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Metamath Proof Explorer

Theorem ltrncnvatb

Description: The converse of the lattice translation of an atom is an atom. (Contributed by NM, 2-Jun-2012)

Ref Expression
Hypotheses ltrnatb.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
ltrnatb.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
ltrnatb.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
ltrnatb.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion ltrncnvatb ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐵 ) → ( 𝑃 ∈ 𝐴 ↔ ( ◡ 𝐹 ‘ 𝑃 ) ∈ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 ltrnatb.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 ltrnatb.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 ltrnatb.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 ltrnatb.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 1 3 4 ltrn1o ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 )
6 f1ocnvdm ⊢ ( ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ 𝑃 ∈ 𝐵 ) → ( ◡ 𝐹 ‘ 𝑃 ) ∈ 𝐵 )
7 5 6 stoic3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐵 ) → ( ◡ 𝐹 ‘ 𝑃 ) ∈ 𝐵 )
8 1 2 3 4 ltrnatb ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ◡ 𝐹 ‘ 𝑃 ) ∈ 𝐵 ) → ( ( ◡ 𝐹 ‘ 𝑃 ) ∈ 𝐴 ↔ ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑃 ) ) ∈ 𝐴 ) )
9 7 8 syld3an3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐵 ) → ( ( ◡ 𝐹 ‘ 𝑃 ) ∈ 𝐴 ↔ ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑃 ) ) ∈ 𝐴 ) )
10 f1ocnvfv2 ⊢ ( ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ 𝑃 ∈ 𝐵 ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑃 ) ) = 𝑃 )
11 5 10 stoic3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐵 ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑃 ) ) = 𝑃 )
12 11 eleq1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐵 ) → ( ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑃 ) ) ∈ 𝐴 ↔ 𝑃 ∈ 𝐴 ) )
13 9 12 bitr2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐵 ) → ( 𝑃 ∈ 𝐴 ↔ ( ◡ 𝐹 ‘ 𝑃 ) ∈ 𝐴 ) )