Theorem ltrnatlw

Description: If the value of an atom equals the atom in a non-identity translation, the atom is under the fiducial hyperplane. (Contributed by NM, 15-May-2013)

	Ref	Expression
Hypotheses	ltrn2eq.l	⊢ ≤ = ( le ‘ 𝐾 )
	ltrn2eq.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	ltrn2eq.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	ltrn2eq.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion	ltrnatlw	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) → 𝑄 ≤ 𝑊 )

Proof

Step	Hyp	Ref	Expression
1		ltrn2eq.l	⊢ ≤ = ( le ‘ 𝐾 )
2		ltrn2eq.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3		ltrn2eq.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		ltrn2eq.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5		simp3r	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) → ( 𝐹 ‘ 𝑄 ) = 𝑄 )
6		simpl1	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 ≤ 𝑊 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
7		simpl21	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 ≤ 𝑊 ) → 𝐹 ∈ 𝑇 )
8		simpl22	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 ≤ 𝑊 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
9		simpl23	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 ≤ 𝑊 ) → 𝑄 ∈ 𝐴 )
10		simpr	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 ≤ 𝑊 ) → ¬ 𝑄 ≤ 𝑊 )
11	9 10	jca	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 ≤ 𝑊 ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
12		simpl3l	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 ≤ 𝑊 ) → ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 )
13	1 2 3 4	ltrnatneq	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝐹 ‘ 𝑄 ) ≠ 𝑄 )
14	6 7 8 11 12 13	syl131anc	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 ≤ 𝑊 ) → ( 𝐹 ‘ 𝑄 ) ≠ 𝑄 )
15	14	ex	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) → ( ¬ 𝑄 ≤ 𝑊 → ( 𝐹 ‘ 𝑄 ) ≠ 𝑄 ) )
16	15	necon4bd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) → ( ( 𝐹 ‘ 𝑄 ) = 𝑄 → 𝑄 ≤ 𝑊 ) )
17	5 16	mpd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) → 𝑄 ≤ 𝑊 )