Theorem fucoisfunc2lem1 50693

Description: Lemma for fucoisfunc2lem2 50694. An equivalence that expands the transformation outputs of a composition-shaped hom-family into exact component functions, component typing, and naturality. The component values are arbitrary; no canonical value is imposed. (Contributed by Zhi Wang, 14-Sep-2026.)

Hypotheses

Ref	Expression
fucoisfunc.w	⊢ 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))
fucoisfunc2lem1.o	⊢ 𝑂 = ( ∘func ↾ 𝑊)
fucoisfunc2lem1.b	⊢ 𝐵 = (Base‘𝐶)
fucoisfunc2lem1.r	⊢ 𝑅 = (Hom ‘𝐶)
fucoisfunc2lem1.s	⊢ 𝑆 = (Hom ‘𝐸)
fucoisfunc2lem1.v	⊢ ∙ = (comp‘𝐸)

Assertion

Ref

Expression

fucoisfunc2lem1

⊢ ((𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝑃𝑛)𝑞) ∈ ((𝑂‘𝑚)(𝐶 Nat 𝐸)(𝑂‘𝑛)))) ↔ (𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))((𝑝(𝑚𝑃𝑛)𝑞) = (𝑥 ∈ 𝐵 ↦ ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥) ∈ (((1st ‘(𝑂‘𝑚))‘𝑥)𝑆((1st ‘(𝑂‘𝑛))‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝑅𝑦)(((𝑝(𝑚𝑃𝑛)𝑞)‘𝑦)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑚))‘𝑦)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑥(2nd ‘(𝑂‘𝑚))𝑦)‘ℎ)) = (((𝑥(2nd ‘(𝑂‘𝑛))𝑦)‘ℎ)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑛))‘𝑥)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥))))))

Distinct variable groups:   𝐶,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦   𝐷,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦   𝐸,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦   𝑊,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦   𝑂,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦   𝑃,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦   𝐵,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,ℎ   𝑅,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,ℎ   𝑆,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,ℎ   ∙ ,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,ℎ   𝐶,ℎ   𝐷,ℎ   ℎ,𝐸   ℎ,𝑊   ℎ,𝑂   𝑃,ℎ

Proof of Theorem fucoisfunc2lem1

Step	Hyp	Ref	Expression
1		eqid 2763	. . . . . . . 8 ⊢ (𝐶 Nat 𝐸) = (𝐶 Nat 𝐸)
2		fucoisfunc2lem1.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
3		fucoisfunc2lem1.r	. . . . . . . 8 ⊢ 𝑅 = (Hom ‘𝐶)
4		fucoisfunc2lem1.s	. . . . . . . 8 ⊢ 𝑆 = (Hom ‘𝐸)
5		fucoisfunc2lem1.v	. . . . . . . 8 ⊢ ∙ = (comp‘𝐸)
6		rescofuf 50413	. . . . . . . . . . 11 ⊢ ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))):((𝐷 Func 𝐸) × (𝐶 Func 𝐷))⟶(𝐶 Func 𝐸)
7		fucoisfunc2lem1.o	. . . . . . . . . . . . 13 ⊢ 𝑂 = ( ∘func ↾ 𝑊)
8		fucoisfunc.w	. . . . . . . . . . . . . 14 ⊢ 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))
9	8	reseq2i 5978	. . . . . . . . . . . . 13 ⊢ ( ∘func ↾ 𝑊) = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
10	7, 9	eqtri 2786	. . . . . . . . . . . 12 ⊢ 𝑂 = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
11	10, 8	feq12i 6724	. . . . . . . . . . 11 ⊢ (𝑂:𝑊⟶(𝐶 Func 𝐸) ↔ ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))):((𝐷 Func 𝐸) × (𝐶 Func 𝐷))⟶(𝐶 Func 𝐸))
12	6, 11	mpbir 234	. . . . . . . . . 10 ⊢ 𝑂:𝑊⟶(𝐶 Func 𝐸)
13	12	a1i 11	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑚 ∈ 𝑊) ∧ 𝑛 ∈ 𝑊) → 𝑂:𝑊⟶(𝐶 Func 𝐸))
14		simplr 781	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑚 ∈ 𝑊) ∧ 𝑛 ∈ 𝑊) → 𝑚 ∈ 𝑊)
15	13, 14	ffvelcdmd 7109	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑚 ∈ 𝑊) ∧ 𝑛 ∈ 𝑊) → (𝑂‘𝑚) ∈ (𝐶 Func 𝐸))
16		simpr 490	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑚 ∈ 𝑊) ∧ 𝑛 ∈ 𝑊) → 𝑛 ∈ 𝑊)
17	13, 16	ffvelcdmd 7109	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑚 ∈ 𝑊) ∧ 𝑛 ∈ 𝑊) → (𝑂‘𝑛) ∈ (𝐶 Func 𝐸))
18	1, 2, 3, 4, 5, 15, 17	isnat3 50547	. . . . . . 7 ⊢ (((⊤ ∧ 𝑚 ∈ 𝑊) ∧ 𝑛 ∈ 𝑊) → ((𝑝(𝑚𝑃𝑛)𝑞) ∈ ((𝑂‘𝑚)(𝐶 Nat 𝐸)(𝑂‘𝑛)) ↔ ((𝑝(𝑚𝑃𝑛)𝑞) = (𝑥 ∈ 𝐵 ↦ ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥) ∈ (((1st ‘(𝑂‘𝑚))‘𝑥)𝑆((1st ‘(𝑂‘𝑛))‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝑅𝑦)(((𝑝(𝑚𝑃𝑛)𝑞)‘𝑦)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑚))‘𝑦)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑥(2nd ‘(𝑂‘𝑚))𝑦)‘ℎ)) = (((𝑥(2nd ‘(𝑂‘𝑛))𝑦)‘ℎ)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑛))‘𝑥)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥)))))
19	18	2ralbidv 3230	. . . . . 6 ⊢ (((⊤ ∧ 𝑚 ∈ 𝑊) ∧ 𝑛 ∈ 𝑊) → (∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝑃𝑛)𝑞) ∈ ((𝑂‘𝑚)(𝐶 Nat 𝐸)(𝑂‘𝑛)) ↔ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))((𝑝(𝑚𝑃𝑛)𝑞) = (𝑥 ∈ 𝐵 ↦ ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥) ∈ (((1st ‘(𝑂‘𝑚))‘𝑥)𝑆((1st ‘(𝑂‘𝑛))‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝑅𝑦)(((𝑝(𝑚𝑃𝑛)𝑞)‘𝑦)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑚))‘𝑦)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑥(2nd ‘(𝑂‘𝑚))𝑦)‘ℎ)) = (((𝑥(2nd ‘(𝑂‘𝑛))𝑦)‘ℎ)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑛))‘𝑥)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥)))))
20	19	anbi2d 642	. . . . 5 ⊢ (((⊤ ∧ 𝑚 ∈ 𝑊) ∧ 𝑛 ∈ 𝑊) → (((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝑃𝑛)𝑞) ∈ ((𝑂‘𝑚)(𝐶 Nat 𝐸)(𝑂‘𝑛))) ↔ ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))((𝑝(𝑚𝑃𝑛)𝑞) = (𝑥 ∈ 𝐵 ↦ ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥) ∈ (((1st ‘(𝑂‘𝑚))‘𝑥)𝑆((1st ‘(𝑂‘𝑛))‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝑅𝑦)(((𝑝(𝑚𝑃𝑛)𝑞)‘𝑦)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑚))‘𝑦)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑥(2nd ‘(𝑂‘𝑚))𝑦)‘ℎ)) = (((𝑥(2nd ‘(𝑂‘𝑛))𝑦)‘ℎ)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑛))‘𝑥)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥))))))
21	20	ralbidva 3187	. . . 4 ⊢ ((⊤ ∧ 𝑚 ∈ 𝑊) → (∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝑃𝑛)𝑞) ∈ ((𝑂‘𝑚)(𝐶 Nat 𝐸)(𝑂‘𝑛))) ↔ ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))((𝑝(𝑚𝑃𝑛)𝑞) = (𝑥 ∈ 𝐵 ↦ ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥) ∈ (((1st ‘(𝑂‘𝑚))‘𝑥)𝑆((1st ‘(𝑂‘𝑛))‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝑅𝑦)(((𝑝(𝑚𝑃𝑛)𝑞)‘𝑦)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑚))‘𝑦)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑥(2nd ‘(𝑂‘𝑚))𝑦)‘ℎ)) = (((𝑥(2nd ‘(𝑂‘𝑛))𝑦)‘ℎ)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑛))‘𝑥)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥))))))
22	21	ralbidva 3187	. . 3 ⊢ (⊤ → (∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝑃𝑛)𝑞) ∈ ((𝑂‘𝑚)(𝐶 Nat 𝐸)(𝑂‘𝑛))) ↔ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))((𝑝(𝑚𝑃𝑛)𝑞) = (𝑥 ∈ 𝐵 ↦ ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥) ∈ (((1st ‘(𝑂‘𝑚))‘𝑥)𝑆((1st ‘(𝑂‘𝑛))‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝑅𝑦)(((𝑝(𝑚𝑃𝑛)𝑞)‘𝑦)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑚))‘𝑦)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑥(2nd ‘(𝑂‘𝑚))𝑦)‘ℎ)) = (((𝑥(2nd ‘(𝑂‘𝑛))𝑦)‘ℎ)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑛))‘𝑥)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥))))))
23	22	anbi2d 642	. 2 ⊢ (⊤ → ((𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝑃𝑛)𝑞) ∈ ((𝑂‘𝑚)(𝐶 Nat 𝐸)(𝑂‘𝑛)))) ↔ (𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))((𝑝(𝑚𝑃𝑛)𝑞) = (𝑥 ∈ 𝐵 ↦ ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥) ∈ (((1st ‘(𝑂‘𝑚))‘𝑥)𝑆((1st ‘(𝑂‘𝑛))‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝑅𝑦)(((𝑝(𝑚𝑃𝑛)𝑞)‘𝑦)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑚))‘𝑦)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑥(2nd ‘(𝑂‘𝑚))𝑦)‘ℎ)) = (((𝑥(2nd ‘(𝑂‘𝑛))𝑦)‘ℎ)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑛))‘𝑥)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥)))))))
24	23	mptru 1577	1 ⊢ ((𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝑃𝑛)𝑞) ∈ ((𝑂‘𝑚)(𝐶 Nat 𝐸)(𝑂‘𝑛)))) ↔ (𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))((𝑝(𝑚𝑃𝑛)𝑞) = (𝑥 ∈ 𝐵 ↦ ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥) ∈ (((1st ‘(𝑂‘𝑚))‘𝑥)𝑆((1st ‘(𝑂‘𝑛))‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝑅𝑦)(((𝑝(𝑚𝑃𝑛)𝑞)‘𝑦)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑚))‘𝑦)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑥(2nd ‘(𝑂‘𝑚))𝑦)‘ℎ)) = (((𝑥(2nd ‘(𝑂‘𝑛))𝑦)‘ℎ)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑛))‘𝑥)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥))))))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ⊤wtru 1571   ∈ wcel 2147  ∀wral 3080  ⟨cop 4838   ↦ cmpt 5189   × cxp 5653   ↾ cres 5657  ⟶wf 6554  ‘cfv 6558  (class class class)co 7443   ∈ cmpo 7445  1^st c1st 8023  2^nd c2nd 8024  Basecbs 17416  Hom chom 17468  compcco 17469   Func cfunc 18058   ∘_func ccofu 18060   Nat cnat 18148

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-map 8877  df-ixp 8954  df-cat 17871  df-cid 17872  df-func 18062  df-cofu 18064  df-nat 18150

This theorem is used by:  fucoisfunc2lem2  50694