Theorem chfacffsupp 23050

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 6903	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (0g‘𝑌) ∈ V)
3		ovex 7456	. . . . 5 ⊢ ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) ∈ V
4		fvex 6901	. . . . . 6 ⊢ (𝑇‘(𝑏‘𝑠)) ∈ V
5		chfacfisf.0	. . . . . . . 8 ⊢ 0 = (0g‘𝑌)
6	5	fvexi 6902	. . . . . . 7 ⊢ 0 ∈ V
7		ovex 7456	. . . . . . 7 ⊢ ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))) ∈ V
8	6, 7	ifex 4543	. . . . . 6 ⊢ if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))) ∈ V
9	4, 8	ifex 4543	. . . . 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 4543	. . . 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 12529	. . . . . 6 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℕ0)
13		peano2nn0 12562	. . . . . 6 ⊢ (𝑠 ∈ ℕ0 → (𝑠 + 1) ∈ ℕ0)
14	12, 13	syl 18	. . . . 5 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℕ0)
15	14	ad2antrl 741	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑠 + 1) ∈ ℕ0)
16		simplr 781	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 𝑘 ∈ ℕ0)
17		0red 11229	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 0 ∈ ℝ)
18		nnre 12258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℝ)
19		peano2re 11401	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ ℝ → (𝑠 + 1) ∈ ℝ)
20	18, 19	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℝ)
21	20	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑠 + 1) ∈ ℝ)
22	21	ad3antlr 744	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → (𝑠 + 1) ∈ ℝ)
23		nn0re 12531	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
24	23	ad2antlr 740	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 𝑘 ∈ ℝ)
25	12	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑠 ∈ ℕ0)
26	25	ad2antlr 740	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) → 𝑠 ∈ ℕ0)
27		nn0p1gt0 12551	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℕ0 → 0 < (𝑠 + 1))
28	26, 27	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) → 0 < (𝑠 + 1))
29	28	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 0 < (𝑠 + 1))
30		simpr 490	. . . . . . . . . . . . . 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 11389	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 0 < 𝑘)
32	31	gt0ne0d 11796	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 𝑘 ≠ 0)
33	32	neneqd 2966	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → ¬ 𝑘 = 0)
34	33	adantr 486	. . . . . . . . . 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 2770	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑛 = 0 ↔ 𝑘 = 0))
36	35	notbid 321	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (¬ 𝑛 = 0 ↔ ¬ 𝑘 = 0))
37	36	adantl 487	. . . . . . . . . 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 260	. . . . . . . . 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 4503	. . . . . . . 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 740	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) → (𝑠 + 1) ∈ ℝ)
41		ltne 11325	. . . . . . . . . . . . 13 ⊢ (((𝑠 + 1) ∈ ℝ ∧ (𝑠 + 1) < 𝑘) → 𝑘 ≠ (𝑠 + 1))
42	40, 41	sylan 592	. . . . . . . . . . . 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 2966	. . . . . . . . . . 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 486	. . . . . . . . . 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 2770	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑛 = (𝑠 + 1) ↔ 𝑘 = (𝑠 + 1)))
46	45	notbid 321	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (¬ 𝑛 = (𝑠 + 1) ↔ ¬ 𝑘 = (𝑠 + 1)))
47	46	adantl 487	. . . . . . . . . 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 260	. . . . . . . . 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 4503	. . . . . . . 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 781	. . . . . . . . . . 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 5118	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → ((𝑠 + 1) < 𝑛 ↔ (𝑠 + 1) < 𝑘))
52	51	adantl 487	. . . . . . . . . . 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 260	. . . . . . . . . 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 4500	. . . . . . . . 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 2817	. . . . . . . 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 2805	. . . . . . 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 3892	. . . . . 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 418	. . . . 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 3160	. . . 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 5117	. . . . 5 ⊢ (𝑙 = (𝑠 + 1) → (𝑙 < 𝑘 ↔ (𝑠 + 1) < 𝑘))
61	60	rspceaimv 3590	. . . 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 596	. . 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 14053	. 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 5151	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺 finSupp (0g‘𝑌))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2961  ∀wral 3082  ∃wrex 3092  Vcvv 3458  ⦋csb 3856  ifcif 4492   class class class wbr 5114   ↦ cmpt 5197  ‘cfv 6543  (class class class)co 7423   ↑_m cmap 8833  Fincfn 8952   finSupp cfsupp 9331  ℝcr 11117  0cc0 11118  1c1 11119   + caddc 11121   < clt 11261   − cmin 11459  ℕcn 12251  ℕ₀cn0 12522  ...cfz 13553  Basecbs 17294  ._rcmulr 17336  0_gc0g 17517  -_gcsg 19033  CRingccrg 20347  Poly₁cpl1 22374   Mat cmat 22601   matToPolyMat cmat2pmat 22898

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-supp 8166  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8703  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-fsupp 9332  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-n0 12523  df-z 12610  df-uz 12881  df-fz 13554

This theorem is used by:  cpmadumatpolylem2  23076  cayhamlem4  23082