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Theorem hlsuprexch 38909
Description: A Hilbert lattice has the superposition and exchange properties. (Contributed by NM, 13-Nov-2011.)
Hypotheses
Ref Expression
hlsuprexch.b 𝐡 = (Baseβ€˜πΎ)
hlsuprexch.l ≀ = (leβ€˜πΎ)
hlsuprexch.j ∨ = (joinβ€˜πΎ)
hlsuprexch.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
hlsuprexch ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ ((𝑃 β‰  𝑄 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  𝑃 ∧ 𝑧 β‰  𝑄 ∧ 𝑧 ≀ (𝑃 ∨ 𝑄))) ∧ βˆ€π‘§ ∈ 𝐡 ((Β¬ 𝑃 ≀ 𝑧 ∧ 𝑃 ≀ (𝑧 ∨ 𝑄)) β†’ 𝑄 ≀ (𝑧 ∨ 𝑃))))
Distinct variable groups:   𝑧,𝐴   𝑧,𝐡   𝑧,𝐾   𝑧,𝑃   𝑧,𝑄
Allowed substitution hints:   ∨ (𝑧)   ≀ (𝑧)

Proof of Theorem hlsuprexch
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hlsuprexch.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
2 hlsuprexch.l . . . . 5 ≀ = (leβ€˜πΎ)
3 eqid 2725 . . . . 5 (ltβ€˜πΎ) = (ltβ€˜πΎ)
4 hlsuprexch.j . . . . 5 ∨ = (joinβ€˜πΎ)
5 eqid 2725 . . . . 5 (0.β€˜πΎ) = (0.β€˜πΎ)
6 eqid 2725 . . . . 5 (1.β€˜πΎ) = (1.β€˜πΎ)
7 hlsuprexch.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
81, 2, 3, 4, 5, 6, 7ishlat2 38880 . . . 4 (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 ((π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦))) ∧ βˆ€π‘§ ∈ 𝐡 ((Β¬ π‘₯ ≀ 𝑧 ∧ π‘₯ ≀ (𝑧 ∨ 𝑦)) β†’ 𝑦 ≀ (𝑧 ∨ π‘₯))) ∧ βˆƒπ‘₯ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐡 (((0.β€˜πΎ)(ltβ€˜πΎ)π‘₯ ∧ π‘₯(ltβ€˜πΎ)𝑦) ∧ (𝑦(ltβ€˜πΎ)𝑧 ∧ 𝑧(ltβ€˜πΎ)(1.β€˜πΎ))))))
9 simprl 769 . . . 4 (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 ((π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦))) ∧ βˆ€π‘§ ∈ 𝐡 ((Β¬ π‘₯ ≀ 𝑧 ∧ π‘₯ ≀ (𝑧 ∨ 𝑦)) β†’ 𝑦 ≀ (𝑧 ∨ π‘₯))) ∧ βˆƒπ‘₯ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐡 (((0.β€˜πΎ)(ltβ€˜πΎ)π‘₯ ∧ π‘₯(ltβ€˜πΎ)𝑦) ∧ (𝑦(ltβ€˜πΎ)𝑧 ∧ 𝑧(ltβ€˜πΎ)(1.β€˜πΎ))))) β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 ((π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦))) ∧ βˆ€π‘§ ∈ 𝐡 ((Β¬ π‘₯ ≀ 𝑧 ∧ π‘₯ ≀ (𝑧 ∨ 𝑦)) β†’ 𝑦 ≀ (𝑧 ∨ π‘₯))))
108, 9sylbi 216 . . 3 (𝐾 ∈ HL β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 ((π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦))) ∧ βˆ€π‘§ ∈ 𝐡 ((Β¬ π‘₯ ≀ 𝑧 ∧ π‘₯ ≀ (𝑧 ∨ 𝑦)) β†’ 𝑦 ≀ (𝑧 ∨ π‘₯))))
11 neeq1 2993 . . . . . 6 (π‘₯ = 𝑃 β†’ (π‘₯ β‰  𝑦 ↔ 𝑃 β‰  𝑦))
12 neeq2 2994 . . . . . . . 8 (π‘₯ = 𝑃 β†’ (𝑧 β‰  π‘₯ ↔ 𝑧 β‰  𝑃))
13 oveq1 7422 . . . . . . . . 9 (π‘₯ = 𝑃 β†’ (π‘₯ ∨ 𝑦) = (𝑃 ∨ 𝑦))
1413breq2d 5155 . . . . . . . 8 (π‘₯ = 𝑃 β†’ (𝑧 ≀ (π‘₯ ∨ 𝑦) ↔ 𝑧 ≀ (𝑃 ∨ 𝑦)))
1512, 143anbi13d 1434 . . . . . . 7 (π‘₯ = 𝑃 β†’ ((𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦)) ↔ (𝑧 β‰  𝑃 ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (𝑃 ∨ 𝑦))))
1615rexbidv 3169 . . . . . 6 (π‘₯ = 𝑃 β†’ (βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦)) ↔ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  𝑃 ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (𝑃 ∨ 𝑦))))
1711, 16imbi12d 343 . . . . 5 (π‘₯ = 𝑃 β†’ ((π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦))) ↔ (𝑃 β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  𝑃 ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (𝑃 ∨ 𝑦)))))
18 breq1 5146 . . . . . . . . 9 (π‘₯ = 𝑃 β†’ (π‘₯ ≀ 𝑧 ↔ 𝑃 ≀ 𝑧))
1918notbid 317 . . . . . . . 8 (π‘₯ = 𝑃 β†’ (Β¬ π‘₯ ≀ 𝑧 ↔ Β¬ 𝑃 ≀ 𝑧))
20 breq1 5146 . . . . . . . 8 (π‘₯ = 𝑃 β†’ (π‘₯ ≀ (𝑧 ∨ 𝑦) ↔ 𝑃 ≀ (𝑧 ∨ 𝑦)))
2119, 20anbi12d 630 . . . . . . 7 (π‘₯ = 𝑃 β†’ ((Β¬ π‘₯ ≀ 𝑧 ∧ π‘₯ ≀ (𝑧 ∨ 𝑦)) ↔ (Β¬ 𝑃 ≀ 𝑧 ∧ 𝑃 ≀ (𝑧 ∨ 𝑦))))
22 oveq2 7423 . . . . . . . 8 (π‘₯ = 𝑃 β†’ (𝑧 ∨ π‘₯) = (𝑧 ∨ 𝑃))
2322breq2d 5155 . . . . . . 7 (π‘₯ = 𝑃 β†’ (𝑦 ≀ (𝑧 ∨ π‘₯) ↔ 𝑦 ≀ (𝑧 ∨ 𝑃)))
2421, 23imbi12d 343 . . . . . 6 (π‘₯ = 𝑃 β†’ (((Β¬ π‘₯ ≀ 𝑧 ∧ π‘₯ ≀ (𝑧 ∨ 𝑦)) β†’ 𝑦 ≀ (𝑧 ∨ π‘₯)) ↔ ((Β¬ 𝑃 ≀ 𝑧 ∧ 𝑃 ≀ (𝑧 ∨ 𝑦)) β†’ 𝑦 ≀ (𝑧 ∨ 𝑃))))
2524ralbidv 3168 . . . . 5 (π‘₯ = 𝑃 β†’ (βˆ€π‘§ ∈ 𝐡 ((Β¬ π‘₯ ≀ 𝑧 ∧ π‘₯ ≀ (𝑧 ∨ 𝑦)) β†’ 𝑦 ≀ (𝑧 ∨ π‘₯)) ↔ βˆ€π‘§ ∈ 𝐡 ((Β¬ 𝑃 ≀ 𝑧 ∧ 𝑃 ≀ (𝑧 ∨ 𝑦)) β†’ 𝑦 ≀ (𝑧 ∨ 𝑃))))
2617, 25anbi12d 630 . . . 4 (π‘₯ = 𝑃 β†’ (((π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦))) ∧ βˆ€π‘§ ∈ 𝐡 ((Β¬ π‘₯ ≀ 𝑧 ∧ π‘₯ ≀ (𝑧 ∨ 𝑦)) β†’ 𝑦 ≀ (𝑧 ∨ π‘₯))) ↔ ((𝑃 β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  𝑃 ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (𝑃 ∨ 𝑦))) ∧ βˆ€π‘§ ∈ 𝐡 ((Β¬ 𝑃 ≀ 𝑧 ∧ 𝑃 ≀ (𝑧 ∨ 𝑦)) β†’ 𝑦 ≀ (𝑧 ∨ 𝑃)))))
27 neeq2 2994 . . . . . 6 (𝑦 = 𝑄 β†’ (𝑃 β‰  𝑦 ↔ 𝑃 β‰  𝑄))
28 neeq2 2994 . . . . . . . 8 (𝑦 = 𝑄 β†’ (𝑧 β‰  𝑦 ↔ 𝑧 β‰  𝑄))
29 oveq2 7423 . . . . . . . . 9 (𝑦 = 𝑄 β†’ (𝑃 ∨ 𝑦) = (𝑃 ∨ 𝑄))
3029breq2d 5155 . . . . . . . 8 (𝑦 = 𝑄 β†’ (𝑧 ≀ (𝑃 ∨ 𝑦) ↔ 𝑧 ≀ (𝑃 ∨ 𝑄)))
3128, 303anbi23d 1435 . . . . . . 7 (𝑦 = 𝑄 β†’ ((𝑧 β‰  𝑃 ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (𝑃 ∨ 𝑦)) ↔ (𝑧 β‰  𝑃 ∧ 𝑧 β‰  𝑄 ∧ 𝑧 ≀ (𝑃 ∨ 𝑄))))
3231rexbidv 3169 . . . . . 6 (𝑦 = 𝑄 β†’ (βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  𝑃 ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (𝑃 ∨ 𝑦)) ↔ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  𝑃 ∧ 𝑧 β‰  𝑄 ∧ 𝑧 ≀ (𝑃 ∨ 𝑄))))
3327, 32imbi12d 343 . . . . 5 (𝑦 = 𝑄 β†’ ((𝑃 β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  𝑃 ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (𝑃 ∨ 𝑦))) ↔ (𝑃 β‰  𝑄 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  𝑃 ∧ 𝑧 β‰  𝑄 ∧ 𝑧 ≀ (𝑃 ∨ 𝑄)))))
34 oveq2 7423 . . . . . . . . 9 (𝑦 = 𝑄 β†’ (𝑧 ∨ 𝑦) = (𝑧 ∨ 𝑄))
3534breq2d 5155 . . . . . . . 8 (𝑦 = 𝑄 β†’ (𝑃 ≀ (𝑧 ∨ 𝑦) ↔ 𝑃 ≀ (𝑧 ∨ 𝑄)))
3635anbi2d 628 . . . . . . 7 (𝑦 = 𝑄 β†’ ((Β¬ 𝑃 ≀ 𝑧 ∧ 𝑃 ≀ (𝑧 ∨ 𝑦)) ↔ (Β¬ 𝑃 ≀ 𝑧 ∧ 𝑃 ≀ (𝑧 ∨ 𝑄))))
37 breq1 5146 . . . . . . 7 (𝑦 = 𝑄 β†’ (𝑦 ≀ (𝑧 ∨ 𝑃) ↔ 𝑄 ≀ (𝑧 ∨ 𝑃)))
3836, 37imbi12d 343 . . . . . 6 (𝑦 = 𝑄 β†’ (((Β¬ 𝑃 ≀ 𝑧 ∧ 𝑃 ≀ (𝑧 ∨ 𝑦)) β†’ 𝑦 ≀ (𝑧 ∨ 𝑃)) ↔ ((Β¬ 𝑃 ≀ 𝑧 ∧ 𝑃 ≀ (𝑧 ∨ 𝑄)) β†’ 𝑄 ≀ (𝑧 ∨ 𝑃))))
3938ralbidv 3168 . . . . 5 (𝑦 = 𝑄 β†’ (βˆ€π‘§ ∈ 𝐡 ((Β¬ 𝑃 ≀ 𝑧 ∧ 𝑃 ≀ (𝑧 ∨ 𝑦)) β†’ 𝑦 ≀ (𝑧 ∨ 𝑃)) ↔ βˆ€π‘§ ∈ 𝐡 ((Β¬ 𝑃 ≀ 𝑧 ∧ 𝑃 ≀ (𝑧 ∨ 𝑄)) β†’ 𝑄 ≀ (𝑧 ∨ 𝑃))))
4033, 39anbi12d 630 . . . 4 (𝑦 = 𝑄 β†’ (((𝑃 β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  𝑃 ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (𝑃 ∨ 𝑦))) ∧ βˆ€π‘§ ∈ 𝐡 ((Β¬ 𝑃 ≀ 𝑧 ∧ 𝑃 ≀ (𝑧 ∨ 𝑦)) β†’ 𝑦 ≀ (𝑧 ∨ 𝑃))) ↔ ((𝑃 β‰  𝑄 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  𝑃 ∧ 𝑧 β‰  𝑄 ∧ 𝑧 ≀ (𝑃 ∨ 𝑄))) ∧ βˆ€π‘§ ∈ 𝐡 ((Β¬ 𝑃 ≀ 𝑧 ∧ 𝑃 ≀ (𝑧 ∨ 𝑄)) β†’ 𝑄 ≀ (𝑧 ∨ 𝑃)))))
4126, 40rspc2v 3613 . . 3 ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 ((π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦))) ∧ βˆ€π‘§ ∈ 𝐡 ((Β¬ π‘₯ ≀ 𝑧 ∧ π‘₯ ≀ (𝑧 ∨ 𝑦)) β†’ 𝑦 ≀ (𝑧 ∨ π‘₯))) β†’ ((𝑃 β‰  𝑄 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  𝑃 ∧ 𝑧 β‰  𝑄 ∧ 𝑧 ≀ (𝑃 ∨ 𝑄))) ∧ βˆ€π‘§ ∈ 𝐡 ((Β¬ 𝑃 ≀ 𝑧 ∧ 𝑃 ≀ (𝑧 ∨ 𝑄)) β†’ 𝑄 ≀ (𝑧 ∨ 𝑃)))))
4210, 41mpan9 505 . 2 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑃 β‰  𝑄 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  𝑃 ∧ 𝑧 β‰  𝑄 ∧ 𝑧 ≀ (𝑃 ∨ 𝑄))) ∧ βˆ€π‘§ ∈ 𝐡 ((Β¬ 𝑃 ≀ 𝑧 ∧ 𝑃 ≀ (𝑧 ∨ 𝑄)) β†’ 𝑄 ≀ (𝑧 ∨ 𝑃))))
43423impb 1112 1 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ ((𝑃 β‰  𝑄 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  𝑃 ∧ 𝑧 β‰  𝑄 ∧ 𝑧 ≀ (𝑃 ∨ 𝑄))) ∧ βˆ€π‘§ ∈ 𝐡 ((Β¬ 𝑃 ≀ 𝑧 ∧ 𝑃 ≀ (𝑧 ∨ 𝑄)) β†’ 𝑄 ≀ (𝑧 ∨ 𝑃))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  βˆ€wral 3051  βˆƒwrex 3060   class class class wbr 5143  β€˜cfv 6542  (class class class)co 7415  Basecbs 17177  lecple 17237  ltcplt 18297  joincjn 18300  0.cp0 18412  1.cp1 18413  CLatccla 18487  OMLcoml 38702  Atomscatm 38790  AtLatcal 38791  HLchlt 38877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-iota 6494  df-fv 6550  df-ov 7418  df-cvlat 38849  df-hlat 38878
This theorem is referenced by:  hlsupr  38914
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