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Metamath Proof Explorer

Theorem lautco

Description: The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013)

Ref Expression
Hypothesis lautco.i ⊢ 𝐼 = ( LAut ‘ 𝐾 )
Assertion lautco ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼 ) → ( 𝐹 ∘ 𝐺 ) ∈ 𝐼 )

Proof

Step Hyp Ref Expression
1 lautco.i ⊢ 𝐼 = ( LAut ‘ 𝐾 )
2 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
3 2 1 laut1o ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ) → 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
4 3 3adant3 ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼 ) → 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
5 2 1 laut1o ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝐺 ∈ 𝐼 ) → 𝐺 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
6 5 3adant2 ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼 ) → 𝐺 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
7 f1oco ⊢ ( ( 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) ∧ 𝐺 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) ) → ( 𝐹 ∘ 𝐺 ) : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
8 4 6 7 syl2anc ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼 ) → ( 𝐹 ∘ 𝐺 ) : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
9 simpl1 ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑦 ∈ ( Base ‘ 𝐾 ) ) ) → 𝐾 ∈ 𝑉 )
10 simpl2 ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑦 ∈ ( Base ‘ 𝐾 ) ) ) → 𝐹 ∈ 𝐼 )
11 simpl3 ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑦 ∈ ( Base ‘ 𝐾 ) ) ) → 𝐺 ∈ 𝐼 )
12 simprl ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑦 ∈ ( Base ‘ 𝐾 ) ) ) → 𝑥 ∈ ( Base ‘ 𝐾 ) )
13 2 1 lautcl ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐺 ∈ 𝐼 ) ∧ 𝑥 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐺 ‘ 𝑥 ) ∈ ( Base ‘ 𝐾 ) )
14 9 11 12 13 syl21anc ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑦 ∈ ( Base ‘ 𝐾 ) ) ) → ( 𝐺 ‘ 𝑥 ) ∈ ( Base ‘ 𝐾 ) )
15 simprr ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑦 ∈ ( Base ‘ 𝐾 ) ) ) → 𝑦 ∈ ( Base ‘ 𝐾 ) )
16 2 1 lautcl ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐺 ∈ 𝐼 ) ∧ 𝑦 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ ( Base ‘ 𝐾 ) )
17 9 11 15 16 syl21anc ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑦 ∈ ( Base ‘ 𝐾 ) ) ) → ( 𝐺 ‘ 𝑦 ) ∈ ( Base ‘ 𝐾 ) )
18 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
19 2 18 1 lautle ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ) ∧ ( ( 𝐺 ‘ 𝑥 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝐺 ‘ 𝑥 ) ( le ‘ 𝐾 ) ( 𝐺 ‘ 𝑦 ) ↔ ( 𝐹 ‘ ( 𝐺 ‘ 𝑥 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( 𝐺 ‘ 𝑦 ) ) ) )
20 9 10 14 17 19 syl22anc ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑦 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝐺 ‘ 𝑥 ) ( le ‘ 𝐾 ) ( 𝐺 ‘ 𝑦 ) ↔ ( 𝐹 ‘ ( 𝐺 ‘ 𝑥 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( 𝐺 ‘ 𝑦 ) ) ) )
21 2 18 1 lautle ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐺 ∈ 𝐼 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑦 ∈ ( Base ‘ 𝐾 ) ) ) → ( 𝑥 ( le ‘ 𝐾 ) 𝑦 ↔ ( 𝐺 ‘ 𝑥 ) ( le ‘ 𝐾 ) ( 𝐺 ‘ 𝑦 ) ) )
22 21 3adantl2 ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑦 ∈ ( Base ‘ 𝐾 ) ) ) → ( 𝑥 ( le ‘ 𝐾 ) 𝑦 ↔ ( 𝐺 ‘ 𝑥 ) ( le ‘ 𝐾 ) ( 𝐺 ‘ 𝑦 ) ) )
23 f1of ⊢ ( 𝐺 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) → 𝐺 : ( Base ‘ 𝐾 ) ⟶ ( Base ‘ 𝐾 ) )
24 6 23 syl ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼 ) → 𝐺 : ( Base ‘ 𝐾 ) ⟶ ( Base ‘ 𝐾 ) )
25 simpl ⊢ ( ( 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑦 ∈ ( Base ‘ 𝐾 ) ) → 𝑥 ∈ ( Base ‘ 𝐾 ) )
26 fvco3 ⊢ ( ( 𝐺 : ( Base ‘ 𝐾 ) ⟶ ( Base ‘ 𝐾 ) ∧ 𝑥 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑥 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑥 ) ) )
27 24 25 26 syl2an ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑦 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑥 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑥 ) ) )
28 simpr ⊢ ( ( 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑦 ∈ ( Base ‘ 𝐾 ) ) → 𝑦 ∈ ( Base ‘ 𝐾 ) )
29 fvco3 ⊢ ( ( 𝐺 : ( Base ‘ 𝐾 ) ⟶ ( Base ‘ 𝐾 ) ∧ 𝑦 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑦 ) ) )
30 24 28 29 syl2an ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑦 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑦 ) ) )
31 27 30 breq12d ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑦 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑥 ) ( le ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑦 ) ↔ ( 𝐹 ‘ ( 𝐺 ‘ 𝑥 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( 𝐺 ‘ 𝑦 ) ) ) )
32 20 22 31 3bitr4d ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑦 ∈ ( Base ‘ 𝐾 ) ) ) → ( 𝑥 ( le ‘ 𝐾 ) 𝑦 ↔ ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑥 ) ( le ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑦 ) ) )
33 32 ralrimivva ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼 ) → ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( le ‘ 𝐾 ) 𝑦 ↔ ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑥 ) ( le ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑦 ) ) )
34 2 18 1 islaut ⊢ ( 𝐾 ∈ 𝑉 → ( ( 𝐹 ∘ 𝐺 ) ∈ 𝐼 ↔ ( ( 𝐹 ∘ 𝐺 ) : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( le ‘ 𝐾 ) 𝑦 ↔ ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑥 ) ( le ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑦 ) ) ) ) )
35 34 3ad2ant1 ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼 ) → ( ( 𝐹 ∘ 𝐺 ) ∈ 𝐼 ↔ ( ( 𝐹 ∘ 𝐺 ) : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( le ‘ 𝐾 ) 𝑦 ↔ ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑥 ) ( le ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑦 ) ) ) ) )
36 8 33 35 mpbir2and ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ∧ 𝐺 ∈ 𝐼 ) → ( 𝐹 ∘ 𝐺 ) ∈ 𝐼 )