Theorem ltrnu

Description: Uniqueness property of a lattice translation value for atoms not under the fiducial co-atom W . Similar to definition of translation in Crawley p. 111. (Contributed by NM, 20-May-2012)

	Ref	Expression
Hypotheses	ltrnu.l	⊢ ≤ = ( le ‘ 𝐾 )
	ltrnu.j	⊢ ∨ = ( join ‘ 𝐾 )
	ltrnu.m	⊢ ∧ = ( meet ‘ 𝐾 )
	ltrnu.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	ltrnu.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	ltrnu.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion	ltrnu	⊢ ( ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) = ( ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ∧ 𝑊 ) )

Proof

Step	Hyp	Ref	Expression
1		ltrnu.l	⊢ ≤ = ( le ‘ 𝐾 )
2		ltrnu.j	⊢ ∨ = ( join ‘ 𝐾 )
3		ltrnu.m	⊢ ∧ = ( meet ‘ 𝐾 )
4		ltrnu.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		ltrnu.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6		ltrnu.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7		an4	⊢ ( ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ↔ ( ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( ¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊 ) ) )
8		simpr	⊢ ( ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) )
9		simplr	⊢ ( ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝐹 ∈ 𝑇 )
10		eqid	⊢ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 )
11	1 2 3 4 5 10 6	isltrn	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 ∈ 𝑇 ↔ ( 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ∧ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) → ( ( 𝑝 ∨ ( 𝐹 ‘ 𝑝 ) ) ∧ 𝑊 ) = ( ( 𝑞 ∨ ( 𝐹 ‘ 𝑞 ) ) ∧ 𝑊 ) ) ) ) )
12	11	ad2antrr	⊢ ( ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝐹 ∈ 𝑇 ↔ ( 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ∧ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) → ( ( 𝑝 ∨ ( 𝐹 ‘ 𝑝 ) ) ∧ 𝑊 ) = ( ( 𝑞 ∨ ( 𝐹 ‘ 𝑞 ) ) ∧ 𝑊 ) ) ) ) )
13		simpr	⊢ ( ( 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ∧ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) → ( ( 𝑝 ∨ ( 𝐹 ‘ 𝑝 ) ) ∧ 𝑊 ) = ( ( 𝑞 ∨ ( 𝐹 ‘ 𝑞 ) ) ∧ 𝑊 ) ) ) → ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) → ( ( 𝑝 ∨ ( 𝐹 ‘ 𝑝 ) ) ∧ 𝑊 ) = ( ( 𝑞 ∨ ( 𝐹 ‘ 𝑞 ) ) ∧ 𝑊 ) ) )
14	12 13	syl6bi	⊢ ( ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝐹 ∈ 𝑇 → ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) → ( ( 𝑝 ∨ ( 𝐹 ‘ 𝑝 ) ) ∧ 𝑊 ) = ( ( 𝑞 ∨ ( 𝐹 ‘ 𝑞 ) ) ∧ 𝑊 ) ) ) )
15	9 14	mpd	⊢ ( ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) → ( ( 𝑝 ∨ ( 𝐹 ‘ 𝑝 ) ) ∧ 𝑊 ) = ( ( 𝑞 ∨ ( 𝐹 ‘ 𝑞 ) ) ∧ 𝑊 ) ) )
16		breq1	⊢ ( 𝑝 = 𝑃 → ( 𝑝 ≤ 𝑊 ↔ 𝑃 ≤ 𝑊 ) )
17	16	notbid	⊢ ( 𝑝 = 𝑃 → ( ¬ 𝑝 ≤ 𝑊 ↔ ¬ 𝑃 ≤ 𝑊 ) )
18	17	anbi1d	⊢ ( 𝑝 = 𝑃 → ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ↔ ( ¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ) )
19		id	⊢ ( 𝑝 = 𝑃 → 𝑝 = 𝑃 )
20		fveq2	⊢ ( 𝑝 = 𝑃 → ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑃 ) )
21	19 20	oveq12d	⊢ ( 𝑝 = 𝑃 → ( 𝑝 ∨ ( 𝐹 ‘ 𝑝 ) ) = ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
22	21	oveq1d	⊢ ( 𝑝 = 𝑃 → ( ( 𝑝 ∨ ( 𝐹 ‘ 𝑝 ) ) ∧ 𝑊 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) )
23	22	eqeq1d	⊢ ( 𝑝 = 𝑃 → ( ( ( 𝑝 ∨ ( 𝐹 ‘ 𝑝 ) ) ∧ 𝑊 ) = ( ( 𝑞 ∨ ( 𝐹 ‘ 𝑞 ) ) ∧ 𝑊 ) ↔ ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) = ( ( 𝑞 ∨ ( 𝐹 ‘ 𝑞 ) ) ∧ 𝑊 ) ) )
24	18 23	imbi12d	⊢ ( 𝑝 = 𝑃 → ( ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) → ( ( 𝑝 ∨ ( 𝐹 ‘ 𝑝 ) ) ∧ 𝑊 ) = ( ( 𝑞 ∨ ( 𝐹 ‘ 𝑞 ) ) ∧ 𝑊 ) ) ↔ ( ( ¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) = ( ( 𝑞 ∨ ( 𝐹 ‘ 𝑞 ) ) ∧ 𝑊 ) ) ) )
25		breq1	⊢ ( 𝑞 = 𝑄 → ( 𝑞 ≤ 𝑊 ↔ 𝑄 ≤ 𝑊 ) )
26	25	notbid	⊢ ( 𝑞 = 𝑄 → ( ¬ 𝑞 ≤ 𝑊 ↔ ¬ 𝑄 ≤ 𝑊 ) )
27	26	anbi2d	⊢ ( 𝑞 = 𝑄 → ( ( ¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ↔ ( ¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊 ) ) )
28		id	⊢ ( 𝑞 = 𝑄 → 𝑞 = 𝑄 )
29		fveq2	⊢ ( 𝑞 = 𝑄 → ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑄 ) )
30	28 29	oveq12d	⊢ ( 𝑞 = 𝑄 → ( 𝑞 ∨ ( 𝐹 ‘ 𝑞 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) )
31	30	oveq1d	⊢ ( 𝑞 = 𝑄 → ( ( 𝑞 ∨ ( 𝐹 ‘ 𝑞 ) ) ∧ 𝑊 ) = ( ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ∧ 𝑊 ) )
32	31	eqeq2d	⊢ ( 𝑞 = 𝑄 → ( ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) = ( ( 𝑞 ∨ ( 𝐹 ‘ 𝑞 ) ) ∧ 𝑊 ) ↔ ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) = ( ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ∧ 𝑊 ) ) )
33	27 32	imbi12d	⊢ ( 𝑞 = 𝑄 → ( ( ( ¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) = ( ( 𝑞 ∨ ( 𝐹 ‘ 𝑞 ) ) ∧ 𝑊 ) ) ↔ ( ( ¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊 ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) = ( ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ∧ 𝑊 ) ) ) )
34	24 33	rspc2v	⊢ ( ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) → ( ( 𝑝 ∨ ( 𝐹 ‘ 𝑝 ) ) ∧ 𝑊 ) = ( ( 𝑞 ∨ ( 𝐹 ‘ 𝑞 ) ) ∧ 𝑊 ) ) → ( ( ¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊 ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) = ( ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ∧ 𝑊 ) ) ) )
35	8 15 34	sylc	⊢ ( ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( ¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊 ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) = ( ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ∧ 𝑊 ) ) )
36	35	impr	⊢ ( ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( ¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) = ( ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ∧ 𝑊 ) )
37	7 36	sylan2b	⊢ ( ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) = ( ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ∧ 𝑊 ) )
38	37	3impb	⊢ ( ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) = ( ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ∧ 𝑊 ) )