Step |
Hyp |
Ref |
Expression |
1 |
|
sectco.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
2 |
|
sectco.o |
⊢ · = ( comp ‘ 𝐶 ) |
3 |
|
sectco.s |
⊢ 𝑆 = ( Sect ‘ 𝐶 ) |
4 |
|
sectco.c |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
5 |
|
sectco.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
6 |
|
sectco.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
7 |
|
sectco.z |
⊢ ( 𝜑 → 𝑍 ∈ 𝐵 ) |
8 |
|
sectco.1 |
⊢ ( 𝜑 → 𝐹 ( 𝑋 𝑆 𝑌 ) 𝐺 ) |
9 |
|
sectco.2 |
⊢ ( 𝜑 → 𝐻 ( 𝑌 𝑆 𝑍 ) 𝐾 ) |
10 |
|
eqid |
⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 ) |
11 |
|
eqid |
⊢ ( Id ‘ 𝐶 ) = ( Id ‘ 𝐶 ) |
12 |
1 10 2 11 3 4 5 6
|
issect |
⊢ ( 𝜑 → ( 𝐹 ( 𝑋 𝑆 𝑌 ) 𝐺 ↔ ( 𝐹 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) ∧ ( 𝐺 ( 〈 𝑋 , 𝑌 〉 · 𝑋 ) 𝐹 ) = ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ) ) ) |
13 |
8 12
|
mpbid |
⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) ∧ ( 𝐺 ( 〈 𝑋 , 𝑌 〉 · 𝑋 ) 𝐹 ) = ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ) ) |
14 |
13
|
simp1d |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ) |
15 |
1 10 2 11 3 4 6 7
|
issect |
⊢ ( 𝜑 → ( 𝐻 ( 𝑌 𝑆 𝑍 ) 𝐾 ↔ ( 𝐻 ∈ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑍 ) ∧ 𝐾 ∈ ( 𝑍 ( Hom ‘ 𝐶 ) 𝑌 ) ∧ ( 𝐾 ( 〈 𝑌 , 𝑍 〉 · 𝑌 ) 𝐻 ) = ( ( Id ‘ 𝐶 ) ‘ 𝑌 ) ) ) ) |
16 |
9 15
|
mpbid |
⊢ ( 𝜑 → ( 𝐻 ∈ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑍 ) ∧ 𝐾 ∈ ( 𝑍 ( Hom ‘ 𝐶 ) 𝑌 ) ∧ ( 𝐾 ( 〈 𝑌 , 𝑍 〉 · 𝑌 ) 𝐻 ) = ( ( Id ‘ 𝐶 ) ‘ 𝑌 ) ) ) |
17 |
16
|
simp1d |
⊢ ( 𝜑 → 𝐻 ∈ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑍 ) ) |
18 |
1 10 2 4 5 6 7 14 17
|
catcocl |
⊢ ( 𝜑 → ( 𝐻 ( 〈 𝑋 , 𝑌 〉 · 𝑍 ) 𝐹 ) ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑍 ) ) |
19 |
16
|
simp2d |
⊢ ( 𝜑 → 𝐾 ∈ ( 𝑍 ( Hom ‘ 𝐶 ) 𝑌 ) ) |
20 |
13
|
simp2d |
⊢ ( 𝜑 → 𝐺 ∈ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) ) |
21 |
1 10 2 4 5 7 6 18 19 5 20
|
catass |
⊢ ( 𝜑 → ( ( 𝐺 ( 〈 𝑍 , 𝑌 〉 · 𝑋 ) 𝐾 ) ( 〈 𝑋 , 𝑍 〉 · 𝑋 ) ( 𝐻 ( 〈 𝑋 , 𝑌 〉 · 𝑍 ) 𝐹 ) ) = ( 𝐺 ( 〈 𝑋 , 𝑌 〉 · 𝑋 ) ( 𝐾 ( 〈 𝑋 , 𝑍 〉 · 𝑌 ) ( 𝐻 ( 〈 𝑋 , 𝑌 〉 · 𝑍 ) 𝐹 ) ) ) ) |
22 |
16
|
simp3d |
⊢ ( 𝜑 → ( 𝐾 ( 〈 𝑌 , 𝑍 〉 · 𝑌 ) 𝐻 ) = ( ( Id ‘ 𝐶 ) ‘ 𝑌 ) ) |
23 |
22
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝐾 ( 〈 𝑌 , 𝑍 〉 · 𝑌 ) 𝐻 ) ( 〈 𝑋 , 𝑌 〉 · 𝑌 ) 𝐹 ) = ( ( ( Id ‘ 𝐶 ) ‘ 𝑌 ) ( 〈 𝑋 , 𝑌 〉 · 𝑌 ) 𝐹 ) ) |
24 |
1 10 2 4 5 6 7 14 17 6 19
|
catass |
⊢ ( 𝜑 → ( ( 𝐾 ( 〈 𝑌 , 𝑍 〉 · 𝑌 ) 𝐻 ) ( 〈 𝑋 , 𝑌 〉 · 𝑌 ) 𝐹 ) = ( 𝐾 ( 〈 𝑋 , 𝑍 〉 · 𝑌 ) ( 𝐻 ( 〈 𝑋 , 𝑌 〉 · 𝑍 ) 𝐹 ) ) ) |
25 |
1 10 11 4 5 2 6 14
|
catlid |
⊢ ( 𝜑 → ( ( ( Id ‘ 𝐶 ) ‘ 𝑌 ) ( 〈 𝑋 , 𝑌 〉 · 𝑌 ) 𝐹 ) = 𝐹 ) |
26 |
23 24 25
|
3eqtr3d |
⊢ ( 𝜑 → ( 𝐾 ( 〈 𝑋 , 𝑍 〉 · 𝑌 ) ( 𝐻 ( 〈 𝑋 , 𝑌 〉 · 𝑍 ) 𝐹 ) ) = 𝐹 ) |
27 |
26
|
oveq2d |
⊢ ( 𝜑 → ( 𝐺 ( 〈 𝑋 , 𝑌 〉 · 𝑋 ) ( 𝐾 ( 〈 𝑋 , 𝑍 〉 · 𝑌 ) ( 𝐻 ( 〈 𝑋 , 𝑌 〉 · 𝑍 ) 𝐹 ) ) ) = ( 𝐺 ( 〈 𝑋 , 𝑌 〉 · 𝑋 ) 𝐹 ) ) |
28 |
13
|
simp3d |
⊢ ( 𝜑 → ( 𝐺 ( 〈 𝑋 , 𝑌 〉 · 𝑋 ) 𝐹 ) = ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ) |
29 |
21 27 28
|
3eqtrd |
⊢ ( 𝜑 → ( ( 𝐺 ( 〈 𝑍 , 𝑌 〉 · 𝑋 ) 𝐾 ) ( 〈 𝑋 , 𝑍 〉 · 𝑋 ) ( 𝐻 ( 〈 𝑋 , 𝑌 〉 · 𝑍 ) 𝐹 ) ) = ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ) |
30 |
1 10 2 4 7 6 5 19 20
|
catcocl |
⊢ ( 𝜑 → ( 𝐺 ( 〈 𝑍 , 𝑌 〉 · 𝑋 ) 𝐾 ) ∈ ( 𝑍 ( Hom ‘ 𝐶 ) 𝑋 ) ) |
31 |
1 10 2 11 3 4 5 7 18 30
|
issect2 |
⊢ ( 𝜑 → ( ( 𝐻 ( 〈 𝑋 , 𝑌 〉 · 𝑍 ) 𝐹 ) ( 𝑋 𝑆 𝑍 ) ( 𝐺 ( 〈 𝑍 , 𝑌 〉 · 𝑋 ) 𝐾 ) ↔ ( ( 𝐺 ( 〈 𝑍 , 𝑌 〉 · 𝑋 ) 𝐾 ) ( 〈 𝑋 , 𝑍 〉 · 𝑋 ) ( 𝐻 ( 〈 𝑋 , 𝑌 〉 · 𝑍 ) 𝐹 ) ) = ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ) ) |
32 |
29 31
|
mpbird |
⊢ ( 𝜑 → ( 𝐻 ( 〈 𝑋 , 𝑌 〉 · 𝑍 ) 𝐹 ) ( 𝑋 𝑆 𝑍 ) ( 𝐺 ( 〈 𝑍 , 𝑌 〉 · 𝑋 ) 𝐾 ) ) |