Step |
Hyp |
Ref |
Expression |
1 |
|
ringcsect.c |
⊢ 𝐶 = ( RingCat ‘ 𝑈 ) |
2 |
|
ringcsect.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
3 |
|
ringcsect.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑉 ) |
4 |
|
ringcsect.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
5 |
|
ringcsect.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
6 |
|
ringcsect.e |
⊢ 𝐸 = ( Base ‘ 𝑋 ) |
7 |
|
ringcsect.n |
⊢ 𝑆 = ( Sect ‘ 𝐶 ) |
8 |
|
eqid |
⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 ) |
9 |
|
eqid |
⊢ ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 ) |
10 |
|
eqid |
⊢ ( Id ‘ 𝐶 ) = ( Id ‘ 𝐶 ) |
11 |
1
|
ringccat |
⊢ ( 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
|
ringchom |
⊢ ( 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
|
ringchom |
⊢ ( 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 |
1 2 3
|
ringcbas |
⊢ ( 𝜑 → 𝐵 = ( 𝑈 ∩ Ring ) ) |
23 |
22
|
eleq2d |
⊢ ( 𝜑 → ( 𝑋 ∈ 𝐵 ↔ 𝑋 ∈ ( 𝑈 ∩ Ring ) ) ) |
24 |
|
inss1 |
⊢ ( 𝑈 ∩ Ring ) ⊆ 𝑈 |
25 |
24
|
a1i |
⊢ ( 𝜑 → ( 𝑈 ∩ Ring ) ⊆ 𝑈 ) |
26 |
25
|
sseld |
⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑈 ∩ Ring ) → 𝑋 ∈ 𝑈 ) ) |
27 |
23 26
|
sylbid |
⊢ ( 𝜑 → ( 𝑋 ∈ 𝐵 → 𝑋 ∈ 𝑈 ) ) |
28 |
27
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → ( 𝑋 ∈ 𝐵 → 𝑋 ∈ 𝑈 ) ) |
29 |
21 28
|
mpd |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → 𝑋 ∈ 𝑈 ) |
30 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → 𝑌 ∈ 𝐵 ) |
31 |
22
|
eleq2d |
⊢ ( 𝜑 → ( 𝑌 ∈ 𝐵 ↔ 𝑌 ∈ ( 𝑈 ∩ Ring ) ) ) |
32 |
25
|
sseld |
⊢ ( 𝜑 → ( 𝑌 ∈ ( 𝑈 ∩ Ring ) → 𝑌 ∈ 𝑈 ) ) |
33 |
31 32
|
sylbid |
⊢ ( 𝜑 → ( 𝑌 ∈ 𝐵 → 𝑌 ∈ 𝑈 ) ) |
34 |
33
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → ( 𝑌 ∈ 𝐵 → 𝑌 ∈ 𝑈 ) ) |
35 |
30 34
|
mpd |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → 𝑌 ∈ 𝑈 ) |
36 |
|
eqid |
⊢ ( Base ‘ 𝑋 ) = ( Base ‘ 𝑋 ) |
37 |
|
eqid |
⊢ ( Base ‘ 𝑌 ) = ( Base ‘ 𝑌 ) |
38 |
36 37
|
rhmf |
⊢ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) → 𝐹 : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑌 ) ) |
39 |
38
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) → 𝐹 : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑌 ) ) |
40 |
39
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → 𝐹 : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑌 ) ) |
41 |
37 36
|
rhmf |
⊢ ( 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) → 𝐺 : ( Base ‘ 𝑌 ) ⟶ ( Base ‘ 𝑋 ) ) |
42 |
41
|
adantl |
⊢ ( ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) → 𝐺 : ( Base ‘ 𝑌 ) ⟶ ( Base ‘ 𝑋 ) ) |
43 |
42
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → 𝐺 : ( Base ‘ 𝑌 ) ⟶ ( Base ‘ 𝑋 ) ) |
44 |
1 20 9 29 35 29 40 43
|
ringcco |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → ( 𝐺 ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑋 ) 𝐹 ) = ( 𝐺 ∘ 𝐹 ) ) |
45 |
1 2 10 3 4 6
|
ringcid |
⊢ ( 𝜑 → ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) = ( I ↾ 𝐸 ) ) |
46 |
45
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) = ( I ↾ 𝐸 ) ) |
47 |
44 46
|
eqeq12d |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → ( ( 𝐺 ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑋 ) 𝐹 ) = ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ↔ ( 𝐺 ∘ 𝐹 ) = ( I ↾ 𝐸 ) ) ) |
48 |
47
|
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 ) ) ) ) |
49 |
19 48
|
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 ) ) ) ) |
50 |
|
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 ) ) ) |
51 |
|
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 ) ) ) |
52 |
49 50 51
|
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 ) ) ) ) |
53 |
13 52
|
bitrd |
⊢ ( 𝜑 → ( 𝐹 ( 𝑋 𝑆 𝑌 ) 𝐺 ↔ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ∧ ( 𝐺 ∘ 𝐹 ) = ( I ↾ 𝐸 ) ) ) ) |