Theorem hlsuprexch 39500

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 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ((𝑃 ≠ 𝑄 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑄 ∧ 𝑧 ≤ (𝑃 ∨ 𝑄))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (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.

Step	Hyp	Ref	Expression
1		hlsuprexch.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		hlsuprexch.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
3		eqid 2733	. . . . 5 ⊢ (lt‘𝐾) = (lt‘𝐾)
4		hlsuprexch.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
5		eqid 2733	. . . . 5 ⊢ (0.‘𝐾) = (0.‘𝐾)
6		eqid 2733	. . . . 5 ⊢ (1.‘𝐾) = (1.‘𝐾)
7		hlsuprexch.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
8	1, 2, 3, 4, 5, 6, 7	ishlat2 39472	. . . 4 ⊢ (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥))) ∧ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (((0.‘𝐾)(lt‘𝐾)𝑥 ∧ 𝑥(lt‘𝐾)𝑦) ∧ (𝑦(lt‘𝐾)𝑧 ∧ 𝑧(lt‘𝐾)(1.‘𝐾))))))
9		simprl 770	. . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥))) ∧ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (((0.‘𝐾)(lt‘𝐾)𝑥 ∧ 𝑥(lt‘𝐾)𝑦) ∧ (𝑦(lt‘𝐾)𝑧 ∧ 𝑧(lt‘𝐾)(1.‘𝐾))))) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥))))
10	8, 9	sylbi 217	. . 3 ⊢ (𝐾 ∈ HL → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥))))
11		neeq1 2991	. . . . . 6 ⊢ (𝑥 = 𝑃 → (𝑥 ≠ 𝑦 ↔ 𝑃 ≠ 𝑦))
12		neeq2 2992	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (𝑧 ≠ 𝑥 ↔ 𝑧 ≠ 𝑃))
13		oveq1 7359	. . . . . . . . 9 ⊢ (𝑥 = 𝑃 → (𝑥 ∨ 𝑦) = (𝑃 ∨ 𝑦))
14	13	breq2d 5105	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (𝑧 ≤ (𝑥 ∨ 𝑦) ↔ 𝑧 ≤ (𝑃 ∨ 𝑦)))
15	12, 14	3anbi13d 1440	. . . . . . 7 ⊢ (𝑥 = 𝑃 → ((𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) ↔ (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑃 ∨ 𝑦))))
16	15	rexbidv 3157	. . . . . 6 ⊢ (𝑥 = 𝑃 → (∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) ↔ ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑃 ∨ 𝑦))))
17	11, 16	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑃 → ((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ↔ (𝑃 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑃 ∨ 𝑦)))))
18		breq1 5096	. . . . . . . . 9 ⊢ (𝑥 = 𝑃 → (𝑥 ≤ 𝑧 ↔ 𝑃 ≤ 𝑧))
19	18	notbid 318	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (¬ 𝑥 ≤ 𝑧 ↔ ¬ 𝑃 ≤ 𝑧))
20		breq1 5096	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (𝑥 ≤ (𝑧 ∨ 𝑦) ↔ 𝑃 ≤ (𝑧 ∨ 𝑦)))
21	19, 20	anbi12d 632	. . . . . . 7 ⊢ (𝑥 = 𝑃 → ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) ↔ (¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑦))))
22		oveq2 7360	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (𝑧 ∨ 𝑥) = (𝑧 ∨ 𝑃))
23	22	breq2d 5105	. . . . . . 7 ⊢ (𝑥 = 𝑃 → (𝑦 ≤ (𝑧 ∨ 𝑥) ↔ 𝑦 ≤ (𝑧 ∨ 𝑃)))
24	21, 23	imbi12d 344	. . . . . 6 ⊢ (𝑥 = 𝑃 → (((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥)) ↔ ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑃))))
25	24	ralbidv 3156	. . . . 5 ⊢ (𝑥 = 𝑃 → (∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥)) ↔ ∀𝑧 ∈ 𝐵 ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑃))))
26	17, 25	anbi12d 632	. . . 4 ⊢ (𝑥 = 𝑃 → (((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥))) ↔ ((𝑃 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑃 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑃)))))
27		neeq2 2992	. . . . . 6 ⊢ (𝑦 = 𝑄 → (𝑃 ≠ 𝑦 ↔ 𝑃 ≠ 𝑄))
28		neeq2 2992	. . . . . . . 8 ⊢ (𝑦 = 𝑄 → (𝑧 ≠ 𝑦 ↔ 𝑧 ≠ 𝑄))
29		oveq2 7360	. . . . . . . . 9 ⊢ (𝑦 = 𝑄 → (𝑃 ∨ 𝑦) = (𝑃 ∨ 𝑄))
30	29	breq2d 5105	. . . . . . . 8 ⊢ (𝑦 = 𝑄 → (𝑧 ≤ (𝑃 ∨ 𝑦) ↔ 𝑧 ≤ (𝑃 ∨ 𝑄)))
31	28, 30	3anbi23d 1441	. . . . . . 7 ⊢ (𝑦 = 𝑄 → ((𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑃 ∨ 𝑦)) ↔ (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑄 ∧ 𝑧 ≤ (𝑃 ∨ 𝑄))))
32	31	rexbidv 3157	. . . . . 6 ⊢ (𝑦 = 𝑄 → (∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑃 ∨ 𝑦)) ↔ ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑄 ∧ 𝑧 ≤ (𝑃 ∨ 𝑄))))
33	27, 32	imbi12d 344	. . . . 5 ⊢ (𝑦 = 𝑄 → ((𝑃 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑃 ∨ 𝑦))) ↔ (𝑃 ≠ 𝑄 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑄 ∧ 𝑧 ≤ (𝑃 ∨ 𝑄)))))
34		oveq2 7360	. . . . . . . . 9 ⊢ (𝑦 = 𝑄 → (𝑧 ∨ 𝑦) = (𝑧 ∨ 𝑄))
35	34	breq2d 5105	. . . . . . . 8 ⊢ (𝑦 = 𝑄 → (𝑃 ≤ (𝑧 ∨ 𝑦) ↔ 𝑃 ≤ (𝑧 ∨ 𝑄)))
36	35	anbi2d 630	. . . . . . 7 ⊢ (𝑦 = 𝑄 → ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑦)) ↔ (¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑄))))
37		breq1 5096	. . . . . . 7 ⊢ (𝑦 = 𝑄 → (𝑦 ≤ (𝑧 ∨ 𝑃) ↔ 𝑄 ≤ (𝑧 ∨ 𝑃)))
38	36, 37	imbi12d 344	. . . . . 6 ⊢ (𝑦 = 𝑄 → (((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑃)) ↔ ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑄)) → 𝑄 ≤ (𝑧 ∨ 𝑃))))
39	38	ralbidv 3156	. . . . 5 ⊢ (𝑦 = 𝑄 → (∀𝑧 ∈ 𝐵 ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑃)) ↔ ∀𝑧 ∈ 𝐵 ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑄)) → 𝑄 ≤ (𝑧 ∨ 𝑃))))
40	33, 39	anbi12d 632	. . . 4 ⊢ (𝑦 = 𝑄 → (((𝑃 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑃 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑃))) ↔ ((𝑃 ≠ 𝑄 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑄 ∧ 𝑧 ≤ (𝑃 ∨ 𝑄))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑄)) → 𝑄 ≤ (𝑧 ∨ 𝑃)))))
41	26, 40	rspc2v 3584	. . 3 ⊢ ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥))) → ((𝑃 ≠ 𝑄 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑄 ∧ 𝑧 ≤ (𝑃 ∨ 𝑄))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑄)) → 𝑄 ≤ (𝑧 ∨ 𝑃)))))
42	10, 41	mpan9 506	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑃 ≠ 𝑄 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑄 ∧ 𝑧 ≤ (𝑃 ∨ 𝑄))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑄)) → 𝑄 ≤ (𝑧 ∨ 𝑃))))
43	42	3impb 1114	1 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ((𝑃 ≠ 𝑄 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑄 ∧ 𝑧 ≤ (𝑃 ∨ 𝑄))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑄)) → 𝑄 ≤ (𝑧 ∨ 𝑃))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  ∀wral 3048  ∃wrex 3057   class class class wbr 5093  ‘cfv 6486  (class class class)co 7352  Basecbs 17122  lecple 17170  ltcplt 18216  joincjn 18219  0.cp0 18329  1.cp1 18330  CLatccla 18406  OMLcoml 39294  Atomscatm 39382  AtLatcal 39383  HLchlt 39469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-iota 6442  df-fv 6494  df-ov 7355  df-cvlat 39441  df-hlat 39470

This theorem is referenced by:  hlsupr  39505