Theorem chfacfisf 22203

Description: The "characteristic factor function" is a function from the nonnegative integers to polynomial matrices. (Contributed by AV, 8-Nov-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
chfacfisf	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶(Base‘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 chfacfisf

Step	Hyp	Ref	Expression
1		chfacfisf.p	. . . . . . . . 9 ⊢ 𝑃 = (Poly1‘𝑅)
2		chfacfisf.y	. . . . . . . . 9 ⊢ 𝑌 = (𝑁 Mat 𝑃)
3	1, 2	pmatring 22041	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
4	3	3adant3 1132	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
5		ringgrp 19969	. . . . . . 7 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp)
6	4, 5	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Grp)
7	6	adantr 481	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑌 ∈ Grp)
8		eqid 2736	. . . . . . . 8 ⊢ (Base‘𝑌) = (Base‘𝑌)
9		chfacfisf.0	. . . . . . . 8 ⊢ 0 = (0g‘𝑌)
10	8, 9	ring0cl 19990	. . . . . . 7 ⊢ (𝑌 ∈ Ring → 0 ∈ (Base‘𝑌))
11	4, 10	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 0 ∈ (Base‘𝑌))
12	11	adantr 481	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 0 ∈ (Base‘𝑌))
13	4	adantr 481	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑌 ∈ Ring)
14		chfacfisf.t	. . . . . . . 8 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
15		chfacfisf.a	. . . . . . . 8 ⊢ 𝐴 = (𝑁 Mat 𝑅)
16		chfacfisf.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐴)
17	14, 15, 16, 1, 2	mat2pmatbas 22075	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
18	17	adantr 481	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑇‘𝑀) ∈ (Base‘𝑌))
19		3simpa 1148	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
20		elmapi 8787	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (𝐵 ↑m (0...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
21	20	adantl 482	. . . . . . . . . 10 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑏:(0...𝑠)⟶𝐵)
22		nnnn0 12420	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℕ0)
23		nn0uz 12805	. . . . . . . . . . . . 13 ⊢ ℕ0 = (ℤ≥‘0)
24	22, 23	eleqtrdi 2848	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ (ℤ≥‘0))
25		eluzfz1 13448	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ (ℤ≥‘0) → 0 ∈ (0...𝑠))
26	24, 25	syl 17	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ℕ → 0 ∈ (0...𝑠))
27	26	adantr 481	. . . . . . . . . 10 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 0 ∈ (0...𝑠))
28	21, 27	ffvelcdmd 7036	. . . . . . . . 9 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑏‘0) ∈ 𝐵)
29	19, 28	anim12i 613	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑏‘0) ∈ 𝐵))
30		df-3an 1089	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑏‘0) ∈ 𝐵))
31	29, 30	sylibr 233	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵))
32	14, 15, 16, 1, 2	mat2pmatbas 22075	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
33	31, 32	syl 17	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
34		chfacfisf.r	. . . . . . 7 ⊢ × = (.r‘𝑌)
35	8, 34	ringcl 19981	. . . . . 6 ⊢ ((𝑌 ∈ Ring ∧ (𝑇‘𝑀) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌)) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
36	13, 18, 33, 35	syl3anc 1371	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
37		chfacfisf.s	. . . . . 6 ⊢ − = (-g‘𝑌)
38	8, 37	grpsubcl 18827	. . . . 5 ⊢ ((𝑌 ∈ Grp ∧ 0 ∈ (Base‘𝑌) ∧ ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌)) → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) ∈ (Base‘𝑌))
39	7, 12, 36, 38	syl3anc 1371	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) ∈ (Base‘𝑌))
40	39	ad2antrr 724	. . 3 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) ∈ (Base‘𝑌))
41	22	adantr 481	. . . . . . . . 9 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑠 ∈ ℕ0)
42	19, 41	anim12i 613	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑠 ∈ ℕ0))
43		df-3an 1089	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑠 ∈ ℕ0))
44	42, 43	sylibr 233	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0))
45		eluzfz2 13449	. . . . . . . . . 10 ⊢ (𝑠 ∈ (ℤ≥‘0) → 𝑠 ∈ (0...𝑠))
46	24, 45	syl 17	. . . . . . . . 9 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ (0...𝑠))
47	46	anim1ci 616	. . . . . . . 8 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑏 ∈ (𝐵 ↑m (0...𝑠)) ∧ 𝑠 ∈ (0...𝑠)))
48	47	adantl 482	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑏 ∈ (𝐵 ↑m (0...𝑠)) ∧ 𝑠 ∈ (0...𝑠)))
49	15, 16, 1, 2, 14	m2pmfzmap 22096	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0) ∧ (𝑏 ∈ (𝐵 ↑m (0...𝑠)) ∧ 𝑠 ∈ (0...𝑠))) → (𝑇‘(𝑏‘𝑠)) ∈ (Base‘𝑌))
50	44, 48, 49	syl2anc 584	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑇‘(𝑏‘𝑠)) ∈ (Base‘𝑌))
51	50	adantr 481	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑏‘𝑠)) ∈ (Base‘𝑌))
52	51	ad2antrr 724	. . . 4 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 = (𝑠 + 1)) → (𝑇‘(𝑏‘𝑠)) ∈ (Base‘𝑌))
53	12	ad4antr 730	. . . . 5 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ (𝑠 + 1) < 𝑛) → 0 ∈ (Base‘𝑌))
54		nn0re 12422	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℝ)
55	54	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℝ)
56		peano2nn 12165	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℕ)
57	56	nnred 12168	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℝ)
58	57	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (𝑠 + 1) ∈ ℝ)
59	55, 58	lenltd 11301	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (𝑛 ≤ (𝑠 + 1) ↔ ¬ (𝑠 + 1) < 𝑛))
60		nesym 3000	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 + 1) ≠ 𝑛 ↔ ¬ 𝑛 = (𝑠 + 1))
61		ltlen 11256	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ) → (𝑛 < (𝑠 + 1) ↔ (𝑛 ≤ (𝑠 + 1) ∧ (𝑠 + 1) ≠ 𝑛)))
62	54, 57, 61	syl2anr 597	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (𝑛 < (𝑠 + 1) ↔ (𝑛 ≤ (𝑠 + 1) ∧ (𝑠 + 1) ≠ 𝑛)))
63	62	biimprd 247	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → ((𝑛 ≤ (𝑠 + 1) ∧ (𝑠 + 1) ≠ 𝑛) → 𝑛 < (𝑠 + 1)))
64	63	expcomd 417	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → ((𝑠 + 1) ≠ 𝑛 → (𝑛 ≤ (𝑠 + 1) → 𝑛 < (𝑠 + 1))))
65	60, 64	biimtrrid 242	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (¬ 𝑛 = (𝑠 + 1) → (𝑛 ≤ (𝑠 + 1) → 𝑛 < (𝑠 + 1))))
66	65	com23 86	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (𝑛 ≤ (𝑠 + 1) → (¬ 𝑛 = (𝑠 + 1) → 𝑛 < (𝑠 + 1))))
67	59, 66	sylbird 259	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (¬ (𝑠 + 1) < 𝑛 → (¬ 𝑛 = (𝑠 + 1) → 𝑛 < (𝑠 + 1))))
68	67	impcomd 412	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → ((¬ 𝑛 = (𝑠 + 1) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 < (𝑠 + 1)))
69	68	ex 413	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℕ → (𝑛 ∈ ℕ0 → ((¬ 𝑛 = (𝑠 + 1) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 < (𝑠 + 1))))
70	69	ad2antrl 726	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑛 ∈ ℕ0 → ((¬ 𝑛 = (𝑠 + 1) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 < (𝑠 + 1))))
71	70	imp 407	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → ((¬ 𝑛 = (𝑠 + 1) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 < (𝑠 + 1)))
72	71	adantr 481	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → ((¬ 𝑛 = (𝑠 + 1) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 < (𝑠 + 1)))
73	3, 5	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Grp)
74	73	3adant3 1132	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Grp)
75	74	ad4antr 730	. . . . . . . . 9 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → 𝑌 ∈ Grp)
76	19	ad4antr 730	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
77	21	ad4antlr 731	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → 𝑏:(0...𝑠)⟶𝐵)
78		neqne 2951	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑛 = 0 → 𝑛 ≠ 0)
79	78	anim2i 617	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → (𝑛 ∈ ℕ0 ∧ 𝑛 ≠ 0))
80		elnnne0 12427	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ0 ∧ 𝑛 ≠ 0))
81	79, 80	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ)
82		nnm1nn0 12454	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
83	81, 82	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → (𝑛 − 1) ∈ ℕ0)
84	83	ad4ant23 751	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → (𝑛 − 1) ∈ ℕ0)
85	41	ad4antlr 731	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → 𝑠 ∈ ℕ0)
86	62	simprbda 499	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑛 ≤ (𝑠 + 1))
87	55	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑛 ∈ ℝ)
88		1red 11156	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 1 ∈ ℝ)
89		nnre 12160	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℝ)
90	89	ad2antrr 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑠 ∈ ℝ)
91	87, 88, 90	lesubaddd 11752	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → ((𝑛 − 1) ≤ 𝑠 ↔ 𝑛 ≤ (𝑠 + 1)))
92	86, 91	mpbird 256	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → (𝑛 − 1) ≤ 𝑠)
93	92	exp31 420	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ ℕ → (𝑛 ∈ ℕ0 → (𝑛 < (𝑠 + 1) → (𝑛 − 1) ≤ 𝑠)))
94	93	ad2antrl 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑛 ∈ ℕ0 → (𝑛 < (𝑠 + 1) → (𝑛 − 1) ≤ 𝑠)))
95	94	imp 407	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (𝑛 < (𝑠 + 1) → (𝑛 − 1) ≤ 𝑠))
96	95	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → (𝑛 < (𝑠 + 1) → (𝑛 − 1) ≤ 𝑠))
97	96	imp 407	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → (𝑛 − 1) ≤ 𝑠)
98		elfz2nn0 13532	. . . . . . . . . . . . 13 ⊢ ((𝑛 − 1) ∈ (0...𝑠) ↔ ((𝑛 − 1) ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ (𝑛 − 1) ≤ 𝑠))
99	84, 85, 97, 98	syl3anbrc 1343	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → (𝑛 − 1) ∈ (0...𝑠))
100	77, 99	ffvelcdmd 7036	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → (𝑏‘(𝑛 − 1)) ∈ 𝐵)
101		df-3an 1089	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑛 − 1)) ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑏‘(𝑛 − 1)) ∈ 𝐵))
102	76, 100, 101	sylanbrc 583	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑛 − 1)) ∈ 𝐵))
103	14, 15, 16, 1, 2	mat2pmatbas 22075	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑛 − 1)) ∈ 𝐵) → (𝑇‘(𝑏‘(𝑛 − 1))) ∈ (Base‘𝑌))
104	102, 103	syl 17	. . . . . . . . 9 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → (𝑇‘(𝑏‘(𝑛 − 1))) ∈ (Base‘𝑌))
105	13	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑌 ∈ Ring)
106	18	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → (𝑇‘𝑀) ∈ (Base‘𝑌))
107	44	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0))
108		simprr 771	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑏 ∈ (𝐵 ↑m (0...𝑠)))
109	108	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑏 ∈ (𝐵 ↑m (0...𝑠)))
110		simplr 767	. . . . . . . . . . . . . . . 16 ⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑛 ∈ ℕ0)
111	22	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑠 ∈ ℕ0)
112		nn0z 12524	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℤ)
113		nnz 12520	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℤ)
114		zleltp1 12554	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ℤ ∧ 𝑠 ∈ ℤ) → (𝑛 ≤ 𝑠 ↔ 𝑛 < (𝑠 + 1)))
115	112, 113, 114	syl2anr 597	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (𝑛 ≤ 𝑠 ↔ 𝑛 < (𝑠 + 1)))
116	115	biimpar 478	. . . . . . . . . . . . . . . 16 ⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑛 ≤ 𝑠)
117		elfz2nn0 13532	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (0...𝑠) ↔ (𝑛 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝑛 ≤ 𝑠))
118	110, 111, 116, 117	syl3anbrc 1343	. . . . . . . . . . . . . . 15 ⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑛 ∈ (0...𝑠))
119	118	exp31 420	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ ℕ → (𝑛 ∈ ℕ0 → (𝑛 < (𝑠 + 1) → 𝑛 ∈ (0...𝑠))))
120	119	ad2antrl 726	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑛 ∈ ℕ0 → (𝑛 < (𝑠 + 1) → 𝑛 ∈ (0...𝑠))))
121	120	imp31 418	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑛 ∈ (0...𝑠))
122	15, 16, 1, 2, 14	m2pmfzmap 22096	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0) ∧ (𝑏 ∈ (𝐵 ↑m (0...𝑠)) ∧ 𝑛 ∈ (0...𝑠))) → (𝑇‘(𝑏‘𝑛)) ∈ (Base‘𝑌))
123	107, 109, 121, 122	syl12anc 835	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → (𝑇‘(𝑏‘𝑛)) ∈ (Base‘𝑌))
124	8, 34	ringcl 19981	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ Ring ∧ (𝑇‘𝑀) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘𝑛)) ∈ (Base‘𝑌)) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))) ∈ (Base‘𝑌))
125	105, 106, 123, 124	syl3anc 1371	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))) ∈ (Base‘𝑌))
126	125	adantlr 713	. . . . . . . . 9 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))) ∈ (Base‘𝑌))
127	8, 37	grpsubcl 18827	. . . . . . . . 9 ⊢ ((𝑌 ∈ Grp ∧ (𝑇‘(𝑏‘(𝑛 − 1))) ∈ (Base‘𝑌) ∧ ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))) ∈ (Base‘𝑌)) → ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))) ∈ (Base‘𝑌))
128	75, 104, 126, 127	syl3anc 1371	. . . . . . . 8 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))) ∈ (Base‘𝑌))
129	128	ex 413	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → (𝑛 < (𝑠 + 1) → ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))) ∈ (Base‘𝑌)))
130	72, 129	syld 47	. . . . . 6 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → ((¬ 𝑛 = (𝑠 + 1) ∧ ¬ (𝑠 + 1) < 𝑛) → ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))) ∈ (Base‘𝑌)))
131	130	impl 456	. . . . 5 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))) ∈ (Base‘𝑌))
132	53, 131	ifclda 4521	. . . 4 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) → if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))) ∈ (Base‘𝑌))
133	52, 132	ifclda 4521	. . 3 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))) ∈ (Base‘𝑌))
134	40, 133	ifclda 4521	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 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))))))) ∈ (Base‘𝑌))
135		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))))))))
136	134, 135	fmptd 7062	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶(Base‘𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ifcif 4486   class class class wbr 5105   ↦ cmpt 5188  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ↑_m cmap 8765  Fincfn 8883  ℝcr 11050  0cc0 11051  1c1 11052   + caddc 11054   < clt 11189   ≤ cle 11190   − cmin 11385  ℕcn 12153  ℕ₀cn0 12413  ℤcz 12499  ℤ_≥cuz 12763  ...cfz 13424  Basecbs 17083  ._rcmulr 17134  0_gc0g 17321  Grpcgrp 18748  -_gcsg 18750  Ringcrg 19964  Poly₁cpl1 21548   Mat cmat 21754   matToPolyMat cmat2pmat 22053

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-ot 4595  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-subrg 20220  df-lmod 20324  df-lss 20393  df-sra 20633  df-rgmod 20634  df-dsmm 21138  df-frlm 21153  df-ascl 21261  df-psr 21311  df-mpl 21313  df-opsr 21315  df-psr1 21551  df-ply1 21553  df-mamu 21733  df-mat 21755  df-mat2pmat 22056

This theorem is referenced by:  chfacfscmulcl  22206  chfacfscmulgsum  22209  chfacfpmmulcl  22210  chfacfpmmulgsum  22213