = (∥r𝐾)
19 eqid 2735 . . 3 (.r𝐾) = (.r𝐾)
2017, 18, 19dvdsrval 20378 . 2 (∥r𝐾) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r𝐾)𝑦) = 𝑧)}
21 eqid 2735 . . 3 (Base‘𝐿) = (Base‘𝐿)
22 eqid 2735 . . 3 (∥r𝐿) = (∥r𝐿)
23 eqid 2735 . . 3 (.r𝐿) = (.r𝐿)
2421, 22, 23dvdsrval 20378 . 2 (∥r𝐿) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r𝐿)𝑦) = 𝑧)}
2516, 20, 243eqtr4g 2800 1 (𝜑 → (∥r𝐾) = (∥r𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  {copab 5210  cfv 6563  (class class class)co 7431  Basecbs 17245  .rcmulr 17299  rcdsr 20371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-dvdsr 20374
This theorem is referenced by:  unitpropd  20434
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