Theorem hlsuprexch 39363

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 2734	. . . . 5 ⊢ (lt‘𝐾) = (lt‘𝐾)
4		hlsuprexch.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
5		eqid 2734	. . . . 5 ⊢ (0.‘𝐾) = (0.‘𝐾)
6		eqid 2734	. . . . 5 ⊢ (1.‘𝐾) = (1.‘𝐾)
7		hlsuprexch.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
8	1, 2, 3, 4, 5, 6, 7	ishlat2 39334	. . . 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 771	. . . 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 3000	. . . . . 6 ⊢ (𝑥 = 𝑃 → (𝑥 ≠ 𝑦 ↔ 𝑃 ≠ 𝑦))
12		neeq2 3001	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (𝑧 ≠ 𝑥 ↔ 𝑧 ≠ 𝑃))
13		oveq1 7437	. . . . . . . . 9 ⊢ (𝑥 = 𝑃 → (𝑥 ∨ 𝑦) = (𝑃 ∨ 𝑦))
14	13	breq2d 5159	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (𝑧 ≤ (𝑥 ∨ 𝑦) ↔ 𝑧 ≤ (𝑃 ∨ 𝑦)))
15	12, 14	3anbi13d 1437	. . . . . . 7 ⊢ (𝑥 = 𝑃 → ((𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) ↔ (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑃 ∨ 𝑦))))
16	15	rexbidv 3176	. . . . . 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 5150	. . . . . . . . 9 ⊢ (𝑥 = 𝑃 → (𝑥 ≤ 𝑧 ↔ 𝑃 ≤ 𝑧))
19	18	notbid 318	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (¬ 𝑥 ≤ 𝑧 ↔ ¬ 𝑃 ≤ 𝑧))
20		breq1 5150	. . . . . . . 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 7438	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (𝑧 ∨ 𝑥) = (𝑧 ∨ 𝑃))
23	22	breq2d 5159	. . . . . . 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 3175	. . . . 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 3001	. . . . . 6 ⊢ (𝑦 = 𝑄 → (𝑃 ≠ 𝑦 ↔ 𝑃 ≠ 𝑄))
28		neeq2 3001	. . . . . . . 8 ⊢ (𝑦 = 𝑄 → (𝑧 ≠ 𝑦 ↔ 𝑧 ≠ 𝑄))
29		oveq2 7438	. . . . . . . . 9 ⊢ (𝑦 = 𝑄 → (𝑃 ∨ 𝑦) = (𝑃 ∨ 𝑄))
30	29	breq2d 5159	. . . . . . . 8 ⊢ (𝑦 = 𝑄 → (𝑧 ≤ (𝑃 ∨ 𝑦) ↔ 𝑧 ≤ (𝑃 ∨ 𝑄)))
31	28, 30	3anbi23d 1438	. . . . . . 7 ⊢ (𝑦 = 𝑄 → ((𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑃 ∨ 𝑦)) ↔ (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑄 ∧ 𝑧 ≤ (𝑃 ∨ 𝑄))))
32	31	rexbidv 3176	. . . . . 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 7438	. . . . . . . . 9 ⊢ (𝑦 = 𝑄 → (𝑧 ∨ 𝑦) = (𝑧 ∨ 𝑄))
35	34	breq2d 5159	. . . . . . . 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 5150	. . . . . . 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 3175	. . . . 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 3632	. . 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 1536   ∈ wcel 2105   ≠ wne 2937  ∀wral 3058  ∃wrex 3067   class class class wbr 5147  ‘cfv 6562  (class class class)co 7430  Basecbs 17244  lecple 17304  ltcplt 18365  joincjn 18368  0.cp0 18480  1.cp1 18481  CLatccla 18555  OMLcoml 39156  Atomscatm 39244  AtLatcal 39245  HLchlt 39331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-ov 7433  df-cvlat 39303  df-hlat 39332

This theorem is referenced by:  hlsupr  39368