Metamath Proof Explorer


Theorem ringcsect

Description: A section in the category of unital rings, written out. (Contributed by AV, 14-Feb-2020)

Ref Expression
Hypotheses ringcsect.c 𝐶 = ( RingCat ‘ 𝑈 )
ringcsect.b 𝐵 = ( Base ‘ 𝐶 )
ringcsect.u ( 𝜑𝑈𝑉 )
ringcsect.x ( 𝜑𝑋𝐵 )
ringcsect.y ( 𝜑𝑌𝐵 )
ringcsect.e 𝐸 = ( Base ‘ 𝑋 )
ringcsect.n 𝑆 = ( Sect ‘ 𝐶 )
Assertion ringcsect ( 𝜑 → ( 𝐹 ( 𝑋 𝑆 𝑌 ) 𝐺 ↔ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ∧ ( 𝐺𝐹 ) = ( I ↾ 𝐸 ) ) ) )

Proof

Step Hyp Ref Expression
1 ringcsect.c 𝐶 = ( RingCat ‘ 𝑈 )
2 ringcsect.b 𝐵 = ( Base ‘ 𝐶 )
3 ringcsect.u ( 𝜑𝑈𝑉 )
4 ringcsect.x ( 𝜑𝑋𝐵 )
5 ringcsect.y ( 𝜑𝑌𝐵 )
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 ( 𝑈𝑉𝐶 ∈ Cat )
12 3 11 syl ( 𝜑𝐶 ∈ 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 ) ) ) → 𝑈𝑉 )
21 4 adantr ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → 𝑋𝐵 )
22 1 2 3 ringcbas ( 𝜑𝐵 = ( 𝑈 ∩ Ring ) )
23 22 eleq2d ( 𝜑 → ( 𝑋𝐵𝑋 ∈ ( 𝑈 ∩ Ring ) ) )
24 inss1 ( 𝑈 ∩ Ring ) ⊆ 𝑈
25 24 a1i ( 𝜑 → ( 𝑈 ∩ Ring ) ⊆ 𝑈 )
26 25 sseld ( 𝜑 → ( 𝑋 ∈ ( 𝑈 ∩ Ring ) → 𝑋𝑈 ) )
27 23 26 sylbid ( 𝜑 → ( 𝑋𝐵𝑋𝑈 ) )
28 27 adantr ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → ( 𝑋𝐵𝑋𝑈 ) )
29 21 28 mpd ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → 𝑋𝑈 )
30 5 adantr ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → 𝑌𝐵 )
31 22 eleq2d ( 𝜑 → ( 𝑌𝐵𝑌 ∈ ( 𝑈 ∩ Ring ) ) )
32 25 sseld ( 𝜑 → ( 𝑌 ∈ ( 𝑈 ∩ Ring ) → 𝑌𝑈 ) )
33 31 32 sylbid ( 𝜑 → ( 𝑌𝐵𝑌𝑈 ) )
34 33 adantr ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → ( 𝑌𝐵𝑌𝑈 ) )
35 30 34 mpd ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ) → 𝑌𝑈 )
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 ) = ( 𝐺𝐹 ) )
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 ) ↔ ( 𝐺𝐹 ) = ( 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 ) ) ∧ ( 𝐺𝐹 ) = ( 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 ) ) ∧ ( 𝐺𝐹 ) = ( 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 ) ∧ ( 𝐺𝐹 ) = ( I ↾ 𝐸 ) ) ↔ ( ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ) ∧ ( 𝐺𝐹 ) = ( 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 ) ∧ ( 𝐺𝐹 ) = ( I ↾ 𝐸 ) ) ) )
53 13 52 bitrd ( 𝜑 → ( 𝐹 ( 𝑋 𝑆 𝑌 ) 𝐺 ↔ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RingHom 𝑋 ) ∧ ( 𝐺𝐹 ) = ( I ↾ 𝐸 ) ) ) )