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Theorem dvdsrpropd 18904
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 453 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ 𝑦𝐵) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))
32eqeq1d 2773 . . . . . . 7 (((𝜑𝑥𝐵) ∧ 𝑦𝐵) → ((𝑥(.r𝐾)𝑦) = 𝑧 ↔ (𝑥(.r𝐿)𝑦) = 𝑧))
43an32s 631 . . . . . 6 (((𝜑𝑦𝐵) ∧ 𝑥𝐵) → ((𝑥(.r𝐾)𝑦) = 𝑧 ↔ (𝑥(.r𝐿)𝑦) = 𝑧))
54rexbidva 3197 . . . . 5 ((𝜑𝑦𝐵) → (∃𝑥𝐵 (𝑥(.r𝐾)𝑦) = 𝑧 ↔ ∃𝑥𝐵 (𝑥(.r𝐿)𝑦) = 𝑧))
65pm5.32da 568 . . . 4 (𝜑 → ((𝑦𝐵 ∧ ∃𝑥𝐵 (𝑥(.r𝐾)𝑦) = 𝑧) ↔ (𝑦𝐵 ∧ ∃𝑥𝐵 (𝑥(.r𝐿)𝑦) = 𝑧)))
7 rngidpropd.1 . . . . . 6 (𝜑𝐵 = (Base‘𝐾))
87eleq2d 2836 . . . . 5 (𝜑 → (𝑦𝐵𝑦 ∈ (Base‘𝐾)))
97rexeqdv 3294 . . . . 5 (𝜑 → (∃𝑥𝐵 (𝑥(.r𝐾)𝑦) = 𝑧 ↔ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r𝐾)𝑦) = 𝑧))
108, 9anbi12d 616 . . . 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 2836 . . . . 5 (𝜑 → (𝑦𝐵𝑦 ∈ (Base‘𝐿)))
1311rexeqdv 3294 . . . . 5 (𝜑 → (∃𝑥𝐵 (𝑥(.r𝐿)𝑦) = 𝑧 ↔ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r𝐿)𝑦) = 𝑧))
1412, 13anbi12d 616 . . . 4 (𝜑 → ((𝑦𝐵 ∧ ∃𝑥𝐵 (𝑥(.r𝐿)𝑦) = 𝑧) ↔ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r𝐿)𝑦) = 𝑧)))
156, 10, 143bitr3d 298 . . 3 (𝜑 → ((𝑦 ∈ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r𝐾)𝑦) = 𝑧) ↔ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r𝐿)𝑦) = 𝑧)))
1615opabbidv 4851 . 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 2771 . . 3 (Base‘𝐾) = (Base‘𝐾)
18 eqid 2771 . . 3 (∥r𝐾) = (∥r𝐾)
19 eqid 2771 . . 3 (.r𝐾) = (.r𝐾)
2017, 18, 19dvdsrval 18853 . 2 (∥r𝐾) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r𝐾)𝑦) = 𝑧)}
21 eqid 2771 . . 3 (Base‘𝐿) = (Base‘𝐿)
22 eqid 2771 . . 3 (∥r𝐿) = (∥r𝐿)
23 eqid 2771 . . 3 (.r𝐿) = (.r𝐿)
2421, 22, 23dvdsrval 18853 . 2 (∥r𝐿) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r𝐿)𝑦) = 𝑧)}
2516, 20, 243eqtr4g 2830 1 (𝜑 → (∥r𝐾) = (∥r𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wrex 3062  {copab 4847  cfv 6030  (class class class)co 6796  Basecbs 16064  .rcmulr 16150  rcdsr 18846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-dvdsr 18849
This theorem is referenced by:  unitpropd  18905
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