MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvdsrpropd Structured version   Visualization version   GIF version

Theorem dvdsrpropd 18468
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 678 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ 𝑦𝐵) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))
32eqeq1d 2612 . . . . . . 7 (((𝜑𝑥𝐵) ∧ 𝑦𝐵) → ((𝑥(.r𝐾)𝑦) = 𝑧 ↔ (𝑥(.r𝐿)𝑦) = 𝑧))
43an32s 842 . . . . . 6 (((𝜑𝑦𝐵) ∧ 𝑥𝐵) → ((𝑥(.r𝐾)𝑦) = 𝑧 ↔ (𝑥(.r𝐿)𝑦) = 𝑧))
54rexbidva 3031 . . . . 5 ((𝜑𝑦𝐵) → (∃𝑥𝐵 (𝑥(.r𝐾)𝑦) = 𝑧 ↔ ∃𝑥𝐵 (𝑥(.r𝐿)𝑦) = 𝑧))
65pm5.32da 671 . . . 4 (𝜑 → ((𝑦𝐵 ∧ ∃𝑥𝐵 (𝑥(.r𝐾)𝑦) = 𝑧) ↔ (𝑦𝐵 ∧ ∃𝑥𝐵 (𝑥(.r𝐿)𝑦) = 𝑧)))
7 rngidpropd.1 . . . . . 6 (𝜑𝐵 = (Base‘𝐾))
87eleq2d 2673 . . . . 5 (𝜑 → (𝑦𝐵𝑦 ∈ (Base‘𝐾)))
97rexeqdv 3122 . . . . 5 (𝜑 → (∃𝑥𝐵 (𝑥(.r𝐾)𝑦) = 𝑧 ↔ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r𝐾)𝑦) = 𝑧))
108, 9anbi12d 743 . . . 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 2673 . . . . 5 (𝜑 → (𝑦𝐵𝑦 ∈ (Base‘𝐿)))
1311rexeqdv 3122 . . . . 5 (𝜑 → (∃𝑥𝐵 (𝑥(.r𝐿)𝑦) = 𝑧 ↔ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r𝐿)𝑦) = 𝑧))
1412, 13anbi12d 743 . . . 4 (𝜑 → ((𝑦𝐵 ∧ ∃𝑥𝐵 (𝑥(.r𝐿)𝑦) = 𝑧) ↔ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r𝐿)𝑦) = 𝑧)))
156, 10, 143bitr3d 297 . . 3 (𝜑 → ((𝑦 ∈ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r𝐾)𝑦) = 𝑧) ↔ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r𝐿)𝑦) = 𝑧)))
1615opabbidv 4643 . 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 2610 . . 3 (Base‘𝐾) = (Base‘𝐾)
18 eqid 2610 . . 3 (∥r𝐾) = (∥r𝐾)
19 eqid 2610 . . 3 (.r𝐾) = (.r𝐾)
2017, 18, 19dvdsrval 18417 . 2 (∥r𝐾) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r𝐾)𝑦) = 𝑧)}
21 eqid 2610 . . 3 (Base‘𝐿) = (Base‘𝐿)
22 eqid 2610 . . 3 (∥r𝐿) = (∥r𝐿)
23 eqid 2610 . . 3 (.r𝐿) = (.r𝐿)
2421, 22, 23dvdsrval 18417 . 2 (∥r𝐿) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r𝐿)𝑦) = 𝑧)}
2516, 20, 243eqtr4g 2669 1 (𝜑 → (∥r𝐾) = (∥r𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wrex 2897  {copab 4637  cfv 5790  (class class class)co 6527  Basecbs 15644  .rcmulr 15718  rcdsr 18410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-dvdsr 18413
This theorem is referenced by:  unitpropd  18469
  Copyright terms: Public domain W3C validator