evlfval

Description: Value of the evaluation functor. (Contributed by Mario Carneiro, 12-Jan-2017)

	Ref	Expression
Hypotheses	evlfval.e	⊢ 𝐸 = ( 𝐶 eval_F 𝐷 )
	evlfval.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
	evlfval.d	⊢ ( 𝜑 → 𝐷 ∈ Cat )
	evlfval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	evlfval.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	evlfval.o	⊢ · = ( comp ‘ 𝐷 )
	evlfval.n	⊢ 𝑁 = ( 𝐶 Nat 𝐷 )
Assertion	evlfval	⊢ ( 𝜑 → 𝐸 = ⟨ ( 𝑓 ∈ ( 𝐶 Func 𝐷 ) , 𝑥 ∈ 𝐵 ↦ ( ( 1^st ‘ 𝑓 ) ‘ 𝑥 ) ) , ( 𝑥 ∈ ( ( 𝐶 Func 𝐷 ) × 𝐵 ) , 𝑦 ∈ ( ( 𝐶 Func 𝐷 ) × 𝐵 ) ↦ ⦋ ( 1^st ‘ 𝑥 ) / 𝑚 ⦌ ⦋ ( 1^st ‘ 𝑦 ) / 𝑛 ⦌ ( 𝑎 ∈ ( 𝑚 𝑁 𝑛 ) , 𝑔 ∈ ( ( 2^nd ‘ 𝑥 ) 𝐻 ( 2^nd ‘ 𝑦 ) ) ↦ ( ( 𝑎 ‘ ( 2^nd ‘ 𝑦 ) ) ( ⟨ ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) , ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) ⟩ · ( ( 1^st ‘ 𝑛 ) ‘ ( 2^nd ‘ 𝑦 ) ) ) ( ( ( 2^nd ‘ 𝑥 ) ( 2^nd ‘ 𝑚 ) ( 2^nd ‘ 𝑦 ) ) ‘ 𝑔 ) ) ) ) ⟩ )

Proof

Step	Hyp	Ref	Expression
1		evlfval.e	⊢ 𝐸 = ( 𝐶 eval_F 𝐷 )
2		evlfval.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
3		evlfval.d	⊢ ( 𝜑 → 𝐷 ∈ Cat )
4		evlfval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
5		evlfval.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
6		evlfval.o	⊢ · = ( comp ‘ 𝐷 )
7		evlfval.n	⊢ 𝑁 = ( 𝐶 Nat 𝐷 )
8		df-evlf	⊢ eval_F = ( 𝑐 ∈ Cat , 𝑑 ∈ Cat ↦ ⟨ ( 𝑓 ∈ ( 𝑐 Func 𝑑 ) , 𝑥 ∈ ( Base ‘ 𝑐 ) ↦ ( ( 1^st ‘ 𝑓 ) ‘ 𝑥 ) ) , ( 𝑥 ∈ ( ( 𝑐 Func 𝑑 ) × ( Base ‘ 𝑐 ) ) , 𝑦 ∈ ( ( 𝑐 Func 𝑑 ) × ( Base ‘ 𝑐 ) ) ↦ ⦋ ( 1^st ‘ 𝑥 ) / 𝑚 ⦌ ⦋ ( 1^st ‘ 𝑦 ) / 𝑛 ⦌ ( 𝑎 ∈ ( 𝑚 ( 𝑐 Nat 𝑑 ) 𝑛 ) , 𝑔 ∈ ( ( 2^nd ‘ 𝑥 ) ( Hom ‘ 𝑐 ) ( 2^nd ‘ 𝑦 ) ) ↦ ( ( 𝑎 ‘ ( 2^nd ‘ 𝑦 ) ) ( ⟨ ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) , ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) ⟩ ( comp ‘ 𝑑 ) ( ( 1^st ‘ 𝑛 ) ‘ ( 2^nd ‘ 𝑦 ) ) ) ( ( ( 2^nd ‘ 𝑥 ) ( 2^nd ‘ 𝑚 ) ( 2^nd ‘ 𝑦 ) ) ‘ 𝑔 ) ) ) ) ⟩ )
9	8	a1i	⊢ ( 𝜑 → eval_F = ( 𝑐 ∈ Cat , 𝑑 ∈ Cat ↦ ⟨ ( 𝑓 ∈ ( 𝑐 Func 𝑑 ) , 𝑥 ∈ ( Base ‘ 𝑐 ) ↦ ( ( 1^st ‘ 𝑓 ) ‘ 𝑥 ) ) , ( 𝑥 ∈ ( ( 𝑐 Func 𝑑 ) × ( Base ‘ 𝑐 ) ) , 𝑦 ∈ ( ( 𝑐 Func 𝑑 ) × ( Base ‘ 𝑐 ) ) ↦ ⦋ ( 1^st ‘ 𝑥 ) / 𝑚 ⦌ ⦋ ( 1^st ‘ 𝑦 ) / 𝑛 ⦌ ( 𝑎 ∈ ( 𝑚 ( 𝑐 Nat 𝑑 ) 𝑛 ) , 𝑔 ∈ ( ( 2^nd ‘ 𝑥 ) ( Hom ‘ 𝑐 ) ( 2^nd ‘ 𝑦 ) ) ↦ ( ( 𝑎 ‘ ( 2^nd ‘ 𝑦 ) ) ( ⟨ ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) , ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) ⟩ ( comp ‘ 𝑑 ) ( ( 1^st ‘ 𝑛 ) ‘ ( 2^nd ‘ 𝑦 ) ) ) ( ( ( 2^nd ‘ 𝑥 ) ( 2^nd ‘ 𝑚 ) ( 2^nd ‘ 𝑦 ) ) ‘ 𝑔 ) ) ) ) ⟩ ) )
10		simprl	⊢ ( ( 𝜑 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → 𝑐 = 𝐶 )
11		simprr	⊢ ( ( 𝜑 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → 𝑑 = 𝐷 )
12	10 11	oveq12d	⊢ ( ( 𝜑 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( 𝑐 Func 𝑑 ) = ( 𝐶 Func 𝐷 ) )
13	10	fveq2d	⊢ ( ( 𝜑 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( Base ‘ 𝑐 ) = ( Base ‘ 𝐶 ) )
14	13 4	eqtr4di	⊢ ( ( 𝜑 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( Base ‘ 𝑐 ) = 𝐵 )
15		eqidd	⊢ ( ( 𝜑 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( ( 1^st ‘ 𝑓 ) ‘ 𝑥 ) = ( ( 1^st ‘ 𝑓 ) ‘ 𝑥 ) )
16	12 14 15	mpoeq123dv	⊢ ( ( 𝜑 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( 𝑓 ∈ ( 𝑐 Func 𝑑 ) , 𝑥 ∈ ( Base ‘ 𝑐 ) ↦ ( ( 1^st ‘ 𝑓 ) ‘ 𝑥 ) ) = ( 𝑓 ∈ ( 𝐶 Func 𝐷 ) , 𝑥 ∈ 𝐵 ↦ ( ( 1^st ‘ 𝑓 ) ‘ 𝑥 ) ) )
17	12 14	xpeq12d	⊢ ( ( 𝜑 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( ( 𝑐 Func 𝑑 ) × ( Base ‘ 𝑐 ) ) = ( ( 𝐶 Func 𝐷 ) × 𝐵 ) )
18	10 11	oveq12d	⊢ ( ( 𝜑 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( 𝑐 Nat 𝑑 ) = ( 𝐶 Nat 𝐷 ) )
19	18 7	eqtr4di	⊢ ( ( 𝜑 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( 𝑐 Nat 𝑑 ) = 𝑁 )
20	19	oveqd	⊢ ( ( 𝜑 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( 𝑚 ( 𝑐 Nat 𝑑 ) 𝑛 ) = ( 𝑚 𝑁 𝑛 ) )
21	10	fveq2d	⊢ ( ( 𝜑 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( Hom ‘ 𝑐 ) = ( Hom ‘ 𝐶 ) )
22	21 5	eqtr4di	⊢ ( ( 𝜑 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( Hom ‘ 𝑐 ) = 𝐻 )
23	22	oveqd	⊢ ( ( 𝜑 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( ( 2^nd ‘ 𝑥 ) ( Hom ‘ 𝑐 ) ( 2^nd ‘ 𝑦 ) ) = ( ( 2^nd ‘ 𝑥 ) 𝐻 ( 2^nd ‘ 𝑦 ) ) )
24	11	fveq2d	⊢ ( ( 𝜑 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( comp ‘ 𝑑 ) = ( comp ‘ 𝐷 ) )
25	24 6	eqtr4di	⊢ ( ( 𝜑 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( comp ‘ 𝑑 ) = · )
26	25	oveqd	⊢ ( ( 𝜑 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( ⟨ ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) , ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) ⟩ ( comp ‘ 𝑑 ) ( ( 1^st ‘ 𝑛 ) ‘ ( 2^nd ‘ 𝑦 ) ) ) = ( ⟨ ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) , ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) ⟩ · ( ( 1^st ‘ 𝑛 ) ‘ ( 2^nd ‘ 𝑦 ) ) ) )
27	26	oveqd	⊢ ( ( 𝜑 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( ( 𝑎 ‘ ( 2^nd ‘ 𝑦 ) ) ( ⟨ ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) , ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) ⟩ ( comp ‘ 𝑑 ) ( ( 1^st ‘ 𝑛 ) ‘ ( 2^nd ‘ 𝑦 ) ) ) ( ( ( 2^nd ‘ 𝑥 ) ( 2^nd ‘ 𝑚 ) ( 2^nd ‘ 𝑦 ) ) ‘ 𝑔 ) ) = ( ( 𝑎 ‘ ( 2^nd ‘ 𝑦 ) ) ( ⟨ ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) , ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) ⟩ · ( ( 1^st ‘ 𝑛 ) ‘ ( 2^nd ‘ 𝑦 ) ) ) ( ( ( 2^nd ‘ 𝑥 ) ( 2^nd ‘ 𝑚 ) ( 2^nd ‘ 𝑦 ) ) ‘ 𝑔 ) ) )
28	20 23 27	mpoeq123dv	⊢ ( ( 𝜑 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( 𝑎 ∈ ( 𝑚 ( 𝑐 Nat 𝑑 ) 𝑛 ) , 𝑔 ∈ ( ( 2^nd ‘ 𝑥 ) ( Hom ‘ 𝑐 ) ( 2^nd ‘ 𝑦 ) ) ↦ ( ( 𝑎 ‘ ( 2^nd ‘ 𝑦 ) ) ( ⟨ ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) , ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) ⟩ ( comp ‘ 𝑑 ) ( ( 1^st ‘ 𝑛 ) ‘ ( 2^nd ‘ 𝑦 ) ) ) ( ( ( 2^nd ‘ 𝑥 ) ( 2^nd ‘ 𝑚 ) ( 2^nd ‘ 𝑦 ) ) ‘ 𝑔 ) ) ) = ( 𝑎 ∈ ( 𝑚 𝑁 𝑛 ) , 𝑔 ∈ ( ( 2^nd ‘ 𝑥 ) 𝐻 ( 2^nd ‘ 𝑦 ) ) ↦ ( ( 𝑎 ‘ ( 2^nd ‘ 𝑦 ) ) ( ⟨ ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) , ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) ⟩ · ( ( 1^st ‘ 𝑛 ) ‘ ( 2^nd ‘ 𝑦 ) ) ) ( ( ( 2^nd ‘ 𝑥 ) ( 2^nd ‘ 𝑚 ) ( 2^nd ‘ 𝑦 ) ) ‘ 𝑔 ) ) ) )
29	28	csbeq2dv	⊢ ( ( 𝜑 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ⦋ ( 1^st ‘ 𝑦 ) / 𝑛 ⦌ ( 𝑎 ∈ ( 𝑚 ( 𝑐 Nat 𝑑 ) 𝑛 ) , 𝑔 ∈ ( ( 2^nd ‘ 𝑥 ) ( Hom ‘ 𝑐 ) ( 2^nd ‘ 𝑦 ) ) ↦ ( ( 𝑎 ‘ ( 2^nd ‘ 𝑦 ) ) ( ⟨ ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) , ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) ⟩ ( comp ‘ 𝑑 ) ( ( 1^st ‘ 𝑛 ) ‘ ( 2^nd ‘ 𝑦 ) ) ) ( ( ( 2^nd ‘ 𝑥 ) ( 2^nd ‘ 𝑚 ) ( 2^nd ‘ 𝑦 ) ) ‘ 𝑔 ) ) ) = ⦋ ( 1^st ‘ 𝑦 ) / 𝑛 ⦌ ( 𝑎 ∈ ( 𝑚 𝑁 𝑛 ) , 𝑔 ∈ ( ( 2^nd ‘ 𝑥 ) 𝐻 ( 2^nd ‘ 𝑦 ) ) ↦ ( ( 𝑎 ‘ ( 2^nd ‘ 𝑦 ) ) ( ⟨ ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) , ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) ⟩ · ( ( 1^st ‘ 𝑛 ) ‘ ( 2^nd ‘ 𝑦 ) ) ) ( ( ( 2^nd ‘ 𝑥 ) ( 2^nd ‘ 𝑚 ) ( 2^nd ‘ 𝑦 ) ) ‘ 𝑔 ) ) ) )
30	29	csbeq2dv	⊢ ( ( 𝜑 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ⦋ ( 1^st ‘ 𝑥 ) / 𝑚 ⦌ ⦋ ( 1^st ‘ 𝑦 ) / 𝑛 ⦌ ( 𝑎 ∈ ( 𝑚 ( 𝑐 Nat 𝑑 ) 𝑛 ) , 𝑔 ∈ ( ( 2^nd ‘ 𝑥 ) ( Hom ‘ 𝑐 ) ( 2^nd ‘ 𝑦 ) ) ↦ ( ( 𝑎 ‘ ( 2^nd ‘ 𝑦 ) ) ( ⟨ ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) , ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) ⟩ ( comp ‘ 𝑑 ) ( ( 1^st ‘ 𝑛 ) ‘ ( 2^nd ‘ 𝑦 ) ) ) ( ( ( 2^nd ‘ 𝑥 ) ( 2^nd ‘ 𝑚 ) ( 2^nd ‘ 𝑦 ) ) ‘ 𝑔 ) ) ) = ⦋ ( 1^st ‘ 𝑥 ) / 𝑚 ⦌ ⦋ ( 1^st ‘ 𝑦 ) / 𝑛 ⦌ ( 𝑎 ∈ ( 𝑚 𝑁 𝑛 ) , 𝑔 ∈ ( ( 2^nd ‘ 𝑥 ) 𝐻 ( 2^nd ‘ 𝑦 ) ) ↦ ( ( 𝑎 ‘ ( 2^nd ‘ 𝑦 ) ) ( ⟨ ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) , ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) ⟩ · ( ( 1^st ‘ 𝑛 ) ‘ ( 2^nd ‘ 𝑦 ) ) ) ( ( ( 2^nd ‘ 𝑥 ) ( 2^nd ‘ 𝑚 ) ( 2^nd ‘ 𝑦 ) ) ‘ 𝑔 ) ) ) )
31	17 17 30	mpoeq123dv	⊢ ( ( 𝜑 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( 𝑥 ∈ ( ( 𝑐 Func 𝑑 ) × ( Base ‘ 𝑐 ) ) , 𝑦 ∈ ( ( 𝑐 Func 𝑑 ) × ( Base ‘ 𝑐 ) ) ↦ ⦋ ( 1^st ‘ 𝑥 ) / 𝑚 ⦌ ⦋ ( 1^st ‘ 𝑦 ) / 𝑛 ⦌ ( 𝑎 ∈ ( 𝑚 ( 𝑐 Nat 𝑑 ) 𝑛 ) , 𝑔 ∈ ( ( 2^nd ‘ 𝑥 ) ( Hom ‘ 𝑐 ) ( 2^nd ‘ 𝑦 ) ) ↦ ( ( 𝑎 ‘ ( 2^nd ‘ 𝑦 ) ) ( ⟨ ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) , ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) ⟩ ( comp ‘ 𝑑 ) ( ( 1^st ‘ 𝑛 ) ‘ ( 2^nd ‘ 𝑦 ) ) ) ( ( ( 2^nd ‘ 𝑥 ) ( 2^nd ‘ 𝑚 ) ( 2^nd ‘ 𝑦 ) ) ‘ 𝑔 ) ) ) ) = ( 𝑥 ∈ ( ( 𝐶 Func 𝐷 ) × 𝐵 ) , 𝑦 ∈ ( ( 𝐶 Func 𝐷 ) × 𝐵 ) ↦ ⦋ ( 1^st ‘ 𝑥 ) / 𝑚 ⦌ ⦋ ( 1^st ‘ 𝑦 ) / 𝑛 ⦌ ( 𝑎 ∈ ( 𝑚 𝑁 𝑛 ) , 𝑔 ∈ ( ( 2^nd ‘ 𝑥 ) 𝐻 ( 2^nd ‘ 𝑦 ) ) ↦ ( ( 𝑎 ‘ ( 2^nd ‘ 𝑦 ) ) ( ⟨ ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) , ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) ⟩ · ( ( 1^st ‘ 𝑛 ) ‘ ( 2^nd ‘ 𝑦 ) ) ) ( ( ( 2^nd ‘ 𝑥 ) ( 2^nd ‘ 𝑚 ) ( 2^nd ‘ 𝑦 ) ) ‘ 𝑔 ) ) ) ) )
32	16 31	opeq12d	⊢ ( ( 𝜑 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ⟨ ( 𝑓 ∈ ( 𝑐 Func 𝑑 ) , 𝑥 ∈ ( Base ‘ 𝑐 ) ↦ ( ( 1^st ‘ 𝑓 ) ‘ 𝑥 ) ) , ( 𝑥 ∈ ( ( 𝑐 Func 𝑑 ) × ( Base ‘ 𝑐 ) ) , 𝑦 ∈ ( ( 𝑐 Func 𝑑 ) × ( Base ‘ 𝑐 ) ) ↦ ⦋ ( 1^st ‘ 𝑥 ) / 𝑚 ⦌ ⦋ ( 1^st ‘ 𝑦 ) / 𝑛 ⦌ ( 𝑎 ∈ ( 𝑚 ( 𝑐 Nat 𝑑 ) 𝑛 ) , 𝑔 ∈ ( ( 2^nd ‘ 𝑥 ) ( Hom ‘ 𝑐 ) ( 2^nd ‘ 𝑦 ) ) ↦ ( ( 𝑎 ‘ ( 2^nd ‘ 𝑦 ) ) ( ⟨ ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) , ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) ⟩ ( comp ‘ 𝑑 ) ( ( 1^st ‘ 𝑛 ) ‘ ( 2^nd ‘ 𝑦 ) ) ) ( ( ( 2^nd ‘ 𝑥 ) ( 2^nd ‘ 𝑚 ) ( 2^nd ‘ 𝑦 ) ) ‘ 𝑔 ) ) ) ) ⟩ = ⟨ ( 𝑓 ∈ ( 𝐶 Func 𝐷 ) , 𝑥 ∈ 𝐵 ↦ ( ( 1^st ‘ 𝑓 ) ‘ 𝑥 ) ) , ( 𝑥 ∈ ( ( 𝐶 Func 𝐷 ) × 𝐵 ) , 𝑦 ∈ ( ( 𝐶 Func 𝐷 ) × 𝐵 ) ↦ ⦋ ( 1^st ‘ 𝑥 ) / 𝑚 ⦌ ⦋ ( 1^st ‘ 𝑦 ) / 𝑛 ⦌ ( 𝑎 ∈ ( 𝑚 𝑁 𝑛 ) , 𝑔 ∈ ( ( 2^nd ‘ 𝑥 ) 𝐻 ( 2^nd ‘ 𝑦 ) ) ↦ ( ( 𝑎 ‘ ( 2^nd ‘ 𝑦 ) ) ( ⟨ ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) , ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) ⟩ · ( ( 1^st ‘ 𝑛 ) ‘ ( 2^nd ‘ 𝑦 ) ) ) ( ( ( 2^nd ‘ 𝑥 ) ( 2^nd ‘ 𝑚 ) ( 2^nd ‘ 𝑦 ) ) ‘ 𝑔 ) ) ) ) ⟩ )
33		opex	⊢ ⟨ ( 𝑓 ∈ ( 𝐶 Func 𝐷 ) , 𝑥 ∈ 𝐵 ↦ ( ( 1^st ‘ 𝑓 ) ‘ 𝑥 ) ) , ( 𝑥 ∈ ( ( 𝐶 Func 𝐷 ) × 𝐵 ) , 𝑦 ∈ ( ( 𝐶 Func 𝐷 ) × 𝐵 ) ↦ ⦋ ( 1^st ‘ 𝑥 ) / 𝑚 ⦌ ⦋ ( 1^st ‘ 𝑦 ) / 𝑛 ⦌ ( 𝑎 ∈ ( 𝑚 𝑁 𝑛 ) , 𝑔 ∈ ( ( 2^nd ‘ 𝑥 ) 𝐻 ( 2^nd ‘ 𝑦 ) ) ↦ ( ( 𝑎 ‘ ( 2^nd ‘ 𝑦 ) ) ( ⟨ ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) , ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) ⟩ · ( ( 1^st ‘ 𝑛 ) ‘ ( 2^nd ‘ 𝑦 ) ) ) ( ( ( 2^nd ‘ 𝑥 ) ( 2^nd ‘ 𝑚 ) ( 2^nd ‘ 𝑦 ) ) ‘ 𝑔 ) ) ) ) ⟩ ∈ V
34	33	a1i	⊢ ( 𝜑 → ⟨ ( 𝑓 ∈ ( 𝐶 Func 𝐷 ) , 𝑥 ∈ 𝐵 ↦ ( ( 1^st ‘ 𝑓 ) ‘ 𝑥 ) ) , ( 𝑥 ∈ ( ( 𝐶 Func 𝐷 ) × 𝐵 ) , 𝑦 ∈ ( ( 𝐶 Func 𝐷 ) × 𝐵 ) ↦ ⦋ ( 1^st ‘ 𝑥 ) / 𝑚 ⦌ ⦋ ( 1^st ‘ 𝑦 ) / 𝑛 ⦌ ( 𝑎 ∈ ( 𝑚 𝑁 𝑛 ) , 𝑔 ∈ ( ( 2^nd ‘ 𝑥 ) 𝐻 ( 2^nd ‘ 𝑦 ) ) ↦ ( ( 𝑎 ‘ ( 2^nd ‘ 𝑦 ) ) ( ⟨ ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) , ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) ⟩ · ( ( 1^st ‘ 𝑛 ) ‘ ( 2^nd ‘ 𝑦 ) ) ) ( ( ( 2^nd ‘ 𝑥 ) ( 2^nd ‘ 𝑚 ) ( 2^nd ‘ 𝑦 ) ) ‘ 𝑔 ) ) ) ) ⟩ ∈ V )
35	9 32 2 3 34	ovmpod	⊢ ( 𝜑 → ( 𝐶 eval_F 𝐷 ) = ⟨ ( 𝑓 ∈ ( 𝐶 Func 𝐷 ) , 𝑥 ∈ 𝐵 ↦ ( ( 1^st ‘ 𝑓 ) ‘ 𝑥 ) ) , ( 𝑥 ∈ ( ( 𝐶 Func 𝐷 ) × 𝐵 ) , 𝑦 ∈ ( ( 𝐶 Func 𝐷 ) × 𝐵 ) ↦ ⦋ ( 1^st ‘ 𝑥 ) / 𝑚 ⦌ ⦋ ( 1^st ‘ 𝑦 ) / 𝑛 ⦌ ( 𝑎 ∈ ( 𝑚 𝑁 𝑛 ) , 𝑔 ∈ ( ( 2^nd ‘ 𝑥 ) 𝐻 ( 2^nd ‘ 𝑦 ) ) ↦ ( ( 𝑎 ‘ ( 2^nd ‘ 𝑦 ) ) ( ⟨ ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) , ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) ⟩ · ( ( 1^st ‘ 𝑛 ) ‘ ( 2^nd ‘ 𝑦 ) ) ) ( ( ( 2^nd ‘ 𝑥 ) ( 2^nd ‘ 𝑚 ) ( 2^nd ‘ 𝑦 ) ) ‘ 𝑔 ) ) ) ) ⟩ )
36	1 35	eqtrid	⊢ ( 𝜑 → 𝐸 = ⟨ ( 𝑓 ∈ ( 𝐶 Func 𝐷 ) , 𝑥 ∈ 𝐵 ↦ ( ( 1^st ‘ 𝑓 ) ‘ 𝑥 ) ) , ( 𝑥 ∈ ( ( 𝐶 Func 𝐷 ) × 𝐵 ) , 𝑦 ∈ ( ( 𝐶 Func 𝐷 ) × 𝐵 ) ↦ ⦋ ( 1^st ‘ 𝑥 ) / 𝑚 ⦌ ⦋ ( 1^st ‘ 𝑦 ) / 𝑛 ⦌ ( 𝑎 ∈ ( 𝑚 𝑁 𝑛 ) , 𝑔 ∈ ( ( 2^nd ‘ 𝑥 ) 𝐻 ( 2^nd ‘ 𝑦 ) ) ↦ ( ( 𝑎 ‘ ( 2^nd ‘ 𝑦 ) ) ( ⟨ ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) , ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) ⟩ · ( ( 1^st ‘ 𝑛 ) ‘ ( 2^nd ‘ 𝑦 ) ) ) ( ( ( 2^nd ‘ 𝑥 ) ( 2^nd ‘ 𝑚 ) ( 2^nd ‘ 𝑦 ) ) ‘ 𝑔 ) ) ) ) ⟩ )

Metamath Proof Explorer

Theorem evlfval

Proof