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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 ∧ 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 )