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

Theorem isprs 16753
 Description: Property of being a preordered set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypotheses
Ref Expression
isprs.b 𝐵 = (Base‘𝐾)
isprs.l = (le‘𝐾)
Assertion
Ref Expression
isprs (𝐾 ∈ Preset ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
Distinct variable groups:   𝑥,𝐾,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥, ,𝑦,𝑧

Proof of Theorem isprs
Dummy variables 𝑓 𝑏 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6103 . . . 4 (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
2 fveq2 6103 . . . . 5 (𝑓 = 𝐾 → (le‘𝑓) = (le‘𝐾))
32sbceq1d 3407 . . . 4 (𝑓 = 𝐾 → ([(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ [(le‘𝐾) / 𝑟]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
41, 3sbceqbid 3409 . . 3 (𝑓 = 𝐾 → ([(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ [(Base‘𝐾) / 𝑏][(le‘𝐾) / 𝑟]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
5 fvex 6113 . . . 4 (Base‘𝐾) ∈ V
6 fvex 6113 . . . 4 (le‘𝐾) ∈ V
7 isprs.b . . . . . . 7 𝐵 = (Base‘𝐾)
8 eqtr3 2631 . . . . . . 7 ((𝑏 = (Base‘𝐾) ∧ 𝐵 = (Base‘𝐾)) → 𝑏 = 𝐵)
97, 8mpan2 703 . . . . . 6 (𝑏 = (Base‘𝐾) → 𝑏 = 𝐵)
10 raleq 3115 . . . . . . . 8 (𝑏 = 𝐵 → (∀𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
1110raleqbi1dv 3123 . . . . . . 7 (𝑏 = 𝐵 → (∀𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑦𝐵𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
1211raleqbi1dv 3123 . . . . . 6 (𝑏 = 𝐵 → (∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
139, 12syl 17 . . . . 5 (𝑏 = (Base‘𝐾) → (∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
14 isprs.l . . . . . . 7 = (le‘𝐾)
15 eqtr3 2631 . . . . . . 7 ((𝑟 = (le‘𝐾) ∧ = (le‘𝐾)) → 𝑟 = )
1614, 15mpan2 703 . . . . . 6 (𝑟 = (le‘𝐾) → 𝑟 = )
17 breq 4585 . . . . . . . . 9 (𝑟 = → (𝑥𝑟𝑥𝑥 𝑥))
18 breq 4585 . . . . . . . . . . 11 (𝑟 = → (𝑥𝑟𝑦𝑥 𝑦))
19 breq 4585 . . . . . . . . . . 11 (𝑟 = → (𝑦𝑟𝑧𝑦 𝑧))
2018, 19anbi12d 743 . . . . . . . . . 10 (𝑟 = → ((𝑥𝑟𝑦𝑦𝑟𝑧) ↔ (𝑥 𝑦𝑦 𝑧)))
21 breq 4585 . . . . . . . . . 10 (𝑟 = → (𝑥𝑟𝑧𝑥 𝑧))
2220, 21imbi12d 333 . . . . . . . . 9 (𝑟 = → (((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
2317, 22anbi12d 743 . . . . . . . 8 (𝑟 = → ((𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
2423ralbidv 2969 . . . . . . 7 (𝑟 = → (∀𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
25242ralbidv 2972 . . . . . 6 (𝑟 = → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
2616, 25syl 17 . . . . 5 (𝑟 = (le‘𝐾) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
2713, 26sylan9bb 732 . . . 4 ((𝑏 = (Base‘𝐾) ∧ 𝑟 = (le‘𝐾)) → (∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
285, 6, 27sbc2ie 3472 . . 3 ([(Base‘𝐾) / 𝑏][(le‘𝐾) / 𝑟]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
294, 28syl6bb 275 . 2 (𝑓 = 𝐾 → ([(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
30 df-preset 16751 . 2 Preset = {𝑓[(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
3129, 30elab4g 3324 1 (𝐾 ∈ Preset ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  [wsbc 3402   class class class wbr 4583  ‘cfv 5804  Basecbs 15695  lecple 15775   Preset cpreset 16749 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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 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-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-preset 16751 This theorem is referenced by:  prslem  16754  ispos2  16771  ressprs  28986  oduprs  28987
 Copyright terms: Public domain W3C validator