Metamath Proof Explorer

Theorem catcocl

Description: Closure of a composition arrow. (Contributed by Mario Carneiro, 2-Jan-2017)

Ref Expression
Hypotheses catcocl.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
catcocl.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
catcocl.o ⊢ · = ( comp ‘ 𝐶 )
catcocl.c ⊢ ( 𝜑 → 𝐶 ∈ Cat )
catcocl.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
catcocl.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
catcocl.z ⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
catcocl.f ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
catcocl.g ⊢ ( 𝜑 → 𝐺 ∈ ( 𝑌 𝐻 𝑍 ) )
Assertion catcocl ( 𝜑 → ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝐹 ) ∈ ( 𝑋 𝐻 𝑍 ) )

Proof

Step Hyp Ref Expression
1 catcocl.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
2 catcocl.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
3 catcocl.o ⊢ · = ( comp ‘ 𝐶 )
4 catcocl.c ⊢ ( 𝜑 → 𝐶 ∈ Cat )
5 catcocl.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6 catcocl.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7 catcocl.z ⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
8 catcocl.f ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
9 catcocl.g ⊢ ( 𝜑 → 𝐺 ∈ ( 𝑌 𝐻 𝑍 ) )
10 1 2 3 iscat ⊢ ( 𝐶 ∈ Cat → ( 𝐶 ∈ Cat ↔ ∀ 𝑥 ∈ 𝐵 ( ∃ 𝑔 ∈ ( 𝑥 𝐻 𝑥 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ · 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) ∧ ∀ 𝑤 ∈ 𝐵 ∀ 𝑣 ∈ ( 𝑧 𝐻 𝑤 ) ( ( 𝑣 ( ⟨ 𝑦 , 𝑧 ⟩ · 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑤 ) 𝑓 ) = ( 𝑣 ( ⟨ 𝑥 , 𝑧 ⟩ · 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ) ) ) ) )
11 10 ibi ⊢ ( 𝐶 ∈ Cat → ∀ 𝑥 ∈ 𝐵 ( ∃ 𝑔 ∈ ( 𝑥 𝐻 𝑥 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ · 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) ∧ ∀ 𝑤 ∈ 𝐵 ∀ 𝑣 ∈ ( 𝑧 𝐻 𝑤 ) ( ( 𝑣 ( ⟨ 𝑦 , 𝑧 ⟩ · 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑤 ) 𝑓 ) = ( 𝑣 ( ⟨ 𝑥 , 𝑧 ⟩ · 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ) ) ) )
12 simpl ⊢ ( ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) ∧ ∀ 𝑤 ∈ 𝐵 ∀ 𝑣 ∈ ( 𝑧 𝐻 𝑤 ) ( ( 𝑣 ( ⟨ 𝑦 , 𝑧 ⟩ · 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑤 ) 𝑓 ) = ( 𝑣 ( ⟨ 𝑥 , 𝑧 ⟩ · 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ) ) → ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) )
13 12 2ralimi ⊢ ( ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) ∧ ∀ 𝑤 ∈ 𝐵 ∀ 𝑣 ∈ ( 𝑧 𝐻 𝑤 ) ( ( 𝑣 ( ⟨ 𝑦 , 𝑧 ⟩ · 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑤 ) 𝑓 ) = ( 𝑣 ( ⟨ 𝑥 , 𝑧 ⟩ · 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ) ) → ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) )
14 13 2ralimi ⊢ ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) ∧ ∀ 𝑤 ∈ 𝐵 ∀ 𝑣 ∈ ( 𝑧 𝐻 𝑤 ) ( ( 𝑣 ( ⟨ 𝑦 , 𝑧 ⟩ · 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑤 ) 𝑓 ) = ( 𝑣 ( ⟨ 𝑥 , 𝑧 ⟩ · 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ) ) → ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) )
15 14 adantl ⊢ ( ( ∃ 𝑔 ∈ ( 𝑥 𝐻 𝑥 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ · 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) ∧ ∀ 𝑤 ∈ 𝐵 ∀ 𝑣 ∈ ( 𝑧 𝐻 𝑤 ) ( ( 𝑣 ( ⟨ 𝑦 , 𝑧 ⟩ · 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑤 ) 𝑓 ) = ( 𝑣 ( ⟨ 𝑥 , 𝑧 ⟩ · 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ) ) ) → ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) )
16 15 ralimi ⊢ ( ∀ 𝑥 ∈ 𝐵 ( ∃ 𝑔 ∈ ( 𝑥 𝐻 𝑥 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ · 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) ∧ ∀ 𝑤 ∈ 𝐵 ∀ 𝑣 ∈ ( 𝑧 𝐻 𝑤 ) ( ( 𝑣 ( ⟨ 𝑦 , 𝑧 ⟩ · 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑤 ) 𝑓 ) = ( 𝑣 ( ⟨ 𝑥 , 𝑧 ⟩ · 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ) ) ) → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) )
17 4 11 16 3syl ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) )
18 6 adantr ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → 𝑌 ∈ 𝐵 )
19 7 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) → 𝑍 ∈ 𝐵 )
20 8 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
21 simpllr ⊢ ( ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) → 𝑥 = 𝑋 )
22 simplr ⊢ ( ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) → 𝑦 = 𝑌 )
23 21 22 oveq12d ⊢ ( ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) → ( 𝑥 𝐻 𝑦 ) = ( 𝑋 𝐻 𝑌 ) )
24 20 23 eleqtrrd ⊢ ( ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) → 𝐹 ∈ ( 𝑥 𝐻 𝑦 ) )
25 9 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) → 𝐺 ∈ ( 𝑌 𝐻 𝑍 ) )
26 simpr ⊢ ( ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) → 𝑧 = 𝑍 )
27 22 26 oveq12d ⊢ ( ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) → ( 𝑦 𝐻 𝑧 ) = ( 𝑌 𝐻 𝑍 ) )
28 25 27 eleqtrrd ⊢ ( ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) → 𝐺 ∈ ( 𝑦 𝐻 𝑧 ) )
29 28 adantr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) ∧ 𝑓 = 𝐹 ) → 𝐺 ∈ ( 𝑦 𝐻 𝑧 ) )
30 simp-5r ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) ∧ 𝑓 = 𝐹 ) ∧ 𝑔 = 𝐺 ) → 𝑥 = 𝑋 )
31 simp-4r ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) ∧ 𝑓 = 𝐹 ) ∧ 𝑔 = 𝐺 ) → 𝑦 = 𝑌 )
32 30 31 opeq12d ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) ∧ 𝑓 = 𝐹 ) ∧ 𝑔 = 𝐺 ) → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑋 , 𝑌 ⟩ )
33 simpllr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) ∧ 𝑓 = 𝐹 ) ∧ 𝑔 = 𝐺 ) → 𝑧 = 𝑍 )
34 32 33 oveq12d ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) ∧ 𝑓 = 𝐹 ) ∧ 𝑔 = 𝐺 ) → ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) = ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) )
35 simpr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) ∧ 𝑓 = 𝐹 ) ∧ 𝑔 = 𝐺 ) → 𝑔 = 𝐺 )
36 simplr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) ∧ 𝑓 = 𝐹 ) ∧ 𝑔 = 𝐺 ) → 𝑓 = 𝐹 )
37 34 35 36 oveq123d ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) ∧ 𝑓 = 𝐹 ) ∧ 𝑔 = 𝐺 ) → ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) = ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝐹 ) )
38 30 33 oveq12d ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) ∧ 𝑓 = 𝐹 ) ∧ 𝑔 = 𝐺 ) → ( 𝑥 𝐻 𝑧 ) = ( 𝑋 𝐻 𝑍 ) )
39 37 38 eleq12d ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) ∧ 𝑓 = 𝐹 ) ∧ 𝑔 = 𝐺 ) → ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) ↔ ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝐹 ) ∈ ( 𝑋 𝐻 𝑍 ) ) )
40 29 39 rspcdv ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) ∧ 𝑓 = 𝐹 ) → ( ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) → ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝐹 ) ∈ ( 𝑋 𝐻 𝑍 ) ) )
41 24 40 rspcimdv ⊢ ( ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) ∧ 𝑧 = 𝑍 ) → ( ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) → ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝐹 ) ∈ ( 𝑋 𝐻 𝑍 ) ) )
42 19 41 rspcimdv ⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 = 𝑌 ) → ( ∀ 𝑧 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) → ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝐹 ) ∈ ( 𝑋 𝐻 𝑍 ) ) )
43 18 42 rspcimdv ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) → ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝐹 ) ∈ ( 𝑋 𝐻 𝑍 ) ) )
44 5 43 rspcimdv ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) → ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝐹 ) ∈ ( 𝑋 𝐻 𝑍 ) ) )
45 17 44 mpd ⊢ ( 𝜑 → ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝐹 ) ∈ ( 𝑋 𝐻 𝑍 ) )