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Theorem hlsuprexch 37890
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 2733 . . . . 5 (ltβ€˜πΎ) = (ltβ€˜πΎ)
4 hlsuprexch.j . . . . 5 ∨ = (joinβ€˜πΎ)
5 eqid 2733 . . . . 5 (0.β€˜πΎ) = (0.β€˜πΎ)
6 eqid 2733 . . . . 5 (1.β€˜πΎ) = (1.β€˜πΎ)
7 hlsuprexch.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
81, 2, 3, 4, 5, 6, 7ishlat2 37861 . . . 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 770 . . . 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 3003 . . . . . 6 (π‘₯ = 𝑃 β†’ (π‘₯ β‰  𝑦 ↔ 𝑃 β‰  𝑦))
12 neeq2 3004 . . . . . . . 8 (π‘₯ = 𝑃 β†’ (𝑧 β‰  π‘₯ ↔ 𝑧 β‰  𝑃))
13 oveq1 7365 . . . . . . . . 9 (π‘₯ = 𝑃 β†’ (π‘₯ ∨ 𝑦) = (𝑃 ∨ 𝑦))
1413breq2d 5118 . . . . . . . 8 (π‘₯ = 𝑃 β†’ (𝑧 ≀ (π‘₯ ∨ 𝑦) ↔ 𝑧 ≀ (𝑃 ∨ 𝑦)))
1512, 143anbi13d 1439 . . . . . . 7 (π‘₯ = 𝑃 β†’ ((𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦)) ↔ (𝑧 β‰  𝑃 ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (𝑃 ∨ 𝑦))))
1615rexbidv 3172 . . . . . 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 345 . . . . 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 5109 . . . . . . . . 9 (π‘₯ = 𝑃 β†’ (π‘₯ ≀ 𝑧 ↔ 𝑃 ≀ 𝑧))
1918notbid 318 . . . . . . . 8 (π‘₯ = 𝑃 β†’ (Β¬ π‘₯ ≀ 𝑧 ↔ Β¬ 𝑃 ≀ 𝑧))
20 breq1 5109 . . . . . . . 8 (π‘₯ = 𝑃 β†’ (π‘₯ ≀ (𝑧 ∨ 𝑦) ↔ 𝑃 ≀ (𝑧 ∨ 𝑦)))
2119, 20anbi12d 632 . . . . . . 7 (π‘₯ = 𝑃 β†’ ((Β¬ π‘₯ ≀ 𝑧 ∧ π‘₯ ≀ (𝑧 ∨ 𝑦)) ↔ (Β¬ 𝑃 ≀ 𝑧 ∧ 𝑃 ≀ (𝑧 ∨ 𝑦))))
22 oveq2 7366 . . . . . . . 8 (π‘₯ = 𝑃 β†’ (𝑧 ∨ π‘₯) = (𝑧 ∨ 𝑃))
2322breq2d 5118 . . . . . . 7 (π‘₯ = 𝑃 β†’ (𝑦 ≀ (𝑧 ∨ π‘₯) ↔ 𝑦 ≀ (𝑧 ∨ 𝑃)))
2421, 23imbi12d 345 . . . . . 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 3171 . . . . 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 632 . . . 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 3004 . . . . . 6 (𝑦 = 𝑄 β†’ (𝑃 β‰  𝑦 ↔ 𝑃 β‰  𝑄))
28 neeq2 3004 . . . . . . . 8 (𝑦 = 𝑄 β†’ (𝑧 β‰  𝑦 ↔ 𝑧 β‰  𝑄))
29 oveq2 7366 . . . . . . . . 9 (𝑦 = 𝑄 β†’ (𝑃 ∨ 𝑦) = (𝑃 ∨ 𝑄))
3029breq2d 5118 . . . . . . . 8 (𝑦 = 𝑄 β†’ (𝑧 ≀ (𝑃 ∨ 𝑦) ↔ 𝑧 ≀ (𝑃 ∨ 𝑄)))
3128, 303anbi23d 1440 . . . . . . 7 (𝑦 = 𝑄 β†’ ((𝑧 β‰  𝑃 ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (𝑃 ∨ 𝑦)) ↔ (𝑧 β‰  𝑃 ∧ 𝑧 β‰  𝑄 ∧ 𝑧 ≀ (𝑃 ∨ 𝑄))))
3231rexbidv 3172 . . . . . 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 345 . . . . 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 7366 . . . . . . . . 9 (𝑦 = 𝑄 β†’ (𝑧 ∨ 𝑦) = (𝑧 ∨ 𝑄))
3534breq2d 5118 . . . . . . . 8 (𝑦 = 𝑄 β†’ (𝑃 ≀ (𝑧 ∨ 𝑦) ↔ 𝑃 ≀ (𝑧 ∨ 𝑄)))
3635anbi2d 630 . . . . . . 7 (𝑦 = 𝑄 β†’ ((Β¬ 𝑃 ≀ 𝑧 ∧ 𝑃 ≀ (𝑧 ∨ 𝑦)) ↔ (Β¬ 𝑃 ≀ 𝑧 ∧ 𝑃 ≀ (𝑧 ∨ 𝑄))))
37 breq1 5109 . . . . . . 7 (𝑦 = 𝑄 β†’ (𝑦 ≀ (𝑧 ∨ 𝑃) ↔ 𝑄 ≀ (𝑧 ∨ 𝑃)))
3836, 37imbi12d 345 . . . . . 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 3171 . . . . 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 632 . . . 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 3589 . . 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 508 . 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 1116 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  lecple 17145  ltcplt 18202  joincjn 18205  0.cp0 18317  1.cp1 18318  CLatccla 18392  OMLcoml 37683  Atomscatm 37771  AtLatcal 37772  HLchlt 37858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-cvlat 37830  df-hlat 37859
This theorem is referenced by:  hlsupr  37895
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