fucof21 - Mathbox for Zhi Wang

	Mathbox for Zhi Wang		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fucof21		Structured version Visualization version GIF version

Theorem fucof21 50399

Description: The morphism part of the functor composition bifunctor maps a hom-set of the product category into a set of natural transformations. (Contributed by Zhi Wang, 30-Sep-2025.)

**Hypotheses**
Ref	Expression
fucof21.o	⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
fucof21.t	⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
fucof21.j	⊢ 𝐽 = (Hom ‘𝑇)
fucof21.w	⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
fucof21.u	⊢ (𝜑 → 𝑈 ∈ 𝑊)
fucof21.v	⊢ (𝜑 → 𝑉 ∈ 𝑊)

**Assertion**
Ref	Expression
fucof21	⊢ (𝜑 → (𝑈𝑃𝑉):(𝑈𝐽𝑉)⟶((𝑂‘𝑈)(𝐶 Nat 𝐸)(𝑂‘𝑉)))

Proof of Theorem fucof21 Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucof21.o	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
2		fucof21.w	. . . . 5 ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
3		fucof21.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑊)
4		relfunc 18017	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
5		relfunc 18017	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
6	2, 3, 4, 5	fuco2eld3 50367	. . . 4 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑈))(𝐷 Func 𝐸)(2^nd ‘(1^st ‘𝑈)) ∧ (1^st ‘(2^nd ‘𝑈))(𝐶 Func 𝐷)(2^nd ‘(2^nd ‘𝑈))))
7	6	simprd 501	. . 3 ⊢ (𝜑 → (1^st ‘(2^nd ‘𝑈))(𝐶 Func 𝐷)(2^nd ‘(2^nd ‘𝑈)))
8	6	simpld 500	. . 3 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑈))(𝐷 Func 𝐸)(2^nd ‘(1^st ‘𝑈)))
9	2, 3, 4, 5	fuco2eld2 50366	. . 3 ⊢ (𝜑 → 𝑈 = ⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩)
10		fucof21.v	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
11	2, 10, 4, 5	fuco2eld3 50367	. . . 4 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑉))(𝐷 Func 𝐸)(2^nd ‘(1^st ‘𝑉)) ∧ (1^st ‘(2^nd ‘𝑉))(𝐶 Func 𝐷)(2^nd ‘(2^nd ‘𝑉))))
12	11	simprd 501	. . 3 ⊢ (𝜑 → (1^st ‘(2^nd ‘𝑉))(𝐶 Func 𝐷)(2^nd ‘(2^nd ‘𝑉)))
13	11	simpld 500	. . 3 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑉))(𝐷 Func 𝐸)(2^nd ‘(1^st ‘𝑉)))
14	2, 10, 4, 5	fuco2eld2 50366	. . 3 ⊢ (𝜑 → 𝑉 = ⟨⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩, ⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩⟩)
15	1, 7, 8, 9, 12, 13, 14	fuco21 50388	. 2 ⊢ (𝜑 → (𝑈𝑃𝑉) = (𝑏 ∈ (⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩), 𝑎 ∈ (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘((1^st ‘(2^nd ‘𝑉))‘𝑥))(⟨((1^st ‘(1^st ‘𝑈))‘((1^st ‘(2^nd ‘𝑈))‘𝑥)), ((1^st ‘(1^st ‘𝑈))‘((1^st ‘(2^nd ‘𝑉))‘𝑥))⟩(comp‘𝐸)((1^st ‘(1^st ‘𝑉))‘((1^st ‘(2^nd ‘𝑉))‘𝑥)))((((1^st ‘(2^nd ‘𝑈))‘𝑥)(2^nd ‘(1^st ‘𝑈))((1^st ‘(2^nd ‘𝑉))‘𝑥))‘(𝑎‘𝑥))))))
16	1	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑏 ∈ (⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩) ∧ 𝑎 ∈ (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩))) → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
17	9	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑏 ∈ (⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩) ∧ 𝑎 ∈ (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩))) → 𝑈 = ⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩)
18	14	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑏 ∈ (⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩) ∧ 𝑎 ∈ (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩))) → 𝑉 = ⟨⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩, ⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩⟩)
19		simprr 785	. . . 4 ⊢ ((𝜑 ∧ (𝑏 ∈ (⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩) ∧ 𝑎 ∈ (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩))) → 𝑎 ∈ (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩))
20		simprl 783	. . . 4 ⊢ ((𝜑 ∧ (𝑏 ∈ (⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩) ∧ 𝑎 ∈ (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩))) → 𝑏 ∈ (⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩))
21	16, 17, 18, 19, 20	fuco22 50391	. . 3 ⊢ ((𝜑 ∧ (𝑏 ∈ (⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩) ∧ 𝑎 ∈ (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩))) → (𝑏(𝑈𝑃𝑉)𝑎) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘((1^st ‘(2^nd ‘𝑉))‘𝑥))(⟨((1^st ‘(1^st ‘𝑈))‘((1^st ‘(2^nd ‘𝑈))‘𝑥)), ((1^st ‘(1^st ‘𝑈))‘((1^st ‘(2^nd ‘𝑉))‘𝑥))⟩(comp‘𝐸)((1^st ‘(1^st ‘𝑉))‘((1^st ‘(2^nd ‘𝑉))‘𝑥)))((((1^st ‘(2^nd ‘𝑈))‘𝑥)(2^nd ‘(1^st ‘𝑈))((1^st ‘(2^nd ‘𝑉))‘𝑥))‘(𝑎‘𝑥)))))
22	16, 19, 20, 17, 18	fuco22nat 50398	. . 3 ⊢ ((𝜑 ∧ (𝑏 ∈ (⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩) ∧ 𝑎 ∈ (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩))) → (𝑏(𝑈𝑃𝑉)𝑎) ∈ ((𝑂‘𝑈)(𝐶 Nat 𝐸)(𝑂‘𝑉)))
23	21, 22	eqeltrrd 2862	. 2 ⊢ ((𝜑 ∧ (𝑏 ∈ (⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩) ∧ 𝑎 ∈ (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩))) → (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘((1^st ‘(2^nd ‘𝑉))‘𝑥))(⟨((1^st ‘(1^st ‘𝑈))‘((1^st ‘(2^nd ‘𝑈))‘𝑥)), ((1^st ‘(1^st ‘𝑈))‘((1^st ‘(2^nd ‘𝑉))‘𝑥))⟩(comp‘𝐸)((1^st ‘(1^st ‘𝑉))‘((1^st ‘(2^nd ‘𝑉))‘𝑥)))((((1^st ‘(2^nd ‘𝑈))‘𝑥)(2^nd ‘(1^st ‘𝑈))((1^st ‘(2^nd ‘𝑉))‘𝑥))‘(𝑎‘𝑥)))) ∈ ((𝑂‘𝑈)(𝐶 Nat 𝐸)(𝑂‘𝑉)))
24		fucof21.t	. . . 4 ⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
25	24	xpcfucbas 50304	. . . 4 ⊢ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = (Base‘𝑇)
26		fucof21.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝑇)
27	3, 2	eleqtrd 2863	. . . 4 ⊢ (𝜑 → 𝑈 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
28	10, 2	eleqtrd 2863	. . . 4 ⊢ (𝜑 → 𝑉 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
29	24, 25, 26, 27, 28	xpcfuchom 50306	. . 3 ⊢ (𝜑 → (𝑈𝐽𝑉) = (((1^st ‘𝑈)(𝐷 Nat 𝐸)(1^st ‘𝑉)) × ((2^nd ‘𝑈)(𝐶 Nat 𝐷)(2^nd ‘𝑉))))
30	9	fveq2d 6881	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝑈) = (1^st ‘⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩))
31		opex 5432	. . . . . . 7 ⊢ ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩ ∈ V
32		opex 5432	. . . . . . 7 ⊢ ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩ ∈ V
33	31, 32	op1st 7998	. . . . . 6 ⊢ (1^st ‘⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩) = ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩
34	30, 33	eqtrdi 2812	. . . . 5 ⊢ (𝜑 → (1^st ‘𝑈) = ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩)
35	14	fveq2d 6881	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝑉) = (1^st ‘⟨⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩, ⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩⟩))
36		opex 5432	. . . . . . 7 ⊢ ⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩ ∈ V
37		opex 5432	. . . . . . 7 ⊢ ⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩ ∈ V
38	36, 37	op1st 7998	. . . . . 6 ⊢ (1^st ‘⟨⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩, ⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩⟩) = ⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩
39	35, 38	eqtrdi 2812	. . . . 5 ⊢ (𝜑 → (1^st ‘𝑉) = ⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩)
40	34, 39	oveq12d 7430	. . . 4 ⊢ (𝜑 → ((1^st ‘𝑈)(𝐷 Nat 𝐸)(1^st ‘𝑉)) = (⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩))
41	9	fveq2d 6881	. . . . . 6 ⊢ (𝜑 → (2^nd ‘𝑈) = (2^nd ‘⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩))
42	31, 32	op2nd 7999	. . . . . 6 ⊢ (2^nd ‘⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩) = ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩
43	41, 42	eqtrdi 2812	. . . . 5 ⊢ (𝜑 → (2^nd ‘𝑈) = ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩)
44	14	fveq2d 6881	. . . . . 6 ⊢ (𝜑 → (2^nd ‘𝑉) = (2^nd ‘⟨⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩, ⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩⟩))
45	36, 37	op2nd 7999	. . . . . 6 ⊢ (2^nd ‘⟨⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩, ⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩⟩) = ⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩
46	44, 45	eqtrdi 2812	. . . . 5 ⊢ (𝜑 → (2^nd ‘𝑉) = ⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩)
47	43, 46	oveq12d 7430	. . . 4 ⊢ (𝜑 → ((2^nd ‘𝑈)(𝐶 Nat 𝐷)(2^nd ‘𝑉)) = (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩))
48	40, 47	xpeq12d 5682	. . 3 ⊢ (𝜑 → (((1^st ‘𝑈)(𝐷 Nat 𝐸)(1^st ‘𝑉)) × ((2^nd ‘𝑈)(𝐶 Nat 𝐷)(2^nd ‘𝑉))) = ((⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩) × (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩)))
49	29, 48	eqtrd 2796	. 2 ⊢ (𝜑 → (𝑈𝐽𝑉) = ((⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩) × (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩)))
50	15, 23, 49	fmpodg 8072	1 ⊢ (𝜑 → (𝑈𝑃𝑉):(𝑈𝐽𝑉)⟶((𝑂‘𝑈)(𝐶 Nat 𝐸)(𝑂‘𝑉)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ⟨cop 4590 class class class wbr 5103 ↦ cmpt 5186 × cxp 5649 ⟶wf 6527 ‘cfv 6531 (class class class)co 7412 1^st c1st 7988 2^nd c2nd 7989 Basecbs 17367 Hom chom 17419 compcco 17420 Func cfunc 18009 Nat cnat 18099 FuncCat cfuc 18100 ×_c cxpc 18322 ∘_F cfuco 50368

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-er 8701 df-map 8833 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-z 12675 df-dec 12796 df-uz 12947 df-fz 13621 df-struct 17305 df-slot 17340 df-ndx 17352 df-base 17368 df-hom 17432 df-cco 17433 df-cat 17822 df-cid 17823 df-func 18013 df-cofu 18015 df-nat 18101 df-fuc 18102 df-xpc 18326 df-fuco 50369

This theorem is used by: fucofunc 50411

W3C validator