Theorem hlsuprexch 40262

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 2762	. . . . 5 ⊢ (lt‘𝐾) = (lt‘𝐾)
4		hlsuprexch.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
5		eqid 2762	. . . . 5 ⊢ (0.‘𝐾) = (0.‘𝐾)
6		eqid 2762	. . . . 5 ⊢ (1.‘𝐾) = (1.‘𝐾)
7		hlsuprexch.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
8	1, 2, 3, 4, 5, 6, 7	ishlat2 40234	. . . 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 783	. . . 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 220	. . 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 3019	. . . . . 6 ⊢ (𝑥 = 𝑃 → (𝑥 ≠ 𝑦 ↔ 𝑃 ≠ 𝑦))
12		neeq2 3020	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (𝑧 ≠ 𝑥 ↔ 𝑧 ≠ 𝑃))
13		oveq1 7424	. . . . . . . . 9 ⊢ (𝑥 = 𝑃 → (𝑥 ∨ 𝑦) = (𝑃 ∨ 𝑦))
14	13	breq2d 5119	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (𝑧 ≤ (𝑥 ∨ 𝑦) ↔ 𝑧 ≤ (𝑃 ∨ 𝑦)))
15	12, 14	3anbi13d 1466	. . . . . . 7 ⊢ (𝑥 = 𝑃 → ((𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) ↔ (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑃 ∨ 𝑦))))
16	15	rexbidv 3188	. . . . . 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 347	. . . . 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 5110	. . . . . . . . 9 ⊢ (𝑥 = 𝑃 → (𝑥 ≤ 𝑧 ↔ 𝑃 ≤ 𝑧))
19	18	notbid 321	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (¬ 𝑥 ≤ 𝑧 ↔ ¬ 𝑃 ≤ 𝑧))
20		breq1 5110	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (𝑥 ≤ (𝑧 ∨ 𝑦) ↔ 𝑃 ≤ (𝑧 ∨ 𝑦)))
21	19, 20	anbi12d 644	. . . . . . 7 ⊢ (𝑥 = 𝑃 → ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) ↔ (¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑦))))
22		oveq2 7425	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (𝑧 ∨ 𝑥) = (𝑧 ∨ 𝑃))
23	22	breq2d 5119	. . . . . . 7 ⊢ (𝑥 = 𝑃 → (𝑦 ≤ (𝑧 ∨ 𝑥) ↔ 𝑦 ≤ (𝑧 ∨ 𝑃)))
24	21, 23	imbi12d 347	. . . . . 6 ⊢ (𝑥 = 𝑃 → (((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥)) ↔ ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑃))))
25	24	ralbidv 3187	. . . . 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 644	. . . 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 3020	. . . . . 6 ⊢ (𝑦 = 𝑄 → (𝑃 ≠ 𝑦 ↔ 𝑃 ≠ 𝑄))
28		neeq2 3020	. . . . . . . 8 ⊢ (𝑦 = 𝑄 → (𝑧 ≠ 𝑦 ↔ 𝑧 ≠ 𝑄))
29		oveq2 7425	. . . . . . . . 9 ⊢ (𝑦 = 𝑄 → (𝑃 ∨ 𝑦) = (𝑃 ∨ 𝑄))
30	29	breq2d 5119	. . . . . . . 8 ⊢ (𝑦 = 𝑄 → (𝑧 ≤ (𝑃 ∨ 𝑦) ↔ 𝑧 ≤ (𝑃 ∨ 𝑄)))
31	28, 30	3anbi23d 1467	. . . . . . 7 ⊢ (𝑦 = 𝑄 → ((𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑃 ∨ 𝑦)) ↔ (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑄 ∧ 𝑧 ≤ (𝑃 ∨ 𝑄))))
32	31	rexbidv 3188	. . . . . 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 347	. . . . 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 7425	. . . . . . . . 9 ⊢ (𝑦 = 𝑄 → (𝑧 ∨ 𝑦) = (𝑧 ∨ 𝑄))
35	34	breq2d 5119	. . . . . . . 8 ⊢ (𝑦 = 𝑄 → (𝑃 ≤ (𝑧 ∨ 𝑦) ↔ 𝑃 ≤ (𝑧 ∨ 𝑄)))
36	35	anbi2d 642	. . . . . . 7 ⊢ (𝑦 = 𝑄 → ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑦)) ↔ (¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑄))))
37		breq1 5110	. . . . . . 7 ⊢ (𝑦 = 𝑄 → (𝑦 ≤ (𝑧 ∨ 𝑃) ↔ 𝑄 ≤ (𝑧 ∨ 𝑃)))
38	36, 37	imbi12d 347	. . . . . 6 ⊢ (𝑦 = 𝑄 → (((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑃)) ↔ ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑄)) → 𝑄 ≤ (𝑧 ∨ 𝑃))))
39	38	ralbidv 3187	. . . . 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 644	. . . 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 3590	. . 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 516	. 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 1132	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))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  ∃wrex 3088   class class class wbr 5107  ‘cfv 6537  (class class class)co 7417  Basecbs 17307  lecple 17355  ltcplt 18402  joincjn 18405  0.cp0 18515  1.cp1 18516  CLatccla 18592  OMLcoml 40056  Atomscatm 40144  AtLatcal 40145  HLchlt 40231

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-ov 7420  df-cvlat 40203  df-hlat 40232

This theorem is used by:  hlsupr  40267