Metamath Proof Explorer


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 ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( ( 𝐹𝑃 ) ≠ 𝑃 ∧ ( 𝐹𝑄 ) = 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 ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( ( 𝐹𝑃 ) ≠ 𝑃 ∧ ( 𝐹𝑄 ) = 𝑄 ) ) → ( 𝐹𝑄 ) = 𝑄 )
6 simpl1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( ( 𝐹𝑃 ) ≠ 𝑃 ∧ ( 𝐹𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 𝑊 ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
7 simpl21 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( ( 𝐹𝑃 ) ≠ 𝑃 ∧ ( 𝐹𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 𝑊 ) → 𝐹𝑇 )
8 simpl22 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( ( 𝐹𝑃 ) ≠ 𝑃 ∧ ( 𝐹𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 𝑊 ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
9 simpl23 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( ( 𝐹𝑃 ) ≠ 𝑃 ∧ ( 𝐹𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 𝑊 ) → 𝑄𝐴 )
10 simpr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( ( 𝐹𝑃 ) ≠ 𝑃 ∧ ( 𝐹𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 𝑊 ) → ¬ 𝑄 𝑊 )
11 9 10 jca ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( ( 𝐹𝑃 ) ≠ 𝑃 ∧ ( 𝐹𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 𝑊 ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
12 simpl3l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( ( 𝐹𝑃 ) ≠ 𝑃 ∧ ( 𝐹𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 𝑊 ) → ( 𝐹𝑃 ) ≠ 𝑃 )
13 1 2 3 4 ltrnatneq ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑃 ) ≠ 𝑃 ) → ( 𝐹𝑄 ) ≠ 𝑄 )
14 6 7 8 11 12 13 syl131anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( ( 𝐹𝑃 ) ≠ 𝑃 ∧ ( 𝐹𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 𝑊 ) → ( 𝐹𝑄 ) ≠ 𝑄 )
15 14 ex ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( ( 𝐹𝑃 ) ≠ 𝑃 ∧ ( 𝐹𝑄 ) = 𝑄 ) ) → ( ¬ 𝑄 𝑊 → ( 𝐹𝑄 ) ≠ 𝑄 ) )
16 15 necon4bd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( ( 𝐹𝑃 ) ≠ 𝑃 ∧ ( 𝐹𝑄 ) = 𝑄 ) ) → ( ( 𝐹𝑄 ) = 𝑄𝑄 𝑊 ) )
17 5 16 mpd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( ( 𝐹𝑃 ) ≠ 𝑃 ∧ ( 𝐹𝑄 ) = 𝑄 ) ) → 𝑄 𝑊 )