Metamath Proof Explorer

Theorem ldilcnv

Description: The converse of a lattice dilation is a lattice dilation. (Contributed by NM, 10-May-2013)

Ref Expression
Hypotheses ldilcnv.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
ldilcnv.d ⊢ 𝐷 = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 )
Assertion ldilcnv ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ) → ◡ 𝐹 ∈ 𝐷 )

Proof

Step Hyp Ref Expression
1 ldilcnv.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 ldilcnv.d ⊢ 𝐷 = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 )
3 simpll ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ) → 𝐾 ∈ HL )
4 eqid ⊢ ( LAut ‘ 𝐾 ) = ( LAut ‘ 𝐾 )
5 1 4 2 ldillaut ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ) → 𝐹 ∈ ( LAut ‘ 𝐾 ) )
6 4 lautcnv ⊢ ( ( 𝐾 ∈ HL ∧ 𝐹 ∈ ( LAut ‘ 𝐾 ) ) → ◡ 𝐹 ∈ ( LAut ‘ 𝐾 ) )
7 3 5 6 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ) → ◡ 𝐹 ∈ ( LAut ‘ 𝐾 ) )
8 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
9 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
10 8 9 1 2 ldilval ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑥 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐹 ‘ 𝑥 ) = 𝑥 )
11 10 3expa ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑥 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐹 ‘ 𝑥 ) = 𝑥 )
12 11 3impb ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ) ∧ 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑥 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝐹 ‘ 𝑥 ) = 𝑥 )
13 12 fveq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ) ∧ 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑥 ( le ‘ 𝐾 ) 𝑊 ) → ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( ◡ 𝐹 ‘ 𝑥 ) )
14 8 1 2 ldil1o ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ) → 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
15 14 3ad2ant1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ) ∧ 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑥 ( le ‘ 𝐾 ) 𝑊 ) → 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
16 simp2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ) ∧ 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑥 ( le ‘ 𝐾 ) 𝑊 ) → 𝑥 ∈ ( Base ‘ 𝐾 ) )
17 f1ocnvfv1 ⊢ ( ( 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) ∧ 𝑥 ∈ ( Base ‘ 𝐾 ) ) → ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = 𝑥 )
18 15 16 17 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ) ∧ 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑥 ( le ‘ 𝐾 ) 𝑊 ) → ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = 𝑥 )
19 13 18 eqtr3d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ) ∧ 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑥 ( le ‘ 𝐾 ) 𝑊 ) → ( ◡ 𝐹 ‘ 𝑥 ) = 𝑥 )
20 19 3exp ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ) → ( 𝑥 ∈ ( Base ‘ 𝐾 ) → ( 𝑥 ( le ‘ 𝐾 ) 𝑊 → ( ◡ 𝐹 ‘ 𝑥 ) = 𝑥 ) ) )
21 20 ralrimiv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ) → ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( le ‘ 𝐾 ) 𝑊 → ( ◡ 𝐹 ‘ 𝑥 ) = 𝑥 ) )
22 8 9 1 4 2 isldil ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ◡ 𝐹 ∈ 𝐷 ↔ ( ◡ 𝐹 ∈ ( LAut ‘ 𝐾 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( le ‘ 𝐾 ) 𝑊 → ( ◡ 𝐹 ‘ 𝑥 ) = 𝑥 ) ) ) )
23 22 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ) → ( ◡ 𝐹 ∈ 𝐷 ↔ ( ◡ 𝐹 ∈ ( LAut ‘ 𝐾 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( le ‘ 𝐾 ) 𝑊 → ( ◡ 𝐹 ‘ 𝑥 ) = 𝑥 ) ) ) )
24 7 21 23 mpbir2and ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ) → ◡ 𝐹 ∈ 𝐷 )