Theorem chfacffsupp 20661

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 6203	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → (0g‘𝑌) ∈ V)
3		ovex 6678	. . . . 5 ⊢ ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) ∈ V
4		fvex 6201	. . . . . 6 ⊢ (𝑇‘(𝑏‘𝑠)) ∈ V
5		chfacfisf.0	. . . . . . . 8 ⊢ 0 = (0g‘𝑌)
6		fvex 6201	. . . . . . . 8 ⊢ (0g‘𝑌) ∈ V
7	5, 6	eqeltri 2697	. . . . . . 7 ⊢ 0 ∈ V
8		ovex 6678	. . . . . . 7 ⊢ ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))) ∈ V
9	7, 8	ifex 4156	. . . . . 6 ⊢ if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))) ∈ V
10	4, 9	ifex 4156	. . . . 5 ⊢ if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))) ∈ V
11	3, 10	ifex 4156	. . . 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
12	11	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)
13		nnnn0 11299	. . . . . 6 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℕ0)
14		peano2nn0 11333	. . . . . 6 ⊢ (𝑠 ∈ ℕ0 → (𝑠 + 1) ∈ ℕ0)
15	13, 14	syl 17	. . . . 5 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℕ0)
16	15	ad2antrl 764	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → (𝑠 + 1) ∈ ℕ0)
17		simplr 792	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 𝑘 ∈ ℕ0)
18		0red 10041	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 0 ∈ ℝ)
19		nnre 11027	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℝ)
20		peano2re 10209	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ ℝ → (𝑠 + 1) ∈ ℝ)
21	19, 20	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℝ)
22	21	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → (𝑠 + 1) ∈ ℝ)
23	22	ad3antlr 767	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → (𝑠 + 1) ∈ ℝ)
24		nn0re 11301	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
25	24	ad2antlr 763	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 𝑘 ∈ ℝ)
26	13	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → 𝑠 ∈ ℕ0)
27	26	ad2antlr 763	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) → 𝑠 ∈ ℕ0)
28		nn0p1gt0 11322	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℕ0 → 0 < (𝑠 + 1))
29	27, 28	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) → 0 < (𝑠 + 1))
30	29	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 0 < (𝑠 + 1))
31		simpr 477	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → (𝑠 + 1) < 𝑘)
32	18, 23, 25, 30, 31	lttrd 10198	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 0 < 𝑘)
33	32	gt0ne0d 10592	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 𝑘 ≠ 0)
34	33	neneqd 2799	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → ¬ 𝑘 = 0)
35	34	adantr 481	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) ∧ 𝑛 = 𝑘) → ¬ 𝑘 = 0)
36		eqeq1 2626	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑛 = 0 ↔ 𝑘 = 0))
37	36	notbid 308	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (¬ 𝑛 = 0 ↔ ¬ 𝑘 = 0))
38	37	adantl 482	. . . . . . . . . 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))
39	35, 38	mpbird 247	. . . . . . . . 9 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) ∧ 𝑛 = 𝑘) → ¬ 𝑛 = 0)
40	39	iffalsed 4097	. . . . . . . 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)))))))
41	22	ad2antlr 763	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) → (𝑠 + 1) ∈ ℝ)
42		ltne 10134	. . . . . . . . . . . . 13 ⊢ (((𝑠 + 1) ∈ ℝ ∧ (𝑠 + 1) < 𝑘) → 𝑘 ≠ (𝑠 + 1))
43	41, 42	sylan 488	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 𝑘 ≠ (𝑠 + 1))
44	43	neneqd 2799	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → ¬ 𝑘 = (𝑠 + 1))
45	44	adantr 481	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) ∧ 𝑛 = 𝑘) → ¬ 𝑘 = (𝑠 + 1))
46		eqeq1 2626	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑛 = (𝑠 + 1) ↔ 𝑘 = (𝑠 + 1)))
47	46	notbid 308	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (¬ 𝑛 = (𝑠 + 1) ↔ ¬ 𝑘 = (𝑠 + 1)))
48	47	adantl 482	. . . . . . . . . 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)))
49	45, 48	mpbird 247	. . . . . . . . 9 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) ∧ 𝑛 = 𝑘) → ¬ 𝑛 = (𝑠 + 1))
50	49	iffalsed 4097	. . . . . . . 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))))))
51		simplr 792	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) ∧ 𝑛 = 𝑘) → (𝑠 + 1) < 𝑘)
52		breq2 4657	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → ((𝑠 + 1) < 𝑛 ↔ (𝑠 + 1) < 𝑘))
53	52	adantl 482	. . . . . . . . . . 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))
54	51, 53	mpbird 247	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) ∧ 𝑛 = 𝑘) → (𝑠 + 1) < 𝑛)
55	54	iftrued 4094	. . . . . . . . 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 )
56	55, 5	syl6eq 2672	. . . . . . . 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))
57	40, 50, 56	3eqtrd 2660	. . . . . . 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))
58	17, 57	csbied 3560	. . . . . 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))
59	58	ex 450	. . . . 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)))
60	59	ralrimiva 2966	. . . 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)))
61		breq1 4656	. . . . . . 7 ⊢ (𝑙 = (𝑠 + 1) → (𝑙 < 𝑘 ↔ (𝑠 + 1) < 𝑘))
62	61	imbi1d 331	. . . . . 6 ⊢ (𝑙 = (𝑠 + 1) → ((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)) ↔ ((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))))
63	62	ralbidv 2986	. . . . 5 ⊢ (𝑙 = (𝑠 + 1) → (∀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)) ↔ ∀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))))
64	63	rspcev 3309	. . . 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)))
65	16, 60, 64	syl2anc 693	. . 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)))
66	2, 12, 65	mptnn0fsupp 12797	. 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))
67	1, 66	syl5eqbr 4688	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200  ⦋csb 3533  ifcif 4086   class class class wbr 4653   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  Fincfn 7955   finSupp cfsupp 8275  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   < clt 10074   − cmin 10266  ℕcn 11020  ℕ₀cn0 11292  ...cfz 12326  Basecbs 15857  ._rcmulr 15942  0_gc0g 16100  -_gcsg 17424  CRingccrg 18548  Poly₁cpl1 19547   Mat cmat 20213   matToPolyMat cmat2pmat 20509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327

This theorem is referenced by:  cpmadumatpolylem2  20687  cayhamlem4  20693