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Theorem dvdsrpropd 19375
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 2820 . . . . . . 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 3293 . . . . 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 2895 . . . . 5 (𝜑 → (𝑦𝐵𝑦 ∈ (Base‘𝐾)))
97rexeqdv 3414 . . . . 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 2895 . . . . 5 (𝜑 → (𝑦𝐵𝑦 ∈ (Base‘𝐿)))
1311rexeqdv 3414 . . . . 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 5123 . 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 2818 . . 3 (Base‘𝐾) = (Base‘𝐾)
18 eqid 2818 . . 3 (∥r𝐾) = (∥r𝐾)
19 eqid 2818 . . 3 (.r𝐾) = (.r𝐾)
2017, 18, 19dvdsrval 19324 . 2 (∥r𝐾) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r𝐾)𝑦) = 𝑧)}
21 eqid 2818 . . 3 (Base‘𝐿) = (Base‘𝐿)
22 eqid 2818 . . 3 (∥r𝐿) = (∥r𝐿)
23 eqid 2818 . . 3 (.r𝐿) = (.r𝐿)
2421, 22, 23dvdsrval 19324 . 2 (∥r𝐿) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r𝐿)𝑦) = 𝑧)}
2516, 20, 243eqtr4g 2878 1 (𝜑 → (∥r𝐾) = (∥r𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wrex 3136  {copab 5119  cfv 6348  (class class class)co 7145  Basecbs 16471  .rcmulr 16554  rcdsr 19317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-dvdsr 19320
This theorem is referenced by:  unitpropd  19376
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