evlfval - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > evlfval		Structured version Visualization version GIF version

Theorem evlfval 18298

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

**Hypotheses**
Ref	Expression
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**
Ref	Expression
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 ∈ (𝑚𝑁𝑛), 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))))⟩)

Distinct variable groups: 𝑓,𝑎,𝑔,𝑚,𝑛,𝑥,𝑦,𝐶 𝐷,𝑎,𝑓,𝑔,𝑚,𝑛,𝑥,𝑦 𝑔,𝐻,𝑚,𝑛,𝑥,𝑦 𝑁,𝑎,𝑔,𝑚,𝑛,𝑥,𝑦 𝜑,𝑎,𝑓,𝑔,𝑚,𝑛,𝑥,𝑦 · ,𝑎,𝑔,𝑚,𝑛,𝑥,𝑦 𝑥,𝐵,𝑦 Allowed substitution hints: 𝐵(𝑓, 𝑔, 𝑚, 𝑛, 𝑎) · (𝑓) 𝐸(𝑥, 𝑦, 𝑓, 𝑔, 𝑚, 𝑛, 𝑎) 𝐻(𝑓, 𝑎) 𝑁(𝑓)

Proof of Theorem evlfval Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evlfval.e	. 2 ⊢ 𝐸 = (𝐶 eval_F 𝐷)
2		df-evlf 18294	. . . 4 ⊢ 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))))⟩)
3	2	a1i 11	. . 3 ⊢ (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))))⟩))
4		simprl 783	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → 𝑐 = 𝐶)
5		simprr 785	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → 𝑑 = 𝐷)
6	4, 5	oveq12d 7441	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑐 Func 𝑑) = (𝐶 Func 𝐷))
7	4	fveq2d 6892	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (Base‘𝑐) = (Base‘𝐶))
8		evlfval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
9	7, 8	eqtr4di 2819	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (Base‘𝑐) = 𝐵)
10		eqidd 2767	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → ((1^st ‘𝑓)‘𝑥) = ((1^st ‘𝑓)‘𝑥))
11	6, 9, 10	mpoeq123dv 7498	. . . 4 ⊢ ((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)))
12	6, 9	xpeq12d 5697	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → ((𝑐 Func 𝑑) × (Base‘𝑐)) = ((𝐶 Func 𝐷) × 𝐵))
13	4, 5	oveq12d 7441	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑐 Nat 𝑑) = (𝐶 Nat 𝐷))
14		evlfval.n	. . . . . . . . . 10 ⊢ 𝑁 = (𝐶 Nat 𝐷)
15	13, 14	eqtr4di 2819	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑐 Nat 𝑑) = 𝑁)
16	15	oveqd 7440	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑚(𝑐 Nat 𝑑)𝑛) = (𝑚𝑁𝑛))
17	4	fveq2d 6892	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (Hom ‘𝑐) = (Hom ‘𝐶))
18		evlfval.h	. . . . . . . . . 10 ⊢ 𝐻 = (Hom ‘𝐶)
19	17, 18	eqtr4di 2819	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (Hom ‘𝑐) = 𝐻)
20	19	oveqd 7440	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → ((2^nd ‘𝑥)(Hom ‘𝑐)(2^nd ‘𝑦)) = ((2^nd ‘𝑥)𝐻(2^nd ‘𝑦)))
21	5	fveq2d 6892	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (comp‘𝑑) = (comp‘𝐷))
22		evlfval.o	. . . . . . . . . . 11 ⊢ · = (comp‘𝐷)
23	21, 22	eqtr4di 2819	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (comp‘𝑑) = · )
24	23	oveqd 7440	. . . . . . . . 9 ⊢ ((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))))
25	24	oveqd 7440	. . . . . . . 8 ⊢ ((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)))
26	16, 20, 25	mpoeq123dv 7498	. . . . . . 7 ⊢ ((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 ∈ (𝑚𝑁𝑛), 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))))
27	26	csbeq2dv 3863	. . . . . 6 ⊢ ((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 ∈ (𝑚𝑁𝑛), 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))))
28	27	csbeq2dv 3863	. . . . 5 ⊢ ((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 ∈ (𝑚𝑁𝑛), 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	12, 12, 28	mpoeq123dv 7498	. . . 4 ⊢ ((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 ∈ (𝑚𝑁𝑛), 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	11, 29	opeq12d 4851	. . 3 ⊢ ((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 ∈ (𝑚𝑁𝑛), 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		evlfval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
32		evlfval.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
33		opex 5450	. . . 4 ⊢ ⟨(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 ∈ (𝑚𝑁𝑛), 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 11	. . 3 ⊢ (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 ∈ (𝑚𝑁𝑛), 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	3, 30, 31, 32, 34	ovmpod 7575	. 2 ⊢ (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 ∈ (𝑚𝑁𝑛), 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 2813	1 ⊢ (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 ∈ (𝑚𝑁𝑛), 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))))⟩)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 Vcvv 3458 ⦋csb 3856 ⟨cop 4600 × cxp 5664 ‘cfv 6543 (class class class)co 7423 ∈ cmpo 7425 1^st c1st 7993 2^nd c2nd 7994 Basecbs 17294 Hom chom 17346 compcco 17347 Catccat 17745 Func cfunc 17936 Nat cnat 18026 eval_F cevlf 18290

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-pr 5409

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ral 3083 df-rex 3093 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4493 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-br 5115 df-opab 5179 df-id 5561 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-iota 6499 df-fun 6545 df-fv 6551 df-ov 7426 df-oprab 7427 df-mpo 7428 df-evlf 18294

This theorem is used by: evlf2 18299 evlf1 18301 evlfcl 18303

W3C validator