Theorem chfacffsupp 22812

Description: The "characteristic factor function" is finitely supported. (Contributed by AV, 20-Nov-2019.) (Proof shortened by AV, 23-Dec-2019.)

Hypotheses

Ref	Expression
chfacfisf.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
chfacfisf.b	⊢ 𝐵 = (Base‘𝐴)
chfacfisf.p	⊢ 𝑃 = (Poly1‘𝑅)
chfacfisf.y	⊢ 𝑌 = (𝑁 Mat 𝑃)
chfacfisf.r	⊢ × = (.r‘𝑌)
chfacfisf.s	⊢ − = (-g‘𝑌)
chfacfisf.0	⊢ 0 = (0g‘𝑌)
chfacfisf.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
chfacfisf.g	⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))

Assertion

Ref	Expression
chfacffsupp	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺 finSupp (0g‘𝑌))

Distinct variable groups:   𝐵,𝑛   𝑛,𝑀   𝑛,𝑁   𝑅,𝑛   𝑛,𝑌   𝑛,𝑏   𝑛,𝑠

Allowed substitution hints:   𝐴(𝑛,𝑠,𝑏)   𝐵(𝑠,𝑏)   𝑃(𝑛,𝑠,𝑏)   𝑅(𝑠,𝑏)   𝑇(𝑛,𝑠,𝑏)   × (𝑛,𝑠,𝑏)   𝐺(𝑛,𝑠,𝑏)   𝑀(𝑠,𝑏)   − (𝑛,𝑠,𝑏)   𝑁(𝑠,𝑏)   𝑌(𝑠,𝑏)   0 (𝑛,𝑠,𝑏)

Proof of Theorem chfacffsupp

Dummy variables 𝑘 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		chfacfisf.g	. 2 ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))
2		fvexd 6857	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (0g‘𝑌) ∈ V)
3		ovex 7401	. . . . 5 ⊢ ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) ∈ V
4		fvex 6855	. . . . . 6 ⊢ (𝑇‘(𝑏‘𝑠)) ∈ V
5		chfacfisf.0	. . . . . . . 8 ⊢ 0 = (0g‘𝑌)
6	5	fvexi 6856	. . . . . . 7 ⊢ 0 ∈ V
7		ovex 7401	. . . . . . 7 ⊢ ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))) ∈ V
8	6, 7	ifex 4532	. . . . . 6 ⊢ if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))) ∈ V
9	4, 8	ifex 4532	. . . . 5 ⊢ if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))) ∈ V
10	3, 9	ifex 4532	. . . 4 ⊢ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))) ∈ V
11	10	a1i 11	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))) ∈ V)
12		nnnn0 12420	. . . . . 6 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℕ0)
13		peano2nn0 12453	. . . . . 6 ⊢ (𝑠 ∈ ℕ0 → (𝑠 + 1) ∈ ℕ0)
14	12, 13	syl 17	. . . . 5 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℕ0)
15	14	ad2antrl 729	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑠 + 1) ∈ ℕ0)
16		simplr 769	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 𝑘 ∈ ℕ0)
17		0red 11147	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 0 ∈ ℝ)
18		nnre 12164	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℝ)
19		peano2re 11318	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ ℝ → (𝑠 + 1) ∈ ℝ)
20	18, 19	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℝ)
21	20	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑠 + 1) ∈ ℝ)
22	21	ad3antlr 732	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → (𝑠 + 1) ∈ ℝ)
23		nn0re 12422	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
24	23	ad2antlr 728	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 𝑘 ∈ ℝ)
25	12	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑠 ∈ ℕ0)
26	25	ad2antlr 728	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) → 𝑠 ∈ ℕ0)
27		nn0p1gt0 12442	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℕ0 → 0 < (𝑠 + 1))
28	26, 27	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) → 0 < (𝑠 + 1))
29	28	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 0 < (𝑠 + 1))
30		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → (𝑠 + 1) < 𝑘)
31	17, 22, 24, 29, 30	lttrd 11306	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 0 < 𝑘)
32	31	gt0ne0d 11713	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 𝑘 ≠ 0)
33	32	neneqd 2938	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → ¬ 𝑘 = 0)
34	33	adantr 480	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) ∧ 𝑛 = 𝑘) → ¬ 𝑘 = 0)
35		eqeq1 2741	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑛 = 0 ↔ 𝑘 = 0))
36	35	notbid 318	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (¬ 𝑛 = 0 ↔ ¬ 𝑘 = 0))
37	36	adantl 481	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) ∧ 𝑛 = 𝑘) → (¬ 𝑛 = 0 ↔ ¬ 𝑘 = 0))
38	34, 37	mpbird 257	. . . . . . . . 9 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) ∧ 𝑛 = 𝑘) → ¬ 𝑛 = 0)
39	38	iffalsed 4492	. . . . . . . 8 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) ∧ 𝑛 = 𝑘) → if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))) = if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))
40	21	ad2antlr 728	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) → (𝑠 + 1) ∈ ℝ)
41		ltne 11242	. . . . . . . . . . . . 13 ⊢ (((𝑠 + 1) ∈ ℝ ∧ (𝑠 + 1) < 𝑘) → 𝑘 ≠ (𝑠 + 1))
42	40, 41	sylan 581	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 𝑘 ≠ (𝑠 + 1))
43	42	neneqd 2938	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → ¬ 𝑘 = (𝑠 + 1))
44	43	adantr 480	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) ∧ 𝑛 = 𝑘) → ¬ 𝑘 = (𝑠 + 1))
45		eqeq1 2741	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑛 = (𝑠 + 1) ↔ 𝑘 = (𝑠 + 1)))
46	45	notbid 318	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (¬ 𝑛 = (𝑠 + 1) ↔ ¬ 𝑘 = (𝑠 + 1)))
47	46	adantl 481	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) ∧ 𝑛 = 𝑘) → (¬ 𝑛 = (𝑠 + 1) ↔ ¬ 𝑘 = (𝑠 + 1)))
48	44, 47	mpbird 257	. . . . . . . . 9 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) ∧ 𝑛 = 𝑘) → ¬ 𝑛 = (𝑠 + 1))
49	48	iffalsed 4492	. . . . . . . 8 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) ∧ 𝑛 = 𝑘) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))) = if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))
50		simplr 769	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) ∧ 𝑛 = 𝑘) → (𝑠 + 1) < 𝑘)
51		breq2 5104	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → ((𝑠 + 1) < 𝑛 ↔ (𝑠 + 1) < 𝑘))
52	51	adantl 481	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) ∧ 𝑛 = 𝑘) → ((𝑠 + 1) < 𝑛 ↔ (𝑠 + 1) < 𝑘))
53	50, 52	mpbird 257	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) ∧ 𝑛 = 𝑘) → (𝑠 + 1) < 𝑛)
54	53	iftrued 4489	. . . . . . . . 9 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) ∧ 𝑛 = 𝑘) → if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))) = 0 )
55	54, 5	eqtrdi 2788	. . . . . . . 8 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) ∧ 𝑛 = 𝑘) → if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))) = (0g‘𝑌))
56	39, 49, 55	3eqtrd 2776	. . . . . . 7 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) ∧ 𝑛 = 𝑘) → if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))) = (0g‘𝑌))
57	16, 56	csbied 3887	. . . . . 6 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → ⦋𝑘 / 𝑛⦌if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))) = (0g‘𝑌))
58	57	ex 412	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) → ((𝑠 + 1) < 𝑘 → ⦋𝑘 / 𝑛⦌if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))) = (0g‘𝑌)))
59	58	ralrimiva 3130	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ∀𝑘 ∈ ℕ0 ((𝑠 + 1) < 𝑘 → ⦋𝑘 / 𝑛⦌if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))) = (0g‘𝑌)))
60		breq1 5103	. . . . 5 ⊢ (𝑙 = (𝑠 + 1) → (𝑙 < 𝑘 ↔ (𝑠 + 1) < 𝑘))
61	60	rspceaimv 3584	. . . 4 ⊢ (((𝑠 + 1) ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 ((𝑠 + 1) < 𝑘 → ⦋𝑘 / 𝑛⦌if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))) = (0g‘𝑌))) → ∃𝑙 ∈ ℕ0 ∀𝑘 ∈ ℕ0 (𝑙 < 𝑘 → ⦋𝑘 / 𝑛⦌if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))) = (0g‘𝑌)))
62	15, 59, 61	syl2anc 585	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ∃𝑙 ∈ ℕ0 ∀𝑘 ∈ ℕ0 (𝑙 < 𝑘 → ⦋𝑘 / 𝑛⦌if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))) = (0g‘𝑌)))
63	2, 11, 62	mptnn0fsupp 13932	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) finSupp (0g‘𝑌))
64	1, 63	eqbrtrid 5135	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺 finSupp (0g‘𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  Vcvv 3442  ⦋csb 3851  ifcif 4481   class class class wbr 5100   ↦ cmpt 5181  ‘cfv 6500  (class class class)co 7368   ↑_m cmap 8775  Fincfn 8895   finSupp cfsupp 9276  ℝcr 11037  0cc0 11038  1c1 11039   + caddc 11041   < clt 11178   − cmin 11376  ℕcn 12157  ℕ₀cn0 12413  ...cfz 13435  Basecbs 17148  ._rcmulr 17190  0_gc0g 17371  -_gcsg 18877  CRingccrg 20181  Poly₁cpl1 22129   Mat cmat 22363   matToPolyMat cmat2pmat 22660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-supp 8113  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9277  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-n0 12414  df-z 12501  df-uz 12764  df-fz 13436

This theorem is referenced by:  cpmadumatpolylem2  22838  cayhamlem4  22844