Theorem fucoisfunc2lem1 50579

Description: Lemma for fucoisfunc2lem2 50580. 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 2760	. . . . . . . 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 50299	. . . . . . . . . . 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 5966	. . . . . . . . . . . . 13 ⊢ ( ∘func ↾ 𝑊) = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
10	7, 9	eqtri 2783	. . . . . . . . . . . 12 ⊢ 𝑂 = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
11	10, 8	feq12i 6702	. . . . . . . . . . 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 7085	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑚 ∈ 𝑊) ∧ 𝑛 ∈ 𝑊) → (𝑂‘𝑚) ∈ (𝐶 Func 𝐸))
16		simpr 490	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑚 ∈ 𝑊) ∧ 𝑛 ∈ 𝑊) → 𝑛 ∈ 𝑊)
17	13, 16	ffvelcdmd 7085	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑚 ∈ 𝑊) ∧ 𝑛 ∈ 𝑊) → (𝑂‘𝑛) ∈ (𝐶 Func 𝐸))
18	1, 2, 3, 4, 5, 15, 17	isnat3 50433	. . . . . . 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 3226	. . . . . 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 3183	. . . 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 3183	. . 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 1576	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 1569  ⊤wtru 1570   ∈ wcel 2144  ∀wral 3076  ⟨cop 4589   ↦ cmpt 5185   × cxp 5648   ↾ cres 5652  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000  Basecbs 17389  Hom chom 17441  compcco 17442   Func cfunc 18031   ∘_func ccofu 18033   Nat cnat 18121

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-map 8851  df-ixp 8928  df-cat 17844  df-cid 17845  df-func 18035  df-cofu 18037  df-nat 18123

This theorem is used by:  fucoisfunc2lem2  50580