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 ) |