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Theorem hlsuprexch 36670
Description: A Hilbert lattice has the superposition and exchange properties. (Contributed by NM, 13-Nov-2011.)
Hypotheses
Ref Expression
hlsuprexch.b 𝐵 = (Base‘𝐾)
hlsuprexch.l = (le‘𝐾)
hlsuprexch.j = (join‘𝐾)
hlsuprexch.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
hlsuprexch ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → ((𝑃𝑄 → ∃𝑧𝐴 (𝑧𝑃𝑧𝑄𝑧 (𝑃 𝑄))) ∧ ∀𝑧𝐵 ((¬ 𝑃 𝑧𝑃 (𝑧 𝑄)) → 𝑄 (𝑧 𝑃))))
Distinct variable groups:   𝑧,𝐴   𝑧,𝐵   𝑧,𝐾   𝑧,𝑃   𝑧,𝑄
Allowed substitution hints:   (𝑧)   (𝑧)

Proof of Theorem hlsuprexch
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hlsuprexch.b . . . . 5 𝐵 = (Base‘𝐾)
2 hlsuprexch.l . . . . 5 = (le‘𝐾)
3 eqid 2801 . . . . 5 (lt‘𝐾) = (lt‘𝐾)
4 hlsuprexch.j . . . . 5 = (join‘𝐾)
5 eqid 2801 . . . . 5 (0.‘𝐾) = (0.‘𝐾)
6 eqid 2801 . . . . 5 (1.‘𝐾) = (1.‘𝐾)
7 hlsuprexch.a . . . . 5 𝐴 = (Atoms‘𝐾)
81, 2, 3, 4, 5, 6, 7ishlat2 36642 . . . 4 (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ (∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∀𝑧𝐵 ((¬ 𝑥 𝑧𝑥 (𝑧 𝑦)) → 𝑦 (𝑧 𝑥))) ∧ ∃𝑥𝐵𝑦𝐵𝑧𝐵 (((0.‘𝐾)(lt‘𝐾)𝑥𝑥(lt‘𝐾)𝑦) ∧ (𝑦(lt‘𝐾)𝑧𝑧(lt‘𝐾)(1.‘𝐾))))))
9 simprl 770 . . . 4 (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ (∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∀𝑧𝐵 ((¬ 𝑥 𝑧𝑥 (𝑧 𝑦)) → 𝑦 (𝑧 𝑥))) ∧ ∃𝑥𝐵𝑦𝐵𝑧𝐵 (((0.‘𝐾)(lt‘𝐾)𝑥𝑥(lt‘𝐾)𝑦) ∧ (𝑦(lt‘𝐾)𝑧𝑧(lt‘𝐾)(1.‘𝐾))))) → ∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∀𝑧𝐵 ((¬ 𝑥 𝑧𝑥 (𝑧 𝑦)) → 𝑦 (𝑧 𝑥))))
108, 9sylbi 220 . . 3 (𝐾 ∈ HL → ∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∀𝑧𝐵 ((¬ 𝑥 𝑧𝑥 (𝑧 𝑦)) → 𝑦 (𝑧 𝑥))))
11 neeq1 3052 . . . . . 6 (𝑥 = 𝑃 → (𝑥𝑦𝑃𝑦))
12 neeq2 3053 . . . . . . . 8 (𝑥 = 𝑃 → (𝑧𝑥𝑧𝑃))
13 oveq1 7146 . . . . . . . . 9 (𝑥 = 𝑃 → (𝑥 𝑦) = (𝑃 𝑦))
1413breq2d 5045 . . . . . . . 8 (𝑥 = 𝑃 → (𝑧 (𝑥 𝑦) ↔ 𝑧 (𝑃 𝑦)))
1512, 143anbi13d 1435 . . . . . . 7 (𝑥 = 𝑃 → ((𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦)) ↔ (𝑧𝑃𝑧𝑦𝑧 (𝑃 𝑦))))
1615rexbidv 3259 . . . . . 6 (𝑥 = 𝑃 → (∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦)) ↔ ∃𝑧𝐴 (𝑧𝑃𝑧𝑦𝑧 (𝑃 𝑦))))
1711, 16imbi12d 348 . . . . 5 (𝑥 = 𝑃 → ((𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ↔ (𝑃𝑦 → ∃𝑧𝐴 (𝑧𝑃𝑧𝑦𝑧 (𝑃 𝑦)))))
18 breq1 5036 . . . . . . . . 9 (𝑥 = 𝑃 → (𝑥 𝑧𝑃 𝑧))
1918notbid 321 . . . . . . . 8 (𝑥 = 𝑃 → (¬ 𝑥 𝑧 ↔ ¬ 𝑃 𝑧))
20 breq1 5036 . . . . . . . 8 (𝑥 = 𝑃 → (𝑥 (𝑧 𝑦) ↔ 𝑃 (𝑧 𝑦)))
2119, 20anbi12d 633 . . . . . . 7 (𝑥 = 𝑃 → ((¬ 𝑥 𝑧𝑥 (𝑧 𝑦)) ↔ (¬ 𝑃 𝑧𝑃 (𝑧 𝑦))))
22 oveq2 7147 . . . . . . . 8 (𝑥 = 𝑃 → (𝑧 𝑥) = (𝑧 𝑃))
2322breq2d 5045 . . . . . . 7 (𝑥 = 𝑃 → (𝑦 (𝑧 𝑥) ↔ 𝑦 (𝑧 𝑃)))
2421, 23imbi12d 348 . . . . . 6 (𝑥 = 𝑃 → (((¬ 𝑥 𝑧𝑥 (𝑧 𝑦)) → 𝑦 (𝑧 𝑥)) ↔ ((¬ 𝑃 𝑧𝑃 (𝑧 𝑦)) → 𝑦 (𝑧 𝑃))))
2524ralbidv 3165 . . . . 5 (𝑥 = 𝑃 → (∀𝑧𝐵 ((¬ 𝑥 𝑧𝑥 (𝑧 𝑦)) → 𝑦 (𝑧 𝑥)) ↔ ∀𝑧𝐵 ((¬ 𝑃 𝑧𝑃 (𝑧 𝑦)) → 𝑦 (𝑧 𝑃))))
2617, 25anbi12d 633 . . . 4 (𝑥 = 𝑃 → (((𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∀𝑧𝐵 ((¬ 𝑥 𝑧𝑥 (𝑧 𝑦)) → 𝑦 (𝑧 𝑥))) ↔ ((𝑃𝑦 → ∃𝑧𝐴 (𝑧𝑃𝑧𝑦𝑧 (𝑃 𝑦))) ∧ ∀𝑧𝐵 ((¬ 𝑃 𝑧𝑃 (𝑧 𝑦)) → 𝑦 (𝑧 𝑃)))))
27 neeq2 3053 . . . . . 6 (𝑦 = 𝑄 → (𝑃𝑦𝑃𝑄))
28 neeq2 3053 . . . . . . . 8 (𝑦 = 𝑄 → (𝑧𝑦𝑧𝑄))
29 oveq2 7147 . . . . . . . . 9 (𝑦 = 𝑄 → (𝑃 𝑦) = (𝑃 𝑄))
3029breq2d 5045 . . . . . . . 8 (𝑦 = 𝑄 → (𝑧 (𝑃 𝑦) ↔ 𝑧 (𝑃 𝑄)))
3128, 303anbi23d 1436 . . . . . . 7 (𝑦 = 𝑄 → ((𝑧𝑃𝑧𝑦𝑧 (𝑃 𝑦)) ↔ (𝑧𝑃𝑧𝑄𝑧 (𝑃 𝑄))))
3231rexbidv 3259 . . . . . 6 (𝑦 = 𝑄 → (∃𝑧𝐴 (𝑧𝑃𝑧𝑦𝑧 (𝑃 𝑦)) ↔ ∃𝑧𝐴 (𝑧𝑃𝑧𝑄𝑧 (𝑃 𝑄))))
3327, 32imbi12d 348 . . . . 5 (𝑦 = 𝑄 → ((𝑃𝑦 → ∃𝑧𝐴 (𝑧𝑃𝑧𝑦𝑧 (𝑃 𝑦))) ↔ (𝑃𝑄 → ∃𝑧𝐴 (𝑧𝑃𝑧𝑄𝑧 (𝑃 𝑄)))))
34 oveq2 7147 . . . . . . . . 9 (𝑦 = 𝑄 → (𝑧 𝑦) = (𝑧 𝑄))
3534breq2d 5045 . . . . . . . 8 (𝑦 = 𝑄 → (𝑃 (𝑧 𝑦) ↔ 𝑃 (𝑧 𝑄)))
3635anbi2d 631 . . . . . . 7 (𝑦 = 𝑄 → ((¬ 𝑃 𝑧𝑃 (𝑧 𝑦)) ↔ (¬ 𝑃 𝑧𝑃 (𝑧 𝑄))))
37 breq1 5036 . . . . . . 7 (𝑦 = 𝑄 → (𝑦 (𝑧 𝑃) ↔ 𝑄 (𝑧 𝑃)))
3836, 37imbi12d 348 . . . . . 6 (𝑦 = 𝑄 → (((¬ 𝑃 𝑧𝑃 (𝑧 𝑦)) → 𝑦 (𝑧 𝑃)) ↔ ((¬ 𝑃 𝑧𝑃 (𝑧 𝑄)) → 𝑄 (𝑧 𝑃))))
3938ralbidv 3165 . . . . 5 (𝑦 = 𝑄 → (∀𝑧𝐵 ((¬ 𝑃 𝑧𝑃 (𝑧 𝑦)) → 𝑦 (𝑧 𝑃)) ↔ ∀𝑧𝐵 ((¬ 𝑃 𝑧𝑃 (𝑧 𝑄)) → 𝑄 (𝑧 𝑃))))
4033, 39anbi12d 633 . . . 4 (𝑦 = 𝑄 → (((𝑃𝑦 → ∃𝑧𝐴 (𝑧𝑃𝑧𝑦𝑧 (𝑃 𝑦))) ∧ ∀𝑧𝐵 ((¬ 𝑃 𝑧𝑃 (𝑧 𝑦)) → 𝑦 (𝑧 𝑃))) ↔ ((𝑃𝑄 → ∃𝑧𝐴 (𝑧𝑃𝑧𝑄𝑧 (𝑃 𝑄))) ∧ ∀𝑧𝐵 ((¬ 𝑃 𝑧𝑃 (𝑧 𝑄)) → 𝑄 (𝑧 𝑃)))))
4126, 40rspc2v 3584 . . 3 ((𝑃𝐴𝑄𝐴) → (∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∀𝑧𝐵 ((¬ 𝑥 𝑧𝑥 (𝑧 𝑦)) → 𝑦 (𝑧 𝑥))) → ((𝑃𝑄 → ∃𝑧𝐴 (𝑧𝑃𝑧𝑄𝑧 (𝑃 𝑄))) ∧ ∀𝑧𝐵 ((¬ 𝑃 𝑧𝑃 (𝑧 𝑄)) → 𝑄 (𝑧 𝑃)))))
4210, 41mpan9 510 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 → ∃𝑧𝐴 (𝑧𝑃𝑧𝑄𝑧 (𝑃 𝑄))) ∧ ∀𝑧𝐵 ((¬ 𝑃 𝑧𝑃 (𝑧 𝑄)) → 𝑄 (𝑧 𝑃))))
43423impb 1112 1 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → ((𝑃𝑄 → ∃𝑧𝐴 (𝑧𝑃𝑧𝑄𝑧 (𝑃 𝑄))) ∧ ∀𝑧𝐵 ((¬ 𝑃 𝑧𝑃 (𝑧 𝑄)) → 𝑄 (𝑧 𝑃))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2112  wne 2990  wral 3109  wrex 3110   class class class wbr 5033  cfv 6328  (class class class)co 7139  Basecbs 16478  lecple 16567  ltcplt 17546  joincjn 17549  0.cp0 17642  1.cp1 17643  CLatccla 17712  OMLcoml 36464  Atomscatm 36552  AtLatcal 36553  HLchlt 36639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336  df-ov 7142  df-cvlat 36611  df-hlat 36640
This theorem is referenced by:  hlsupr  36675
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