Metamath Proof Explorer

Theorem rngcsectALTV

Description: A section in the category of non-unital rings, written out. (Contributed by AV, 28-Feb-2020) (New usage is discouraged.)

Ref Expression
Hypotheses rngcsectALTV.c ⊢ 𝐶 = ( RngCatALTV ‘ 𝑈 )
rngcsectALTV.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
rngcsectALTV.u ⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
rngcsectALTV.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
rngcsectALTV.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
rngcsectALTV.e ⊢ 𝐸 = ( Base ‘ 𝑋 )
rngcsectALTV.n ⊢ 𝑆 = ( Sect ‘ 𝐶 )
Assertion rngcsectALTV ( 𝜑 → ( 𝐹 ( 𝑋 𝑆 𝑌 ) 𝐺 ↔ ( 𝐹 ∈ ( 𝑋 RngHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RngHom 𝑋 ) ∧ ( 𝐺 ∘ 𝐹 ) = ( I ↾ 𝐸 ) ) ) )

Proof

Step Hyp Ref Expression
1 rngcsectALTV.c ⊢ 𝐶 = ( RngCatALTV ‘ 𝑈 )
2 rngcsectALTV.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
3 rngcsectALTV.u ⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
4 rngcsectALTV.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5 rngcsectALTV.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6 rngcsectALTV.e ⊢ 𝐸 = ( Base ‘ 𝑋 )
7 rngcsectALTV.n ⊢ 𝑆 = ( Sect ‘ 𝐶 )
8 eqid ⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
9 eqid ⊢ ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 )
10 eqid ⊢ ( Id ‘ 𝐶 ) = ( Id ‘ 𝐶 )
11 1 rngccatALTV ⊢ ( 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 rngchomALTV ⊢ ( 𝜑 → ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) = ( 𝑋 RngHom 𝑌 ) )
15 14 eleq2d ⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ↔ 𝐹 ∈ ( 𝑋 RngHom 𝑌 ) ) )
16 1 2 3 8 5 4 rngchomALTV ⊢ ( 𝜑 → ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) = ( 𝑌 RngHom 𝑋 ) )
17 16 eleq2d ⊢ ( 𝜑 → ( 𝐺 ∈ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) ↔ 𝐺 ∈ ( 𝑌 RngHom 𝑋 ) ) )
18 15 17 anbi12d ⊢ ( 𝜑 → ( ( 𝐹 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) ) ↔ ( 𝐹 ∈ ( 𝑋 RngHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RngHom 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 RngHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RngHom 𝑋 ) ) ∧ ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ ( comp ‘ 𝐶 ) 𝑋 ) 𝐹 ) = ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ) ) )
20 3 adantr ⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RngHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RngHom 𝑋 ) ) ) → 𝑈 ∈ 𝑉 )
21 4 adantr ⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RngHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RngHom 𝑋 ) ) ) → 𝑋 ∈ 𝐵 )
22 5 adantr ⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RngHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RngHom 𝑋 ) ) ) → 𝑌 ∈ 𝐵 )
23 simprl ⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RngHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RngHom 𝑋 ) ) ) → 𝐹 ∈ ( 𝑋 RngHom 𝑌 ) )
24 simprr ⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RngHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RngHom 𝑋 ) ) ) → 𝐺 ∈ ( 𝑌 RngHom 𝑋 ) )
25 1 2 20 9 21 22 21 23 24 rngccoALTV ⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RngHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RngHom 𝑋 ) ) ) → ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ ( comp ‘ 𝐶 ) 𝑋 ) 𝐹 ) = ( 𝐺 ∘ 𝐹 ) )
26 1 2 10 3 4 6 rngcidALTV ⊢ ( 𝜑 → ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) = ( I ↾ 𝐸 ) )
27 26 adantr ⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RngHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RngHom 𝑋 ) ) ) → ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) = ( I ↾ 𝐸 ) )
28 25 27 eqeq12d ⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( 𝑋 RngHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RngHom 𝑋 ) ) ) → ( ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ ( comp ‘ 𝐶 ) 𝑋 ) 𝐹 ) = ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ↔ ( 𝐺 ∘ 𝐹 ) = ( I ↾ 𝐸 ) ) )
29 28 pm5.32da ⊢ ( 𝜑 → ( ( ( 𝐹 ∈ ( 𝑋 RngHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RngHom 𝑋 ) ) ∧ ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ ( comp ‘ 𝐶 ) 𝑋 ) 𝐹 ) = ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ) ↔ ( ( 𝐹 ∈ ( 𝑋 RngHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RngHom 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 RngHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RngHom 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 RngHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RngHom 𝑋 ) ∧ ( 𝐺 ∘ 𝐹 ) = ( I ↾ 𝐸 ) ) ↔ ( ( 𝐹 ∈ ( 𝑋 RngHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RngHom 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 RngHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RngHom 𝑋 ) ∧ ( 𝐺 ∘ 𝐹 ) = ( I ↾ 𝐸 ) ) ) )
34 13 33 bitrd ⊢ ( 𝜑 → ( 𝐹 ( 𝑋 𝑆 𝑌 ) 𝐺 ↔ ( 𝐹 ∈ ( 𝑋 RngHom 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 RngHom 𝑋 ) ∧ ( 𝐺 ∘ 𝐹 ) = ( I ↾ 𝐸 ) ) ) )