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Theorem hlsuprexch 39383
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 2737 . . . . 5 (lt‘𝐾) = (lt‘𝐾)
4 hlsuprexch.j . . . . 5 = (join‘𝐾)
5 eqid 2737 . . . . 5 (0.‘𝐾) = (0.‘𝐾)
6 eqid 2737 . . . . 5 (1.‘𝐾) = (1.‘𝐾)
7 hlsuprexch.a . . . . 5 𝐴 = (Atoms‘𝐾)
81, 2, 3, 4, 5, 6, 7ishlat2 39354 . . . 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 771 . . . 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 217 . . 3 (𝐾 ∈ HL → ∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∀𝑧𝐵 ((¬ 𝑥 𝑧𝑥 (𝑧 𝑦)) → 𝑦 (𝑧 𝑥))))
11 neeq1 3003 . . . . . 6 (𝑥 = 𝑃 → (𝑥𝑦𝑃𝑦))
12 neeq2 3004 . . . . . . . 8 (𝑥 = 𝑃 → (𝑧𝑥𝑧𝑃))
13 oveq1 7438 . . . . . . . . 9 (𝑥 = 𝑃 → (𝑥 𝑦) = (𝑃 𝑦))
1413breq2d 5155 . . . . . . . 8 (𝑥 = 𝑃 → (𝑧 (𝑥 𝑦) ↔ 𝑧 (𝑃 𝑦)))
1512, 143anbi13d 1440 . . . . . . 7 (𝑥 = 𝑃 → ((𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦)) ↔ (𝑧𝑃𝑧𝑦𝑧 (𝑃 𝑦))))
1615rexbidv 3179 . . . . . 6 (𝑥 = 𝑃 → (∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦)) ↔ ∃𝑧𝐴 (𝑧𝑃𝑧𝑦𝑧 (𝑃 𝑦))))
1711, 16imbi12d 344 . . . . 5 (𝑥 = 𝑃 → ((𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ↔ (𝑃𝑦 → ∃𝑧𝐴 (𝑧𝑃𝑧𝑦𝑧 (𝑃 𝑦)))))
18 breq1 5146 . . . . . . . . 9 (𝑥 = 𝑃 → (𝑥 𝑧𝑃 𝑧))
1918notbid 318 . . . . . . . 8 (𝑥 = 𝑃 → (¬ 𝑥 𝑧 ↔ ¬ 𝑃 𝑧))
20 breq1 5146 . . . . . . . 8 (𝑥 = 𝑃 → (𝑥 (𝑧 𝑦) ↔ 𝑃 (𝑧 𝑦)))
2119, 20anbi12d 632 . . . . . . 7 (𝑥 = 𝑃 → ((¬ 𝑥 𝑧𝑥 (𝑧 𝑦)) ↔ (¬ 𝑃 𝑧𝑃 (𝑧 𝑦))))
22 oveq2 7439 . . . . . . . 8 (𝑥 = 𝑃 → (𝑧 𝑥) = (𝑧 𝑃))
2322breq2d 5155 . . . . . . 7 (𝑥 = 𝑃 → (𝑦 (𝑧 𝑥) ↔ 𝑦 (𝑧 𝑃)))
2421, 23imbi12d 344 . . . . . 6 (𝑥 = 𝑃 → (((¬ 𝑥 𝑧𝑥 (𝑧 𝑦)) → 𝑦 (𝑧 𝑥)) ↔ ((¬ 𝑃 𝑧𝑃 (𝑧 𝑦)) → 𝑦 (𝑧 𝑃))))
2524ralbidv 3178 . . . . 5 (𝑥 = 𝑃 → (∀𝑧𝐵 ((¬ 𝑥 𝑧𝑥 (𝑧 𝑦)) → 𝑦 (𝑧 𝑥)) ↔ ∀𝑧𝐵 ((¬ 𝑃 𝑧𝑃 (𝑧 𝑦)) → 𝑦 (𝑧 𝑃))))
2617, 25anbi12d 632 . . . 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 3004 . . . . . 6 (𝑦 = 𝑄 → (𝑃𝑦𝑃𝑄))
28 neeq2 3004 . . . . . . . 8 (𝑦 = 𝑄 → (𝑧𝑦𝑧𝑄))
29 oveq2 7439 . . . . . . . . 9 (𝑦 = 𝑄 → (𝑃 𝑦) = (𝑃 𝑄))
3029breq2d 5155 . . . . . . . 8 (𝑦 = 𝑄 → (𝑧 (𝑃 𝑦) ↔ 𝑧 (𝑃 𝑄)))
3128, 303anbi23d 1441 . . . . . . 7 (𝑦 = 𝑄 → ((𝑧𝑃𝑧𝑦𝑧 (𝑃 𝑦)) ↔ (𝑧𝑃𝑧𝑄𝑧 (𝑃 𝑄))))
3231rexbidv 3179 . . . . . 6 (𝑦 = 𝑄 → (∃𝑧𝐴 (𝑧𝑃𝑧𝑦𝑧 (𝑃 𝑦)) ↔ ∃𝑧𝐴 (𝑧𝑃𝑧𝑄𝑧 (𝑃 𝑄))))
3327, 32imbi12d 344 . . . . 5 (𝑦 = 𝑄 → ((𝑃𝑦 → ∃𝑧𝐴 (𝑧𝑃𝑧𝑦𝑧 (𝑃 𝑦))) ↔ (𝑃𝑄 → ∃𝑧𝐴 (𝑧𝑃𝑧𝑄𝑧 (𝑃 𝑄)))))
34 oveq2 7439 . . . . . . . . 9 (𝑦 = 𝑄 → (𝑧 𝑦) = (𝑧 𝑄))
3534breq2d 5155 . . . . . . . 8 (𝑦 = 𝑄 → (𝑃 (𝑧 𝑦) ↔ 𝑃 (𝑧 𝑄)))
3635anbi2d 630 . . . . . . 7 (𝑦 = 𝑄 → ((¬ 𝑃 𝑧𝑃 (𝑧 𝑦)) ↔ (¬ 𝑃 𝑧𝑃 (𝑧 𝑄))))
37 breq1 5146 . . . . . . 7 (𝑦 = 𝑄 → (𝑦 (𝑧 𝑃) ↔ 𝑄 (𝑧 𝑃)))
3836, 37imbi12d 344 . . . . . 6 (𝑦 = 𝑄 → (((¬ 𝑃 𝑧𝑃 (𝑧 𝑦)) → 𝑦 (𝑧 𝑃)) ↔ ((¬ 𝑃 𝑧𝑃 (𝑧 𝑄)) → 𝑄 (𝑧 𝑃))))
3938ralbidv 3178 . . . . 5 (𝑦 = 𝑄 → (∀𝑧𝐵 ((¬ 𝑃 𝑧𝑃 (𝑧 𝑦)) → 𝑦 (𝑧 𝑃)) ↔ ∀𝑧𝐵 ((¬ 𝑃 𝑧𝑃 (𝑧 𝑄)) → 𝑄 (𝑧 𝑃))))
4033, 39anbi12d 632 . . . 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 3633 . . 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 506 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 → ∃𝑧𝐴 (𝑧𝑃𝑧𝑄𝑧 (𝑃 𝑄))) ∧ ∀𝑧𝐵 ((¬ 𝑃 𝑧𝑃 (𝑧 𝑄)) → 𝑄 (𝑧 𝑃))))
43423impb 1115 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 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  ltcplt 18354  joincjn 18357  0.cp0 18468  1.cp1 18469  CLatccla 18543  OMLcoml 39176  Atomscatm 39264  AtLatcal 39265  HLchlt 39351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-cvlat 39323  df-hlat 39352
This theorem is referenced by:  hlsupr  39388
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