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Metamath Proof Explorer

Theorem isdilN

Description: The predicate "is a dilation". (Contributed by NM, 4-Feb-2012) (New usage is discouraged.)

Ref Expression
Hypotheses dilset.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dilset.s ⊢ 𝑆 = ( PSubSp ‘ 𝐾 )
dilset.w ⊢ 𝑊 = ( WAtoms ‘ 𝐾 )
dilset.m ⊢ 𝑀 = ( PAut ‘ 𝐾 )
dilset.l ⊢ 𝐿 = ( Dil ‘ 𝐾 )
Assertion isdilN ( ( 𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴 ) → ( 𝐹 ∈ ( 𝐿 ‘ 𝐷 ) ↔ ( 𝐹 ∈ 𝑀 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝑥 ⊆ ( 𝑊 ‘ 𝐷 ) → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) ) )

Proof

Step Hyp Ref Expression
1 dilset.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
2 dilset.s ⊢ 𝑆 = ( PSubSp ‘ 𝐾 )
3 dilset.w ⊢ 𝑊 = ( WAtoms ‘ 𝐾 )
4 dilset.m ⊢ 𝑀 = ( PAut ‘ 𝐾 )
5 dilset.l ⊢ 𝐿 = ( Dil ‘ 𝐾 )
6 1 2 3 4 5 dilsetN ⊢ ( ( 𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴 ) → ( 𝐿 ‘ 𝐷 ) = { 𝑓 ∈ 𝑀 ∣ ∀ 𝑥 ∈ 𝑆 ( 𝑥 ⊆ ( 𝑊 ‘ 𝐷 ) → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) } )
7 6 eleq2d ⊢ ( ( 𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴 ) → ( 𝐹 ∈ ( 𝐿 ‘ 𝐷 ) ↔ 𝐹 ∈ { 𝑓 ∈ 𝑀 ∣ ∀ 𝑥 ∈ 𝑆 ( 𝑥 ⊆ ( 𝑊 ‘ 𝐷 ) → ( 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 ‘ 𝐷 ) → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) ↔ ( 𝑥 ⊆ ( 𝑊 ‘ 𝐷 ) → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) )
11 10 ralbidv ⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑥 ∈ 𝑆 ( 𝑥 ⊆ ( 𝑊 ‘ 𝐷 ) → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) ↔ ∀ 𝑥 ∈ 𝑆 ( 𝑥 ⊆ ( 𝑊 ‘ 𝐷 ) → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) )
12 11 elrab ⊢ ( 𝐹 ∈ { 𝑓 ∈ 𝑀 ∣ ∀ 𝑥 ∈ 𝑆 ( 𝑥 ⊆ ( 𝑊 ‘ 𝐷 ) → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) } ↔ ( 𝐹 ∈ 𝑀 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝑥 ⊆ ( 𝑊 ‘ 𝐷 ) → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) )
13 7 12 bitrdi ⊢ ( ( 𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴 ) → ( 𝐹 ∈ ( 𝐿 ‘ 𝐷 ) ↔ ( 𝐹 ∈ 𝑀 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝑥 ⊆ ( 𝑊 ‘ 𝐷 ) → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) ) )