Step |
Hyp |
Ref |
Expression |
1 |
|
ringcsectALTV.c |
⊢ 𝐶 = ( RingCatALTV ‘ 𝑈 ) |
2 |
|
ringcsectALTV.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
3 |
|
ringcsectALTV.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑉 ) |
4 |
|
ringcsectALTV.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
5 |
|
ringcsectALTV.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
6 |
|
ringcsectALTV.e |
⊢ 𝐸 = ( Base ‘ 𝑋 ) |
7 |
|
ringcsectALTV.n |
⊢ 𝑆 = ( Sect ‘ 𝐶 ) |
8 |
|
eqid |
⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 ) |
9 |
|
eqid |
⊢ ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 ) |
10 |
|
eqid |
⊢ ( Id ‘ 𝐶 ) = ( Id ‘ 𝐶 ) |
11 |
1
|
ringccatALTV |
⊢ ( 𝑈 ∈ 𝑉 → 𝐶 ∈ Cat ) |
12 |
3 11
|
syl |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
13 |
2 8 9 10 7 12 4 5
|
issect |
⊢ ( 𝜑 → ( 𝐹 ( 𝑋 𝑆 𝑌 ) 𝐺 ↔ ( 𝐹 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) ∧ ( 𝐺 ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑋 ) 𝐹 ) = ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ) ) ) |
14 |
1 2 3 8 4 5
|
ringchomALTV |
⊢ ( 𝜑 → ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) = ( 𝑋 RingHom 𝑌 ) ) |
15 |
14
|
eleq2d |
⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ↔ 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ) ) |
16 |
1 2 3 8 5 4
|
ringchomALTV |
⊢ ( 𝜑 → ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) = ( 𝑌 RingHom 𝑋 ) ) |
17 |
16
|
eleq2d |
⊢ ( 𝜑 → ( 𝐺 ∈ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) ↔ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) |
18 |
15 17
|
anbi12d |
⊢ ( 𝜑 → ( ( 𝐹 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) ) ↔ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) ) |
19 |
18
|
anbi1d |
⊢ ( 𝜑 → ( ( ( 𝐹 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) ) ∧ ( 𝐺 ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑋 ) 𝐹 ) = ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ) ↔ ( ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ∧ ( 𝐺 ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑋 ) 𝐹 ) = ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ) ) ) |
20 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → 𝑈 ∈ 𝑉 ) |
21 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → 𝑋 ∈ 𝐵 ) |
22 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → 𝑌 ∈ 𝐵 ) |
23 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ) |
24 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) |
25 |
1 2 20 9 21 22 21 23 24
|
ringccoALTV |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → ( 𝐺 ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑋 ) 𝐹 ) = ( 𝐺 ∘ 𝐹 ) ) |
26 |
1 2 10 3 4 6
|
ringcidALTV |
⊢ ( 𝜑 → ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) = ( I ↾ 𝐸 ) ) |
27 |
26
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) = ( I ↾ 𝐸 ) ) |
28 |
25 27
|
eqeq12d |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → ( ( 𝐺 ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑋 ) 𝐹 ) = ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ↔ ( 𝐺 ∘ 𝐹 ) = ( I ↾ 𝐸 ) ) ) |
29 |
28
|
pm5.32da |
⊢ ( 𝜑 → ( ( ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ∧ ( 𝐺 ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑋 ) 𝐹 ) = ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ) ↔ ( ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ∧ ( 𝐺 ∘ 𝐹 ) = ( I ↾ 𝐸 ) ) ) ) |
30 |
19 29
|
bitrd |
⊢ ( 𝜑 → ( ( ( 𝐹 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) ) ∧ ( 𝐺 ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑋 ) 𝐹 ) = ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ) ↔ ( ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ∧ ( 𝐺 ∘ 𝐹 ) = ( I ↾ 𝐸 ) ) ) ) |
31 |
|
df-3an |
⊢ ( ( 𝐹 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) ∧ ( 𝐺 ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑋 ) 𝐹 ) = ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ) ↔ ( ( 𝐹 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) ) ∧ ( 𝐺 ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑋 ) 𝐹 ) = ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ) ) |
32 |
|
df-3an |
⊢ ( ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ∧ ( 𝐺 ∘ 𝐹 ) = ( I ↾ 𝐸 ) ) ↔ ( ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ∧ ( 𝐺 ∘ 𝐹 ) = ( I ↾ 𝐸 ) ) ) |
33 |
30 31 32
|
3bitr4g |
⊢ ( 𝜑 → ( ( 𝐹 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) ∧ ( 𝐺 ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑋 ) 𝐹 ) = ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ) ↔ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ∧ ( 𝐺 ∘ 𝐹 ) = ( I ↾ 𝐸 ) ) ) ) |
34 |
13 33
|
bitrd |
⊢ ( 𝜑 → ( 𝐹 ( 𝑋 𝑆 𝑌 ) 𝐺 ↔ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ∧ ( 𝐺 ∘ 𝐹 ) = ( I ↾ 𝐸 ) ) ) ) |