Definition df-psr

Description: Define the algebra of power series over the index set i and with coefficients from the ring r . (Contributed by Mario Carneiro, 21-Mar-2015)

	Ref	Expression
Assertion	df-psr	⊢ mPwSer = ( 𝑖 ∈ V , 𝑟 ∈ V ↦ ⦋ { ℎ ∈ ( ℕ0 ↑m 𝑖 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } / 𝑑 ⦌ ⦋ ( ( Base ‘ 𝑟 ) ↑m 𝑑 ) / 𝑏 ⦌ ( { ⟨ ( Base ‘ ndx ) , 𝑏 ⟩ , ⟨ ( +g ‘ ndx ) , ( ∘f ( +g ‘ 𝑟 ) ↾ ( 𝑏 × 𝑏 ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑓 ∈ 𝑏 , 𝑔 ∈ 𝑏 ↦ ( 𝑘 ∈ 𝑑 ↦ ( 𝑟 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝑑 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑟 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑟 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑟 ) , 𝑓 ∈ 𝑏 ↦ ( ( 𝑑 × { 𝑥 } ) ∘f ( .r ‘ 𝑟 ) 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝑑 × { ( TopOpen ‘ 𝑟 ) } ) ) ⟩ } ) )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cmps	⊢ mPwSer
1		vi	⊢ 𝑖
2		cvv	⊢ V
3		vr	⊢ 𝑟
4		vh	⊢ ℎ
5		cn0	⊢ ℕ0
6		cmap	⊢ ↑m
7	1	cv	⊢ 𝑖
8	5 7 6	co	⊢ ( ℕ0 ↑m 𝑖 )
9	4	cv	⊢ ℎ
10	9	ccnv	⊢ ◡ ℎ
11		cn	⊢ ℕ
12	10 11	cima	⊢ ( ◡ ℎ “ ℕ )
13		cfn	⊢ Fin
14	12 13	wcel	⊢ ( ◡ ℎ “ ℕ ) ∈ Fin
15	14 4 8	crab	⊢ { ℎ ∈ ( ℕ0 ↑m 𝑖 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
16		vd	⊢ 𝑑
17		cbs	⊢ Base
18	3	cv	⊢ 𝑟
19	18 17	cfv	⊢ ( Base ‘ 𝑟 )
20	16	cv	⊢ 𝑑
21	19 20 6	co	⊢ ( ( Base ‘ 𝑟 ) ↑m 𝑑 )
22		vb	⊢ 𝑏
23		cnx	⊢ ndx
24	23 17	cfv	⊢ ( Base ‘ ndx )
25	22	cv	⊢ 𝑏
26	24 25	cop	⊢ ⟨ ( Base ‘ ndx ) , 𝑏 ⟩
27		cplusg	⊢ +g
28	23 27	cfv	⊢ ( +g ‘ ndx )
29	18 27	cfv	⊢ ( +g ‘ 𝑟 )
30	29	cof	⊢ ∘f ( +g ‘ 𝑟 )
31	25 25	cxp	⊢ ( 𝑏 × 𝑏 )
32	30 31	cres	⊢ ( ∘f ( +g ‘ 𝑟 ) ↾ ( 𝑏 × 𝑏 ) )
33	28 32	cop	⊢ ⟨ ( +g ‘ ndx ) , ( ∘f ( +g ‘ 𝑟 ) ↾ ( 𝑏 × 𝑏 ) ) ⟩
34		cmulr	⊢ .r
35	23 34	cfv	⊢ ( .r ‘ ndx )
36		vf	⊢ 𝑓
37		vg	⊢ 𝑔
38		vk	⊢ 𝑘
39		cgsu	⊢ Σg
40		vx	⊢ 𝑥
41		vy	⊢ 𝑦
42	41	cv	⊢ 𝑦
43		cle	⊢ ≤
44	43	cofr	⊢ ∘r ≤
45	38	cv	⊢ 𝑘
46	42 45 44	wbr	⊢ 𝑦 ∘r ≤ 𝑘
47	46 41 20	crab	⊢ { 𝑦 ∈ 𝑑 ∣ 𝑦 ∘r ≤ 𝑘 }
48	36	cv	⊢ 𝑓
49	40	cv	⊢ 𝑥
50	49 48	cfv	⊢ ( 𝑓 ‘ 𝑥 )
51	18 34	cfv	⊢ ( .r ‘ 𝑟 )
52	37	cv	⊢ 𝑔
53		cmin	⊢ −
54	53	cof	⊢ ∘f −
55	45 49 54	co	⊢ ( 𝑘 ∘f − 𝑥 )
56	55 52	cfv	⊢ ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) )
57	50 56 51	co	⊢ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑟 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) )
58	40 47 57	cmpt	⊢ ( 𝑥 ∈ { 𝑦 ∈ 𝑑 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑟 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) )
59	18 58 39	co	⊢ ( 𝑟 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝑑 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑟 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) )
60	38 20 59	cmpt	⊢ ( 𝑘 ∈ 𝑑 ↦ ( 𝑟 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝑑 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑟 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) )
61	36 37 25 25 60	cmpo	⊢ ( 𝑓 ∈ 𝑏 , 𝑔 ∈ 𝑏 ↦ ( 𝑘 ∈ 𝑑 ↦ ( 𝑟 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝑑 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑟 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) )
62	35 61	cop	⊢ ⟨ ( .r ‘ ndx ) , ( 𝑓 ∈ 𝑏 , 𝑔 ∈ 𝑏 ↦ ( 𝑘 ∈ 𝑑 ↦ ( 𝑟 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝑑 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑟 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) ⟩
63	26 33 62	ctp	⊢ { ⟨ ( Base ‘ ndx ) , 𝑏 ⟩ , ⟨ ( +g ‘ ndx ) , ( ∘f ( +g ‘ 𝑟 ) ↾ ( 𝑏 × 𝑏 ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑓 ∈ 𝑏 , 𝑔 ∈ 𝑏 ↦ ( 𝑘 ∈ 𝑑 ↦ ( 𝑟 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝑑 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑟 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) ⟩ }
64		csca	⊢ Scalar
65	23 64	cfv	⊢ ( Scalar ‘ ndx )
66	65 18	cop	⊢ ⟨ ( Scalar ‘ ndx ) , 𝑟 ⟩
67		cvsca	⊢ ·𝑠
68	23 67	cfv	⊢ ( ·𝑠 ‘ ndx )
69	49	csn	⊢ { 𝑥 }
70	20 69	cxp	⊢ ( 𝑑 × { 𝑥 } )
71	51	cof	⊢ ∘f ( .r ‘ 𝑟 )
72	70 48 71	co	⊢ ( ( 𝑑 × { 𝑥 } ) ∘f ( .r ‘ 𝑟 ) 𝑓 )
73	40 36 19 25 72	cmpo	⊢ ( 𝑥 ∈ ( Base ‘ 𝑟 ) , 𝑓 ∈ 𝑏 ↦ ( ( 𝑑 × { 𝑥 } ) ∘f ( .r ‘ 𝑟 ) 𝑓 ) )
74	68 73	cop	⊢ ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑟 ) , 𝑓 ∈ 𝑏 ↦ ( ( 𝑑 × { 𝑥 } ) ∘f ( .r ‘ 𝑟 ) 𝑓 ) ) ⟩
75		cts	⊢ TopSet
76	23 75	cfv	⊢ ( TopSet ‘ ndx )
77		cpt	⊢ ∏t
78		ctopn	⊢ TopOpen
79	18 78	cfv	⊢ ( TopOpen ‘ 𝑟 )
80	79	csn	⊢ { ( TopOpen ‘ 𝑟 ) }
81	20 80	cxp	⊢ ( 𝑑 × { ( TopOpen ‘ 𝑟 ) } )
82	81 77	cfv	⊢ ( ∏t ‘ ( 𝑑 × { ( TopOpen ‘ 𝑟 ) } ) )
83	76 82	cop	⊢ ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝑑 × { ( TopOpen ‘ 𝑟 ) } ) ) ⟩
84	66 74 83	ctp	⊢ { ⟨ ( Scalar ‘ ndx ) , 𝑟 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑟 ) , 𝑓 ∈ 𝑏 ↦ ( ( 𝑑 × { 𝑥 } ) ∘f ( .r ‘ 𝑟 ) 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝑑 × { ( TopOpen ‘ 𝑟 ) } ) ) ⟩ }
85	63 84	cun	⊢ ( { ⟨ ( Base ‘ ndx ) , 𝑏 ⟩ , ⟨ ( +g ‘ ndx ) , ( ∘f ( +g ‘ 𝑟 ) ↾ ( 𝑏 × 𝑏 ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑓 ∈ 𝑏 , 𝑔 ∈ 𝑏 ↦ ( 𝑘 ∈ 𝑑 ↦ ( 𝑟 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝑑 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑟 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑟 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑟 ) , 𝑓 ∈ 𝑏 ↦ ( ( 𝑑 × { 𝑥 } ) ∘f ( .r ‘ 𝑟 ) 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝑑 × { ( TopOpen ‘ 𝑟 ) } ) ) ⟩ } )
86	22 21 85	csb	⊢ ⦋ ( ( Base ‘ 𝑟 ) ↑m 𝑑 ) / 𝑏 ⦌ ( { ⟨ ( Base ‘ ndx ) , 𝑏 ⟩ , ⟨ ( +g ‘ ndx ) , ( ∘f ( +g ‘ 𝑟 ) ↾ ( 𝑏 × 𝑏 ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑓 ∈ 𝑏 , 𝑔 ∈ 𝑏 ↦ ( 𝑘 ∈ 𝑑 ↦ ( 𝑟 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝑑 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑟 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑟 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑟 ) , 𝑓 ∈ 𝑏 ↦ ( ( 𝑑 × { 𝑥 } ) ∘f ( .r ‘ 𝑟 ) 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝑑 × { ( TopOpen ‘ 𝑟 ) } ) ) ⟩ } )
87	16 15 86	csb	⊢ ⦋ { ℎ ∈ ( ℕ0 ↑m 𝑖 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } / 𝑑 ⦌ ⦋ ( ( Base ‘ 𝑟 ) ↑m 𝑑 ) / 𝑏 ⦌ ( { ⟨ ( Base ‘ ndx ) , 𝑏 ⟩ , ⟨ ( +g ‘ ndx ) , ( ∘f ( +g ‘ 𝑟 ) ↾ ( 𝑏 × 𝑏 ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑓 ∈ 𝑏 , 𝑔 ∈ 𝑏 ↦ ( 𝑘 ∈ 𝑑 ↦ ( 𝑟 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝑑 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑟 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑟 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑟 ) , 𝑓 ∈ 𝑏 ↦ ( ( 𝑑 × { 𝑥 } ) ∘f ( .r ‘ 𝑟 ) 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝑑 × { ( TopOpen ‘ 𝑟 ) } ) ) ⟩ } )
88	1 3 2 2 87	cmpo	⊢ ( 𝑖 ∈ V , 𝑟 ∈ V ↦ ⦋ { ℎ ∈ ( ℕ0 ↑m 𝑖 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } / 𝑑 ⦌ ⦋ ( ( Base ‘ 𝑟 ) ↑m 𝑑 ) / 𝑏 ⦌ ( { ⟨ ( Base ‘ ndx ) , 𝑏 ⟩ , ⟨ ( +g ‘ ndx ) , ( ∘f ( +g ‘ 𝑟 ) ↾ ( 𝑏 × 𝑏 ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑓 ∈ 𝑏 , 𝑔 ∈ 𝑏 ↦ ( 𝑘 ∈ 𝑑 ↦ ( 𝑟 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝑑 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑟 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑟 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑟 ) , 𝑓 ∈ 𝑏 ↦ ( ( 𝑑 × { 𝑥 } ) ∘f ( .r ‘ 𝑟 ) 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝑑 × { ( TopOpen ‘ 𝑟 ) } ) ) ⟩ } ) )
89	0 88	wceq	⊢ mPwSer = ( 𝑖 ∈ V , 𝑟 ∈ V ↦ ⦋ { ℎ ∈ ( ℕ0 ↑m 𝑖 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } / 𝑑 ⦌ ⦋ ( ( Base ‘ 𝑟 ) ↑m 𝑑 ) / 𝑏 ⦌ ( { ⟨ ( Base ‘ ndx ) , 𝑏 ⟩ , ⟨ ( +g ‘ ndx ) , ( ∘f ( +g ‘ 𝑟 ) ↾ ( 𝑏 × 𝑏 ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑓 ∈ 𝑏 , 𝑔 ∈ 𝑏 ↦ ( 𝑘 ∈ 𝑑 ↦ ( 𝑟 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝑑 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑟 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑟 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑟 ) , 𝑓 ∈ 𝑏 ↦ ( ( 𝑑 × { 𝑥 } ) ∘f ( .r ‘ 𝑟 ) 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝑑 × { ( TopOpen ‘ 𝑟 ) } ) ) ⟩ } ) )