df-xpc - Metamath Proof Explorer

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

Definition df-xpc 18218

Description: Define the binary product of categories, which has objects for each pair of objects of the factors, and morphisms for each pair of morphisms of the factors. Composition is componentwise. (Contributed by Mario Carneiro, 10-Jan-2017.)

**Assertion**
Ref	Expression
df-xpc	⊢ ×_c = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩})

Distinct variable group: 𝑓,𝑏,𝑔,ℎ,𝑟,𝑠,𝑢,𝑣,𝑥,𝑦

**Detailed syntax breakdown of Definition df-xpc**
Step	Hyp	Ref	Expression
1		cxpc 18214	. 2 class ×_c
2		vr	. . 3 setvar 𝑟
3		vs	. . 3 setvar 𝑠
4		cvv 3457	. . 3 class V
5		vb	. . . 4 setvar 𝑏
6	2	cv 1562	. . . . . 6 class 𝑟
7		cbs 17259	. . . . . 6 class Base
8	6, 7	cfv 6525	. . . . 5 class (Base‘𝑟)
9	3	cv 1562	. . . . . 6 class 𝑠
10	9, 7	cfv 6525	. . . . 5 class (Base‘𝑠)
11	8, 10	cxp 5650	. . . 4 class ((Base‘𝑟) × (Base‘𝑠))
12		vh	. . . . 5 setvar ℎ
13		vu	. . . . . 6 setvar 𝑢
14		vv	. . . . . 6 setvar 𝑣
15	5	cv 1562	. . . . . 6 class 𝑏
16	13	cv 1562	. . . . . . . . 9 class 𝑢
17		c1st 7972	. . . . . . . . 9 class 1^st
18	16, 17	cfv 6525	. . . . . . . 8 class (1^st ‘𝑢)
19	14	cv 1562	. . . . . . . . 9 class 𝑣
20	19, 17	cfv 6525	. . . . . . . 8 class (1^st ‘𝑣)
21		chom 17311	. . . . . . . . 9 class Hom
22	6, 21	cfv 6525	. . . . . . . 8 class (Hom ‘𝑟)
23	18, 20, 22	co 7400	. . . . . . 7 class ((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣))
24		c2nd 7973	. . . . . . . . 9 class 2^nd
25	16, 24	cfv 6525	. . . . . . . 8 class (2^nd ‘𝑢)
26	19, 24	cfv 6525	. . . . . . . 8 class (2^nd ‘𝑣)
27	9, 21	cfv 6525	. . . . . . . 8 class (Hom ‘𝑠)
28	25, 26, 27	co 7400	. . . . . . 7 class ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣))
29	23, 28	cxp 5650	. . . . . 6 class (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))
30	13, 14, 15, 15, 29	cmpo 7402	. . . . 5 class (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣))))
31		cnx 17243	. . . . . . . 8 class ndx
32	31, 7	cfv 6525	. . . . . . 7 class (Base‘ndx)
33	32, 15	cop 4591	. . . . . 6 class ⟨(Base‘ndx), 𝑏⟩
34	31, 21	cfv 6525	. . . . . . 7 class (Hom ‘ndx)
35	12	cv 1562	. . . . . . 7 class ℎ
36	34, 35	cop 4591	. . . . . 6 class ⟨(Hom ‘ndx), ℎ⟩
37		cco 17312	. . . . . . . 8 class comp
38	31, 37	cfv 6525	. . . . . . 7 class (comp‘ndx)
39		vx	. . . . . . . 8 setvar 𝑥
40		vy	. . . . . . . 8 setvar 𝑦
41	15, 15	cxp 5650	. . . . . . . 8 class (𝑏 × 𝑏)
42		vg	. . . . . . . . 9 setvar 𝑔
43		vf	. . . . . . . . 9 setvar 𝑓
44	39	cv 1562	. . . . . . . . . . 11 class 𝑥
45	44, 24	cfv 6525	. . . . . . . . . 10 class (2^nd ‘𝑥)
46	40	cv 1562	. . . . . . . . . 10 class 𝑦
47	45, 46, 35	co 7400	. . . . . . . . 9 class ((2^nd ‘𝑥)ℎ𝑦)
48	44, 35	cfv 6525	. . . . . . . . 9 class (ℎ‘𝑥)
49	42	cv 1562	. . . . . . . . . . . 12 class 𝑔
50	49, 17	cfv 6525	. . . . . . . . . . 11 class (1^st ‘𝑔)
51	43	cv 1562	. . . . . . . . . . . 12 class 𝑓
52	51, 17	cfv 6525	. . . . . . . . . . 11 class (1^st ‘𝑓)
53	44, 17	cfv 6525	. . . . . . . . . . . . . 14 class (1^st ‘𝑥)
54	53, 17	cfv 6525	. . . . . . . . . . . . 13 class (1^st ‘(1^st ‘𝑥))
55	45, 17	cfv 6525	. . . . . . . . . . . . 13 class (1^st ‘(2^nd ‘𝑥))
56	54, 55	cop 4591	. . . . . . . . . . . 12 class ⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩
57	46, 17	cfv 6525	. . . . . . . . . . . 12 class (1^st ‘𝑦)
58	6, 37	cfv 6525	. . . . . . . . . . . 12 class (comp‘𝑟)
59	56, 57, 58	co 7400	. . . . . . . . . . 11 class (⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))
60	50, 52, 59	co 7400	. . . . . . . . . 10 class ((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓))
61	49, 24	cfv 6525	. . . . . . . . . . 11 class (2^nd ‘𝑔)
62	51, 24	cfv 6525	. . . . . . . . . . 11 class (2^nd ‘𝑓)
63	53, 24	cfv 6525	. . . . . . . . . . . . 13 class (2^nd ‘(1^st ‘𝑥))
64	45, 24	cfv 6525	. . . . . . . . . . . . 13 class (2^nd ‘(2^nd ‘𝑥))
65	63, 64	cop 4591	. . . . . . . . . . . 12 class ⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩
66	46, 24	cfv 6525	. . . . . . . . . . . 12 class (2^nd ‘𝑦)
67	9, 37	cfv 6525	. . . . . . . . . . . 12 class (comp‘𝑠)
68	65, 66, 67	co 7400	. . . . . . . . . . 11 class (⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))
69	61, 62, 68	co 7400	. . . . . . . . . 10 class ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))
70	60, 69	cop 4591	. . . . . . . . 9 class ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩
71	42, 43, 47, 48, 70	cmpo 7402	. . . . . . . 8 class (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩)
72	39, 40, 41, 15, 71	cmpo 7402	. . . . . . 7 class (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))
73	38, 72	cop 4591	. . . . . 6 class ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩
74	33, 36, 73	ctp 4589	. . . . 5 class {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩}
75	12, 30, 74	csb 3855	. . . 4 class ⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩}
76	5, 11, 75	csb 3855	. . 3 class ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩}
77	2, 3, 4, 4, 76	cmpo 7402	. 2 class (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩})
78	1, 77	wceq 1563	1 wff ×_c = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩})

Colors of variables: wff setvar class

This definition is referenced by: fnxpc 18222 xpcval 18223 reldmxpcALT 49876

W3C validator