ltrncnvel

Description: The converse of the lattice translation of an atom not under the fiducial co-atom. (Contributed by NM, 10-May-2013)

	Ref	Expression
Hypotheses	ltrnel.l	⊢ ≤ = ( le ‘ 𝐾 )
	ltrnel.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	ltrnel.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	ltrnel.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion	ltrncnvel	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ^◡ 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( ^◡ 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) )

Proof

Step	Hyp	Ref	Expression
1		ltrnel.l	⊢ ≤ = ( le ‘ 𝐾 )
2		ltrnel.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3		ltrnel.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		ltrnel.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5	1 2 3 4	ltrncnvat	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴 ) → ( ^◡ 𝐹 ‘ 𝑃 ) ∈ 𝐴 )
6	5	3adant3r	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ^◡ 𝐹 ‘ 𝑃 ) ∈ 𝐴 )
7		simp3r	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ¬ 𝑃 ≤ 𝑊 )
8		simp1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9		simp2	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 )
10		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
11	10 2	atbase	⊢ ( ( ^◡ 𝐹 ‘ 𝑃 ) ∈ 𝐴 → ( ^◡ 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
12	6 11	syl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ^◡ 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
13		simp1r	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑊 ∈ 𝐻 )
14	10 3	lhpbase	⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
15	13 14	syl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
16	10 1 3 4	ltrnle	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( ^◡ 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ^◡ 𝐹 ‘ 𝑃 ) ≤ 𝑊 ↔ ( 𝐹 ‘ ( ^◡ 𝐹 ‘ 𝑃 ) ) ≤ ( 𝐹 ‘ 𝑊 ) ) )
17	8 9 12 15 16	syl112anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ^◡ 𝐹 ‘ 𝑃 ) ≤ 𝑊 ↔ ( 𝐹 ‘ ( ^◡ 𝐹 ‘ 𝑃 ) ) ≤ ( 𝐹 ‘ 𝑊 ) ) )
18	10 3 4	ltrn1o	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
19	18	3adant3	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
20		simp3l	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑃 ∈ 𝐴 )
21	10 2	atbase	⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
22	20 21	syl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
23		f1ocnvfv2	⊢ ( ( 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐹 ‘ ( ^◡ 𝐹 ‘ 𝑃 ) ) = 𝑃 )
24	19 22 23	syl2anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 ‘ ( ^◡ 𝐹 ‘ 𝑃 ) ) = 𝑃 )
25		simp1l	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐾 ∈ HL )
26	25	hllatd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐾 ∈ Lat )
27	10 1	latref	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → 𝑊 ≤ 𝑊 )
28	26 15 27	syl2anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑊 ≤ 𝑊 )
29	10 1 3 4	ltrnval1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑊 ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑊 ) = 𝑊 )
30	8 9 15 28 29	syl112anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑊 ) = 𝑊 )
31	24 30	breq12d	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ ( ^◡ 𝐹 ‘ 𝑃 ) ) ≤ ( 𝐹 ‘ 𝑊 ) ↔ 𝑃 ≤ 𝑊 ) )
32	17 31	bitrd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ^◡ 𝐹 ‘ 𝑃 ) ≤ 𝑊 ↔ 𝑃 ≤ 𝑊 ) )
33	7 32	mtbird	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ¬ ( ^◡ 𝐹 ‘ 𝑃 ) ≤ 𝑊 )
34	6 33	jca	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ^◡ 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( ^◡ 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) )

Metamath Proof Explorer

Theorem ltrncnvel

Proof