Theorem chfacfisf 22347

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 22185	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
4	3	3adant3 1132	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
5		ringgrp 20054	. . . . . . 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 2732	. . . . . . . 8 ⊢ (Base‘𝑌) = (Base‘𝑌)
9		chfacfisf.0	. . . . . . . 8 ⊢ 0 = (0g‘𝑌)
10	8, 9	ring0cl 20077	. . . . . . 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 22219	. . . . . . 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 8839	. . . . . . . . . . 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 12475	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℕ0)
23		nn0uz 12860	. . . . . . . . . . . . 13 ⊢ ℕ0 = (ℤ≥‘0)
24	22, 23	eleqtrdi 2843	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ (ℤ≥‘0))
25		eluzfz1 13504	. . . . . . . . . . . 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 7084	. . . . . . . . 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 22219	. . . . . . 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 20066	. . . . . 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 18899	. . . . 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 13505	. . . . . . . . . 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 22240	. . . . . . 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 12477	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℝ)
55	54	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℝ)
56		peano2nn 12220	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℕ)
57	56	nnred 12223	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℝ)
58	57	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (𝑠 + 1) ∈ ℝ)
59	55, 58	lenltd 11356	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (𝑛 ≤ (𝑠 + 1) ↔ ¬ (𝑠 + 1) < 𝑛))
60		nesym 2997	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 + 1) ≠ 𝑛 ↔ ¬ 𝑛 = (𝑠 + 1))
61		ltlen 11311	. . . . . . . . . . . . . . . . . 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 2948	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑛 = 0 → 𝑛 ≠ 0)
79	78	anim2i 617	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → (𝑛 ∈ ℕ0 ∧ 𝑛 ≠ 0))
80		elnnne0 12482	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ0 ∧ 𝑛 ≠ 0))
81	79, 80	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ)
82		nnm1nn0 12509	. . . . . . . . . . . . . . 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 11211	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 1 ∈ ℝ)
89		nnre 12215	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℝ)
90	89	ad2antrr 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑠 ∈ ℝ)
91	87, 88, 90	lesubaddd 11807	. . . . . . . . . . . . . . . . . . 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 13588	. . . . . . . . . . . . 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 7084	. . . . . . . . . . 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 22219	. . . . . . . . . 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 12579	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℤ)
113		nnz 12575	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℤ)
114		zleltp1 12609	. . . . . . . . . . . . . . . . . 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 13588	. . . . . . . . . . . . . . . 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 22240	. . . . . . . . . . . 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 20066	. . . . . . . . . . 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 18899	. . . . . . . . 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 4562	. . . 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 4562	. . 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 4562	. 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 7110	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 2940  ifcif 4527   class class class wbr 5147   ↦ cmpt 5230  ⟶wf 6536  ‘cfv 6540  (class class class)co 7405   ↑_m cmap 8816  Fincfn 8935  ℝcr 11105  0cc0 11106  1c1 11107   + caddc 11109   < clt 11244   ≤ cle 11245   − cmin 11440  ℕcn 12208  ℕ₀cn0 12468  ℤcz 12554  ℤ_≥cuz 12818  ...cfz 13480  Basecbs 17140  ._rcmulr 17194  0_gc0g 17381  Grpcgrp 18815  -_gcsg 18817  Ringcrg 20049  Poly₁cpl1 21692   Mat cmat 21898   matToPolyMat cmat2pmat 22197

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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-ot 4636  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-ofr 7667  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-pm 8819  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-sup 9433  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-fz 13481  df-fzo 13624  df-seq 13963  df-hash 14287  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-sca 17209  df-vsca 17210  df-ip 17211  df-tset 17212  df-ple 17213  df-ds 17215  df-hom 17217  df-cco 17218  df-0g 17383  df-gsum 17384  df-prds 17389  df-pws 17391  df-mre 17526  df-mrc 17527  df-acs 17529  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-mhm 18667  df-submnd 18668  df-grp 18818  df-minusg 18819  df-sbg 18820  df-mulg 18945  df-subg 18997  df-ghm 19084  df-cntz 19175  df-cmn 19644  df-abl 19645  df-mgp 19982  df-ur 19999  df-ring 20051  df-subrg 20353  df-lmod 20465  df-lss 20535  df-sra 20777  df-rgmod 20778  df-dsmm 21278  df-frlm 21293  df-ascl 21401  df-psr 21453  df-mpl 21455  df-opsr 21457  df-psr1 21695  df-ply1 21697  df-mamu 21877  df-mat 21899  df-mat2pmat 22200

This theorem is referenced by:  chfacfscmulcl  22350  chfacfscmulgsum  22353  chfacfpmmulcl  22354  chfacfpmmulgsum  22357