Step |
Hyp |
Ref |
Expression |
1 |
|
ldilco.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
ldilco.d |
⊢ 𝐷 = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
simp1l |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → 𝐾 ∈ 𝑉 ) |
4 |
|
eqid |
⊢ ( LAut ‘ 𝐾 ) = ( LAut ‘ 𝐾 ) |
5 |
1 4 2
|
ldillaut |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ) → 𝐹 ∈ ( LAut ‘ 𝐾 ) ) |
6 |
5
|
3adant3 |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → 𝐹 ∈ ( LAut ‘ 𝐾 ) ) |
7 |
1 4 2
|
ldillaut |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝐷 ) → 𝐺 ∈ ( LAut ‘ 𝐾 ) ) |
8 |
7
|
3adant2 |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → 𝐺 ∈ ( LAut ‘ 𝐾 ) ) |
9 |
4
|
lautco |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ ( LAut ‘ 𝐾 ) ∧ 𝐺 ∈ ( LAut ‘ 𝐾 ) ) → ( 𝐹 ∘ 𝐺 ) ∈ ( LAut ‘ 𝐾 ) ) |
10 |
3 6 8 9
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( 𝐹 ∘ 𝐺 ) ∈ ( LAut ‘ 𝐾 ) ) |
11 |
|
simp11 |
⊢ ( ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) ∧ 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑥 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ) |
12 |
|
simp13 |
⊢ ( ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) ∧ 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑥 ( le ‘ 𝐾 ) 𝑊 ) → 𝐺 ∈ 𝐷 ) |
13 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
14 |
13 1 2
|
ldil1o |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝐷 ) → 𝐺 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) ) |
15 |
11 12 14
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) ∧ 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑥 ( le ‘ 𝐾 ) 𝑊 ) → 𝐺 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) ) |
16 |
|
f1of |
⊢ ( 𝐺 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) → 𝐺 : ( Base ‘ 𝐾 ) ⟶ ( Base ‘ 𝐾 ) ) |
17 |
15 16
|
syl |
⊢ ( ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) ∧ 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑥 ( le ‘ 𝐾 ) 𝑊 ) → 𝐺 : ( Base ‘ 𝐾 ) ⟶ ( Base ‘ 𝐾 ) ) |
18 |
|
simp2 |
⊢ ( ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) ∧ 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑥 ( le ‘ 𝐾 ) 𝑊 ) → 𝑥 ∈ ( Base ‘ 𝐾 ) ) |
19 |
|
fvco3 |
⊢ ( ( 𝐺 : ( Base ‘ 𝐾 ) ⟶ ( Base ‘ 𝐾 ) ∧ 𝑥 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑥 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑥 ) ) ) |
20 |
17 18 19
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) ∧ 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑥 ( le ‘ 𝐾 ) 𝑊 ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑥 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑥 ) ) ) |
21 |
|
simp3 |
⊢ ( ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) ∧ 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑥 ( le ‘ 𝐾 ) 𝑊 ) → 𝑥 ( le ‘ 𝐾 ) 𝑊 ) |
22 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
23 |
13 22 1 2
|
ldilval |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝐷 ∧ ( 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑥 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐺 ‘ 𝑥 ) = 𝑥 ) |
24 |
11 12 18 21 23
|
syl112anc |
⊢ ( ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) ∧ 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑥 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝐺 ‘ 𝑥 ) = 𝑥 ) |
25 |
24
|
fveq2d |
⊢ ( ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) ∧ 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑥 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
26 |
|
simp12 |
⊢ ( ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) ∧ 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑥 ( le ‘ 𝐾 ) 𝑊 ) → 𝐹 ∈ 𝐷 ) |
27 |
13 22 1 2
|
ldilval |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑥 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) |
28 |
11 26 18 21 27
|
syl112anc |
⊢ ( ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) ∧ 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑥 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) |
29 |
20 25 28
|
3eqtrd |
⊢ ( ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) ∧ 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑥 ( le ‘ 𝐾 ) 𝑊 ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑥 ) = 𝑥 ) |
30 |
29
|
3exp |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( 𝑥 ∈ ( Base ‘ 𝐾 ) → ( 𝑥 ( le ‘ 𝐾 ) 𝑊 → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑥 ) = 𝑥 ) ) ) |
31 |
30
|
ralrimiv |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( le ‘ 𝐾 ) 𝑊 → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑥 ) = 𝑥 ) ) |
32 |
13 22 1 4 2
|
isldil |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( ( 𝐹 ∘ 𝐺 ) ∈ 𝐷 ↔ ( ( 𝐹 ∘ 𝐺 ) ∈ ( LAut ‘ 𝐾 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( le ‘ 𝐾 ) 𝑊 → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑥 ) = 𝑥 ) ) ) ) |
33 |
32
|
3ad2ant1 |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( ( 𝐹 ∘ 𝐺 ) ∈ 𝐷 ↔ ( ( 𝐹 ∘ 𝐺 ) ∈ ( LAut ‘ 𝐾 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( le ‘ 𝐾 ) 𝑊 → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑥 ) = 𝑥 ) ) ) ) |
34 |
10 31 33
|
mpbir2and |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( 𝐹 ∘ 𝐺 ) ∈ 𝐷 ) |