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Metamath Proof Explorer

Theorem isldil

Description: The predicate "is a lattice dilation". Similar to definition of dilation in Crawley p. 111. (Contributed by NM, 11-May-2012)

Ref Expression
Hypotheses ldilset.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
ldilset.l ⊢ ≤ = ( le ‘ 𝐾 )
ldilset.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
ldilset.i ⊢ 𝐼 = ( LAut ‘ 𝐾 )
ldilset.d ⊢ 𝐷 = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 )
Assertion isldil ( ( 𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 ∈ 𝐷 ↔ ( 𝐹 ∈ 𝐼 ∧ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑊 → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) ) )

Proof

Step Hyp Ref Expression
1 ldilset.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 ldilset.l ⊢ ≤ = ( le ‘ 𝐾 )
3 ldilset.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 ldilset.i ⊢ 𝐼 = ( LAut ‘ 𝐾 )
5 ldilset.d ⊢ 𝐷 = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 )
6 1 2 3 4 5 ldilset ⊢ ( ( 𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻 ) → 𝐷 = { 𝑓 ∈ 𝐼 ∣ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑊 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) } )
7 6 eleq2d ⊢ ( ( 𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 ∈ 𝐷 ↔ 𝐹 ∈ { 𝑓 ∈ 𝐼 ∣ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑊 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) } ) )
8 fveq1 ⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) )
9 8 eqeq1d ⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 𝑥 ) = 𝑥 ↔ ( 𝐹 ‘ 𝑥 ) = 𝑥 ) )
10 9 imbi2d ⊢ ( 𝑓 = 𝐹 → ( ( 𝑥 ≤ 𝑊 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) ↔ ( 𝑥 ≤ 𝑊 → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) )
11 10 ralbidv ⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑊 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑊 → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) )
12 11 elrab ⊢ ( 𝐹 ∈ { 𝑓 ∈ 𝐼 ∣ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑊 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) } ↔ ( 𝐹 ∈ 𝐼 ∧ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑊 → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) )
13 7 12 bitrdi ⊢ ( ( 𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 ∈ 𝐷 ↔ ( 𝐹 ∈ 𝐼 ∧ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑊 → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) ) )