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Metamath Proof Explorer

Theorem ltrniotaidvalN

Description: Value of the unique translation specified by identity value. (Contributed by NM, 25-Aug-2014) (New usage is discouraged.)

Ref Expression
Hypotheses ltrniotaidval.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
ltrniotaidval.l ⊢ ≤ = ( le ‘ 𝐾 )
ltrniotaidval.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
ltrniotaidval.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
ltrniotaidval.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
ltrniotaidval.f ⊢ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 )
Assertion ltrniotaidvalN ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐹 = ( I ↾ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 ltrniotaidval.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 ltrniotaidval.l ⊢ ≤ = ( le ‘ 𝐾 )
3 ltrniotaidval.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 ltrniotaidval.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
5 ltrniotaidval.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6 ltrniotaidval.f ⊢ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 )
7 2 3 4 5 6 ltrniotaval ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑃 ) = 𝑃 )
8 7 3anidm23 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑃 ) = 𝑃 )
9 simpl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10 2 3 4 5 6 ltrniotacl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 )
11 10 3anidm23 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 )
12 simpr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
13 1 2 3 4 5 ltrnideq ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 = ( I ↾ 𝐵 ) ↔ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) )
14 9 11 12 13 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 = ( I ↾ 𝐵 ) ↔ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) )
15 8 14 mpbird ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐹 = ( I ↾ 𝐵 ) )