evlf2

Description: Value of the evaluation functor at a morphism. (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 )
	evlf2.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝐶 Func 𝐷 ) )
	evlf2.g	⊢ ( 𝜑 → 𝐺 ∈ ( 𝐶 Func 𝐷 ) )
	evlf2.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	evlf2.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	evlf2.l	⊢ 𝐿 = ( ⟨ 𝐹 , 𝑋 ⟩ ( 2^nd ‘ 𝐸 ) ⟨ 𝐺 , 𝑌 ⟩ )
Assertion	evlf2	⊢ ( 𝜑 → 𝐿 = ( 𝑎 ∈ ( 𝐹 𝑁 𝐺 ) , 𝑔 ∈ ( 𝑋 𝐻 𝑌 ) ↦ ( ( 𝑎 ‘ 𝑌 ) ( ⟨ ( ( 1^st ‘ 𝐹 ) ‘ 𝑋 ) , ( ( 1^st ‘ 𝐹 ) ‘ 𝑌 ) ⟩ · ( ( 1^st ‘ 𝐺 ) ‘ 𝑌 ) ) ( ( 𝑋 ( 2^nd ‘ 𝐹 ) 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		evlf2.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝐶 Func 𝐷 ) )
9		evlf2.g	⊢ ( 𝜑 → 𝐺 ∈ ( 𝐶 Func 𝐷 ) )
10		evlf2.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
11		evlf2.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
12		evlf2.l	⊢ 𝐿 = ( ⟨ 𝐹 , 𝑋 ⟩ ( 2^nd ‘ 𝐸 ) ⟨ 𝐺 , 𝑌 ⟩ )
13	1 2 3 4 5 6 7	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 ) ) ) ) ⟩ )
14		ovex	⊢ ( 𝐶 Func 𝐷 ) ∈ V
15	4	fvexi	⊢ 𝐵 ∈ V
16	14 15	mpoex	⊢ ( 𝑓 ∈ ( 𝐶 Func 𝐷 ) , 𝑥 ∈ 𝐵 ↦ ( ( 1^st ‘ 𝑓 ) ‘ 𝑥 ) ) ∈ V
17	14 15	xpex	⊢ ( ( 𝐶 Func 𝐷 ) × 𝐵 ) ∈ V
18	17 17	mpoex	⊢ ( 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
19	16 18	op2ndd	⊢ ( 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 ) ) ) ) ⟩ → ( 2^nd ‘ 𝐸 ) = ( 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 ) ) ) ) )
20	13 19	syl	⊢ ( 𝜑 → ( 2^nd ‘ 𝐸 ) = ( 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 ) ) ) ) )
21		fvexd	⊢ ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) → ( 1^st ‘ 𝑥 ) ∈ V )
22		simprl	⊢ ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) → 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ )
23	22	fveq2d	⊢ ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) → ( 1^st ‘ 𝑥 ) = ( 1^st ‘ ⟨ 𝐹 , 𝑋 ⟩ ) )
24		op1stg	⊢ ( ( 𝐹 ∈ ( 𝐶 Func 𝐷 ) ∧ 𝑋 ∈ 𝐵 ) → ( 1^st ‘ ⟨ 𝐹 , 𝑋 ⟩ ) = 𝐹 )
25	8 10 24	syl2anc	⊢ ( 𝜑 → ( 1^st ‘ ⟨ 𝐹 , 𝑋 ⟩ ) = 𝐹 )
26	25	adantr	⊢ ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) → ( 1^st ‘ ⟨ 𝐹 , 𝑋 ⟩ ) = 𝐹 )
27	23 26	eqtrd	⊢ ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) → ( 1^st ‘ 𝑥 ) = 𝐹 )
28		fvexd	⊢ ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) → ( 1^st ‘ 𝑦 ) ∈ V )
29		simplrr	⊢ ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) → 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ )
30	29	fveq2d	⊢ ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) → ( 1^st ‘ 𝑦 ) = ( 1^st ‘ ⟨ 𝐺 , 𝑌 ⟩ ) )
31		op1stg	⊢ ( ( 𝐺 ∈ ( 𝐶 Func 𝐷 ) ∧ 𝑌 ∈ 𝐵 ) → ( 1^st ‘ ⟨ 𝐺 , 𝑌 ⟩ ) = 𝐺 )
32	9 11 31	syl2anc	⊢ ( 𝜑 → ( 1^st ‘ ⟨ 𝐺 , 𝑌 ⟩ ) = 𝐺 )
33	32	ad2antrr	⊢ ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) → ( 1^st ‘ ⟨ 𝐺 , 𝑌 ⟩ ) = 𝐺 )
34	30 33	eqtrd	⊢ ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) → ( 1^st ‘ 𝑦 ) = 𝐺 )
35		simplr	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 𝐺 ) → 𝑚 = 𝐹 )
36		simpr	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 𝐺 ) → 𝑛 = 𝐺 )
37	35 36	oveq12d	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 𝐺 ) → ( 𝑚 𝑁 𝑛 ) = ( 𝐹 𝑁 𝐺 ) )
38	22	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 𝐺 ) → 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ )
39	38	fveq2d	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 𝐺 ) → ( 2^nd ‘ 𝑥 ) = ( 2^nd ‘ ⟨ 𝐹 , 𝑋 ⟩ ) )
40		op2ndg	⊢ ( ( 𝐹 ∈ ( 𝐶 Func 𝐷 ) ∧ 𝑋 ∈ 𝐵 ) → ( 2^nd ‘ ⟨ 𝐹 , 𝑋 ⟩ ) = 𝑋 )
41	8 10 40	syl2anc	⊢ ( 𝜑 → ( 2^nd ‘ ⟨ 𝐹 , 𝑋 ⟩ ) = 𝑋 )
42	41	ad3antrrr	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 𝐺 ) → ( 2^nd ‘ ⟨ 𝐹 , 𝑋 ⟩ ) = 𝑋 )
43	39 42	eqtrd	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 𝐺 ) → ( 2^nd ‘ 𝑥 ) = 𝑋 )
44	29	adantr	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 𝐺 ) → 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ )
45	44	fveq2d	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 𝐺 ) → ( 2^nd ‘ 𝑦 ) = ( 2^nd ‘ ⟨ 𝐺 , 𝑌 ⟩ ) )
46		op2ndg	⊢ ( ( 𝐺 ∈ ( 𝐶 Func 𝐷 ) ∧ 𝑌 ∈ 𝐵 ) → ( 2^nd ‘ ⟨ 𝐺 , 𝑌 ⟩ ) = 𝑌 )
47	9 11 46	syl2anc	⊢ ( 𝜑 → ( 2^nd ‘ ⟨ 𝐺 , 𝑌 ⟩ ) = 𝑌 )
48	47	ad3antrrr	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 𝐺 ) → ( 2^nd ‘ ⟨ 𝐺 , 𝑌 ⟩ ) = 𝑌 )
49	45 48	eqtrd	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 𝐺 ) → ( 2^nd ‘ 𝑦 ) = 𝑌 )
50	43 49	oveq12d	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 𝐺 ) → ( ( 2^nd ‘ 𝑥 ) 𝐻 ( 2^nd ‘ 𝑦 ) ) = ( 𝑋 𝐻 𝑌 ) )
51	35	fveq2d	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 𝐺 ) → ( 1^st ‘ 𝑚 ) = ( 1^st ‘ 𝐹 ) )
52	51 43	fveq12d	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 𝐺 ) → ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) = ( ( 1^st ‘ 𝐹 ) ‘ 𝑋 ) )
53	51 49	fveq12d	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 𝐺 ) → ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) = ( ( 1^st ‘ 𝐹 ) ‘ 𝑌 ) )
54	52 53	opeq12d	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 𝐺 ) → ⟨ ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) , ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) ⟩ = ⟨ ( ( 1^st ‘ 𝐹 ) ‘ 𝑋 ) , ( ( 1^st ‘ 𝐹 ) ‘ 𝑌 ) ⟩ )
55	36	fveq2d	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 𝐺 ) → ( 1^st ‘ 𝑛 ) = ( 1^st ‘ 𝐺 ) )
56	55 49	fveq12d	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 𝐺 ) → ( ( 1^st ‘ 𝑛 ) ‘ ( 2^nd ‘ 𝑦 ) ) = ( ( 1^st ‘ 𝐺 ) ‘ 𝑌 ) )
57	54 56	oveq12d	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 𝐺 ) → ( ⟨ ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑥 ) ) , ( ( 1^st ‘ 𝑚 ) ‘ ( 2^nd ‘ 𝑦 ) ) ⟩ · ( ( 1^st ‘ 𝑛 ) ‘ ( 2^nd ‘ 𝑦 ) ) ) = ( ⟨ ( ( 1^st ‘ 𝐹 ) ‘ 𝑋 ) , ( ( 1^st ‘ 𝐹 ) ‘ 𝑌 ) ⟩ · ( ( 1^st ‘ 𝐺 ) ‘ 𝑌 ) ) )
58	49	fveq2d	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 𝐺 ) → ( 𝑎 ‘ ( 2^nd ‘ 𝑦 ) ) = ( 𝑎 ‘ 𝑌 ) )
59	35	fveq2d	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 𝐺 ) → ( 2^nd ‘ 𝑚 ) = ( 2^nd ‘ 𝐹 ) )
60	59 43 49	oveq123d	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 𝐺 ) → ( ( 2^nd ‘ 𝑥 ) ( 2^nd ‘ 𝑚 ) ( 2^nd ‘ 𝑦 ) ) = ( 𝑋 ( 2^nd ‘ 𝐹 ) 𝑌 ) )
61	60	fveq1d	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 𝐺 ) → ( ( ( 2^nd ‘ 𝑥 ) ( 2^nd ‘ 𝑚 ) ( 2^nd ‘ 𝑦 ) ) ‘ 𝑔 ) = ( ( 𝑋 ( 2^nd ‘ 𝐹 ) 𝑌 ) ‘ 𝑔 ) )
62	57 58 61	oveq123d	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 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 ) ) = ( ( 𝑎 ‘ 𝑌 ) ( ⟨ ( ( 1^st ‘ 𝐹 ) ‘ 𝑋 ) , ( ( 1^st ‘ 𝐹 ) ‘ 𝑌 ) ⟩ · ( ( 1^st ‘ 𝐺 ) ‘ 𝑌 ) ) ( ( 𝑋 ( 2^nd ‘ 𝐹 ) 𝑌 ) ‘ 𝑔 ) ) )
63	37 50 62	mpoeq123dv	⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) ∧ 𝑛 = 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 ) ) ) = ( 𝑎 ∈ ( 𝐹 𝑁 𝐺 ) , 𝑔 ∈ ( 𝑋 𝐻 𝑌 ) ↦ ( ( 𝑎 ‘ 𝑌 ) ( ⟨ ( ( 1^st ‘ 𝐹 ) ‘ 𝑋 ) , ( ( 1^st ‘ 𝐹 ) ‘ 𝑌 ) ⟩ · ( ( 1^st ‘ 𝐺 ) ‘ 𝑌 ) ) ( ( 𝑋 ( 2^nd ‘ 𝐹 ) 𝑌 ) ‘ 𝑔 ) ) ) )
64	28 34 63	csbied2	⊢ ( ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) ∧ 𝑚 = 𝐹 ) → ⦋ ( 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 ) ) ) = ( 𝑎 ∈ ( 𝐹 𝑁 𝐺 ) , 𝑔 ∈ ( 𝑋 𝐻 𝑌 ) ↦ ( ( 𝑎 ‘ 𝑌 ) ( ⟨ ( ( 1^st ‘ 𝐹 ) ‘ 𝑋 ) , ( ( 1^st ‘ 𝐹 ) ‘ 𝑌 ) ⟩ · ( ( 1^st ‘ 𝐺 ) ‘ 𝑌 ) ) ( ( 𝑋 ( 2^nd ‘ 𝐹 ) 𝑌 ) ‘ 𝑔 ) ) ) )
65	21 27 64	csbied2	⊢ ( ( 𝜑 ∧ ( 𝑥 = ⟨ 𝐹 , 𝑋 ⟩ ∧ 𝑦 = ⟨ 𝐺 , 𝑌 ⟩ ) ) → ⦋ ( 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 ) ) ) = ( 𝑎 ∈ ( 𝐹 𝑁 𝐺 ) , 𝑔 ∈ ( 𝑋 𝐻 𝑌 ) ↦ ( ( 𝑎 ‘ 𝑌 ) ( ⟨ ( ( 1^st ‘ 𝐹 ) ‘ 𝑋 ) , ( ( 1^st ‘ 𝐹 ) ‘ 𝑌 ) ⟩ · ( ( 1^st ‘ 𝐺 ) ‘ 𝑌 ) ) ( ( 𝑋 ( 2^nd ‘ 𝐹 ) 𝑌 ) ‘ 𝑔 ) ) ) )
66	8 10	opelxpd	⊢ ( 𝜑 → ⟨ 𝐹 , 𝑋 ⟩ ∈ ( ( 𝐶 Func 𝐷 ) × 𝐵 ) )
67	9 11	opelxpd	⊢ ( 𝜑 → ⟨ 𝐺 , 𝑌 ⟩ ∈ ( ( 𝐶 Func 𝐷 ) × 𝐵 ) )
68		ovex	⊢ ( 𝐹 𝑁 𝐺 ) ∈ V
69		ovex	⊢ ( 𝑋 𝐻 𝑌 ) ∈ V
70	68 69	mpoex	⊢ ( 𝑎 ∈ ( 𝐹 𝑁 𝐺 ) , 𝑔 ∈ ( 𝑋 𝐻 𝑌 ) ↦ ( ( 𝑎 ‘ 𝑌 ) ( ⟨ ( ( 1^st ‘ 𝐹 ) ‘ 𝑋 ) , ( ( 1^st ‘ 𝐹 ) ‘ 𝑌 ) ⟩ · ( ( 1^st ‘ 𝐺 ) ‘ 𝑌 ) ) ( ( 𝑋 ( 2^nd ‘ 𝐹 ) 𝑌 ) ‘ 𝑔 ) ) ) ∈ V
71	70	a1i	⊢ ( 𝜑 → ( 𝑎 ∈ ( 𝐹 𝑁 𝐺 ) , 𝑔 ∈ ( 𝑋 𝐻 𝑌 ) ↦ ( ( 𝑎 ‘ 𝑌 ) ( ⟨ ( ( 1^st ‘ 𝐹 ) ‘ 𝑋 ) , ( ( 1^st ‘ 𝐹 ) ‘ 𝑌 ) ⟩ · ( ( 1^st ‘ 𝐺 ) ‘ 𝑌 ) ) ( ( 𝑋 ( 2^nd ‘ 𝐹 ) 𝑌 ) ‘ 𝑔 ) ) ) ∈ V )
72	20 65 66 67 71	ovmpod	⊢ ( 𝜑 → ( ⟨ 𝐹 , 𝑋 ⟩ ( 2^nd ‘ 𝐸 ) ⟨ 𝐺 , 𝑌 ⟩ ) = ( 𝑎 ∈ ( 𝐹 𝑁 𝐺 ) , 𝑔 ∈ ( 𝑋 𝐻 𝑌 ) ↦ ( ( 𝑎 ‘ 𝑌 ) ( ⟨ ( ( 1^st ‘ 𝐹 ) ‘ 𝑋 ) , ( ( 1^st ‘ 𝐹 ) ‘ 𝑌 ) ⟩ · ( ( 1^st ‘ 𝐺 ) ‘ 𝑌 ) ) ( ( 𝑋 ( 2^nd ‘ 𝐹 ) 𝑌 ) ‘ 𝑔 ) ) ) )
73	12 72	eqtrid	⊢ ( 𝜑 → 𝐿 = ( 𝑎 ∈ ( 𝐹 𝑁 𝐺 ) , 𝑔 ∈ ( 𝑋 𝐻 𝑌 ) ↦ ( ( 𝑎 ‘ 𝑌 ) ( ⟨ ( ( 1^st ‘ 𝐹 ) ‘ 𝑋 ) , ( ( 1^st ‘ 𝐹 ) ‘ 𝑌 ) ⟩ · ( ( 1^st ‘ 𝐺 ) ‘ 𝑌 ) ) ( ( 𝑋 ( 2^nd ‘ 𝐹 ) 𝑌 ) ‘ 𝑔 ) ) ) )

Metamath Proof Explorer

Theorem evlf2

Proof