Metamath Proof Explorer

Theorem funcringcsetcALTV2lem8

Description: Lemma 8 for funcringcsetcALTV2 . (Contributed by AV, 15-Feb-2020) (New usage is discouraged.)

Ref Expression
Hypotheses funcringcsetcALTV2.r ⊢ 𝑅 = ( RingCat ‘ 𝑈 )
funcringcsetcALTV2.s ⊢ 𝑆 = ( SetCat ‘ 𝑈 )
funcringcsetcALTV2.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
funcringcsetcALTV2.c ⊢ 𝐶 = ( Base ‘ 𝑆 )
funcringcsetcALTV2.u ⊢ ( 𝜑 → 𝑈 ∈ WUni )
funcringcsetcALTV2.f ⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) )
funcringcsetcALTV2.g ⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ) )
Assertion funcringcsetcALTV2lem8 ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 𝐺 𝑌 ) : ( 𝑋 ( Hom ‘ 𝑅 ) 𝑌 ) ⟶ ( ( 𝐹 ‘ 𝑋 ) ( Hom ‘ 𝑆 ) ( 𝐹 ‘ 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 funcringcsetcALTV2.r ⊢ 𝑅 = ( RingCat ‘ 𝑈 )
2 funcringcsetcALTV2.s ⊢ 𝑆 = ( SetCat ‘ 𝑈 )
3 funcringcsetcALTV2.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
4 funcringcsetcALTV2.c ⊢ 𝐶 = ( Base ‘ 𝑆 )
5 funcringcsetcALTV2.u ⊢ ( 𝜑 → 𝑈 ∈ WUni )
6 funcringcsetcALTV2.f ⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) )
7 funcringcsetcALTV2.g ⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ) )
8 f1oi ⊢ ( I ↾ ( 𝑋 RingHom 𝑌 ) ) : ( 𝑋 RingHom 𝑌 ) –1-1-onto→ ( 𝑋 RingHom 𝑌 )
9 f1of ⊢ ( ( I ↾ ( 𝑋 RingHom 𝑌 ) ) : ( 𝑋 RingHom 𝑌 ) –1-1-onto→ ( 𝑋 RingHom 𝑌 ) → ( I ↾ ( 𝑋 RingHom 𝑌 ) ) : ( 𝑋 RingHom 𝑌 ) ⟶ ( 𝑋 RingHom 𝑌 ) )
10 8 9 mp1i ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( I ↾ ( 𝑋 RingHom 𝑌 ) ) : ( 𝑋 RingHom 𝑌 ) ⟶ ( 𝑋 RingHom 𝑌 ) )
11 eqid ⊢ ( Base ‘ 𝑋 ) = ( Base ‘ 𝑋 )
12 eqid ⊢ ( Base ‘ 𝑌 ) = ( Base ‘ 𝑌 )
13 11 12 rhmf ⊢ ( 𝑓 ∈ ( 𝑋 RingHom 𝑌 ) → 𝑓 : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑌 ) )
14 fvex ⊢ ( Base ‘ 𝑌 ) ∈ V
15 fvex ⊢ ( Base ‘ 𝑋 ) ∈ V
16 14 15 pm3.2i ⊢ ( ( Base ‘ 𝑌 ) ∈ V ∧ ( Base ‘ 𝑋 ) ∈ V )
17 elmapg ⊢ ( ( ( Base ‘ 𝑌 ) ∈ V ∧ ( Base ‘ 𝑋 ) ∈ V ) → ( 𝑓 ∈ ( ( Base ‘ 𝑌 ) ↑m ( Base ‘ 𝑋 ) ) ↔ 𝑓 : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑌 ) ) )
18 17 bicomd ⊢ ( ( ( Base ‘ 𝑌 ) ∈ V ∧ ( Base ‘ 𝑋 ) ∈ V ) → ( 𝑓 : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑌 ) ↔ 𝑓 ∈ ( ( Base ‘ 𝑌 ) ↑m ( Base ‘ 𝑋 ) ) ) )
19 16 18 mp1i ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑓 : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑌 ) ↔ 𝑓 ∈ ( ( Base ‘ 𝑌 ) ↑m ( Base ‘ 𝑋 ) ) ) )
20 19 biimpa ⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑓 : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑌 ) ) → 𝑓 ∈ ( ( Base ‘ 𝑌 ) ↑m ( Base ‘ 𝑋 ) ) )
21 simpr ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 )
22 1 2 3 4 5 6 funcringcsetcALTV2lem1 ⊢ ( ( 𝜑 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑌 ) = ( Base ‘ 𝑌 ) )
23 21 22 sylan2 ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑌 ) = ( Base ‘ 𝑌 ) )
24 simpl ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 )
25 1 2 3 4 5 6 funcringcsetcALTV2lem1 ⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑋 ) = ( Base ‘ 𝑋 ) )
26 24 25 sylan2 ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑋 ) = ( Base ‘ 𝑋 ) )
27 23 26 oveq12d ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑌 ) ↑m ( 𝐹 ‘ 𝑋 ) ) = ( ( Base ‘ 𝑌 ) ↑m ( Base ‘ 𝑋 ) ) )
28 27 adantr ⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑓 : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑌 ) ) → ( ( 𝐹 ‘ 𝑌 ) ↑m ( 𝐹 ‘ 𝑋 ) ) = ( ( Base ‘ 𝑌 ) ↑m ( Base ‘ 𝑋 ) ) )
29 20 28 eleqtrrd ⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑓 : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑌 ) ) → 𝑓 ∈ ( ( 𝐹 ‘ 𝑌 ) ↑m ( 𝐹 ‘ 𝑋 ) ) )
30 29 ex ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑓 : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑌 ) → 𝑓 ∈ ( ( 𝐹 ‘ 𝑌 ) ↑m ( 𝐹 ‘ 𝑋 ) ) ) )
31 13 30 syl5 ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑓 ∈ ( 𝑋 RingHom 𝑌 ) → 𝑓 ∈ ( ( 𝐹 ‘ 𝑌 ) ↑m ( 𝐹 ‘ 𝑋 ) ) ) )
32 31 ssrdv ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 RingHom 𝑌 ) ⊆ ( ( 𝐹 ‘ 𝑌 ) ↑m ( 𝐹 ‘ 𝑋 ) ) )
33 10 32 fssd ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( I ↾ ( 𝑋 RingHom 𝑌 ) ) : ( 𝑋 RingHom 𝑌 ) ⟶ ( ( 𝐹 ‘ 𝑌 ) ↑m ( 𝐹 ‘ 𝑋 ) ) )
34 1 2 3 4 5 6 7 funcringcsetcALTV2lem5 ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 𝐺 𝑌 ) = ( I ↾ ( 𝑋 RingHom 𝑌 ) ) )
35 5 adantr ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑈 ∈ WUni )
36 eqid ⊢ ( Hom ‘ 𝑅 ) = ( Hom ‘ 𝑅 )
37 24 adantl ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
38 21 adantl ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
39 1 3 35 36 37 38 ringchom ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( Hom ‘ 𝑅 ) 𝑌 ) = ( 𝑋 RingHom 𝑌 ) )
40 eqid ⊢ ( Hom ‘ 𝑆 ) = ( Hom ‘ 𝑆 )
41 1 2 3 4 5 6 funcringcsetcALTV2lem2 ⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 )
42 24 41 sylan2 ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 )
43 1 2 3 4 5 6 funcringcsetcALTV2lem2 ⊢ ( ( 𝜑 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑌 ) ∈ 𝑈 )
44 21 43 sylan2 ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑌 ) ∈ 𝑈 )
45 2 35 40 42 44 setchom ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑋 ) ( Hom ‘ 𝑆 ) ( 𝐹 ‘ 𝑌 ) ) = ( ( 𝐹 ‘ 𝑌 ) ↑m ( 𝐹 ‘ 𝑋 ) ) )
46 34 39 45 feq123d ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 𝐺 𝑌 ) : ( 𝑋 ( Hom ‘ 𝑅 ) 𝑌 ) ⟶ ( ( 𝐹 ‘ 𝑋 ) ( Hom ‘ 𝑆 ) ( 𝐹 ‘ 𝑌 ) ) ↔ ( I ↾ ( 𝑋 RingHom 𝑌 ) ) : ( 𝑋 RingHom 𝑌 ) ⟶ ( ( 𝐹 ‘ 𝑌 ) ↑m ( 𝐹 ‘ 𝑋 ) ) ) )
47 33 46 mpbird ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 𝐺 𝑌 ) : ( 𝑋 ( Hom ‘ 𝑅 ) 𝑌 ) ⟶ ( ( 𝐹 ‘ 𝑋 ) ( Hom ‘ 𝑆 ) ( 𝐹 ‘ 𝑌 ) ) )