Theorem chfacfisfcpmat 22882

Description: The "characteristic factor function" is a function from the nonnegative integers to constant polynomial matrices. (Contributed by AV, 19-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))))))))
chfacfisfcpmat.s	⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)

Assertion

Ref	Expression
chfacfisfcpmat	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶𝑆)

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

Allowed substitution hints:   𝐴(𝑛,𝑠,𝑏)   𝐵(𝑠,𝑏)   𝑃(𝑛,𝑠,𝑏)   𝑅(𝑠,𝑏)   𝑆(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 chfacfisfcpmat

Step	Hyp	Ref	Expression
1		chfacfisfcpmat.s	. . . . . . . 8 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
2		chfacfisf.p	. . . . . . . 8 ⊢ 𝑃 = (Poly1‘𝑅)
3		chfacfisf.y	. . . . . . . 8 ⊢ 𝑌 = (𝑁 Mat 𝑃)
4	1, 2, 3	cpmatsubgpmat 22747	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝑌))
5	4	3adant3 1132	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑆 ∈ (SubGrp‘𝑌))
6	5	adantr 480	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑆 ∈ (SubGrp‘𝑌))
7		subgsubm 19188	. . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝑌) → 𝑆 ∈ (SubMnd‘𝑌))
8		chfacfisf.0	. . . . . . . 8 ⊢ 0 = (0g‘𝑌)
9	8	subm0cl 18846	. . . . . . 7 ⊢ (𝑆 ∈ (SubMnd‘𝑌) → 0 ∈ 𝑆)
10	5, 7, 9	3syl 18	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 0 ∈ 𝑆)
11	10	adantr 480	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 0 ∈ 𝑆)
12	1, 2, 3	cpmatsrgpmat 22748	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubRing‘𝑌))
13	12	3adant3 1132	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑆 ∈ (SubRing‘𝑌))
14	13	adantr 480	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑆 ∈ (SubRing‘𝑌))
15		chfacfisf.t	. . . . . . . 8 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
16		chfacfisf.a	. . . . . . . 8 ⊢ 𝐴 = (𝑁 Mat 𝑅)
17		chfacfisf.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐴)
18	1, 15, 16, 17	m2cpm 22768	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ 𝑆)
19	18	adantr 480	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑇‘𝑀) ∈ 𝑆)
20		3simpa 1148	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
21		elmapi 8907	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (𝐵 ↑m (0...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
22	21	adantl 481	. . . . . . . . . 10 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑏:(0...𝑠)⟶𝐵)
23		nnnn0 12560	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℕ0)
24		nn0uz 12945	. . . . . . . . . . . . 13 ⊢ ℕ0 = (ℤ≥‘0)
25	23, 24	eleqtrdi 2854	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ (ℤ≥‘0))
26		eluzfz1 13591	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ (ℤ≥‘0) → 0 ∈ (0...𝑠))
27	25, 26	syl 17	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ℕ → 0 ∈ (0...𝑠))
28	27	adantr 480	. . . . . . . . . 10 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 0 ∈ (0...𝑠))
29	22, 28	ffvelcdmd 7119	. . . . . . . . 9 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑏‘0) ∈ 𝐵)
30	20, 29	anim12i 612	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑏‘0) ∈ 𝐵))
31		df-3an 1089	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑏‘0) ∈ 𝐵))
32	30, 31	sylibr 234	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵))
33	1, 15, 16, 17	m2cpm 22768	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵) → (𝑇‘(𝑏‘0)) ∈ 𝑆)
34	32, 33	syl 17	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑇‘(𝑏‘0)) ∈ 𝑆)
35		chfacfisf.r	. . . . . . 7 ⊢ × = (.r‘𝑌)
36	35	subrgmcl 20612	. . . . . 6 ⊢ ((𝑆 ∈ (SubRing‘𝑌) ∧ (𝑇‘𝑀) ∈ 𝑆 ∧ (𝑇‘(𝑏‘0)) ∈ 𝑆) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ 𝑆)
37	14, 19, 34, 36	syl3anc 1371	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ 𝑆)
38		chfacfisf.s	. . . . . 6 ⊢ − = (-g‘𝑌)
39	38	subgsubcl 19177	. . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝑌) ∧ 0 ∈ 𝑆 ∧ ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ 𝑆) → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) ∈ 𝑆)
40	6, 11, 37, 39	syl3anc 1371	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) ∈ 𝑆)
41	40	ad2antrr 725	. . 3 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) ∈ 𝑆)
42		simpl1 1191	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑁 ∈ Fin)
43		simpl2 1192	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑅 ∈ Ring)
44	22	adantl 481	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑏:(0...𝑠)⟶𝐵)
45		eluzfz2 13592	. . . . . . . . . 10 ⊢ (𝑠 ∈ (ℤ≥‘0) → 𝑠 ∈ (0...𝑠))
46	25, 45	syl 17	. . . . . . . . 9 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ (0...𝑠))
47	46	ad2antrl 727	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑠 ∈ (0...𝑠))
48	44, 47	ffvelcdmd 7119	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑏‘𝑠) ∈ 𝐵)
49	1, 15, 16, 17	m2cpm 22768	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘𝑠) ∈ 𝐵) → (𝑇‘(𝑏‘𝑠)) ∈ 𝑆)
50	42, 43, 48, 49	syl3anc 1371	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑇‘(𝑏‘𝑠)) ∈ 𝑆)
51	50	adantr 480	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑏‘𝑠)) ∈ 𝑆)
52	51	ad2antrr 725	. . . 4 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 = (𝑠 + 1)) → (𝑇‘(𝑏‘𝑠)) ∈ 𝑆)
53	11	ad4antr 731	. . . . 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 ∈ 𝑆)
54		nn0re 12562	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℝ)
55	54	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℝ)
56		peano2nn 12305	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℕ)
57	56	nnred 12308	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℝ)
58	57	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (𝑠 + 1) ∈ ℝ)
59	55, 58	lenltd 11436	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (𝑛 ≤ (𝑠 + 1) ↔ ¬ (𝑠 + 1) < 𝑛))
60		nesym 3003	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 + 1) ≠ 𝑛 ↔ ¬ 𝑛 = (𝑠 + 1))
61		ltlen 11391	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ) → (𝑛 < (𝑠 + 1) ↔ (𝑛 ≤ (𝑠 + 1) ∧ (𝑠 + 1) ≠ 𝑛)))
62	54, 57, 61	syl2anr 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (𝑛 < (𝑠 + 1) ↔ (𝑛 ≤ (𝑠 + 1) ∧ (𝑠 + 1) ≠ 𝑛)))
63	62	biimprd 248	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → ((𝑛 ≤ (𝑠 + 1) ∧ (𝑠 + 1) ≠ 𝑛) → 𝑛 < (𝑠 + 1)))
64	63	expcomd 416	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → ((𝑠 + 1) ≠ 𝑛 → (𝑛 ≤ (𝑠 + 1) → 𝑛 < (𝑠 + 1))))
65	60, 64	biimtrrid 243	. . . . . . . . . . . . . 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 260	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (¬ (𝑠 + 1) < 𝑛 → (¬ 𝑛 = (𝑠 + 1) → 𝑛 < (𝑠 + 1))))
68	67	impcomd 411	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → ((¬ 𝑛 = (𝑠 + 1) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 < (𝑠 + 1)))
69	68	ex 412	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℕ → (𝑛 ∈ ℕ0 → ((¬ 𝑛 = (𝑠 + 1) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 < (𝑠 + 1))))
70	69	ad2antrl 727	. . . . . . . . 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 406	. . . . . . . 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 480	. . . . . . 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	5	ad4antr 731	. . . . . . . . 9 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → 𝑆 ∈ (SubGrp‘𝑌))
74	20	ad4antr 731	. . . . . . . . . . 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))
75	22	ad4antlr 732	. . . . . . . . . . . 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)
76		neqne 2954	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑛 = 0 → 𝑛 ≠ 0)
77	76	anim2i 616	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → (𝑛 ∈ ℕ0 ∧ 𝑛 ≠ 0))
78		elnnne0 12567	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ0 ∧ 𝑛 ≠ 0))
79	77, 78	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ)
80		nnm1nn0 12594	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
81	79, 80	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → (𝑛 − 1) ∈ ℕ0)
82	81	ad4ant23 752	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → (𝑛 − 1) ∈ ℕ0)
83	23	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑠 ∈ ℕ0)
84	83	ad4antlr 732	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → 𝑠 ∈ ℕ0)
85	62	simprbda 498	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑛 ≤ (𝑠 + 1))
86	55	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑛 ∈ ℝ)
87		1red 11291	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 1 ∈ ℝ)
88		nnre 12300	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℝ)
89	88	ad2antrr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑠 ∈ ℝ)
90	86, 87, 89	lesubaddd 11887	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → ((𝑛 − 1) ≤ 𝑠 ↔ 𝑛 ≤ (𝑠 + 1)))
91	85, 90	mpbird 257	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → (𝑛 − 1) ≤ 𝑠)
92	91	exp31 419	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ ℕ → (𝑛 ∈ ℕ0 → (𝑛 < (𝑠 + 1) → (𝑛 − 1) ≤ 𝑠)))
93	92	ad2antrl 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑛 ∈ ℕ0 → (𝑛 < (𝑠 + 1) → (𝑛 − 1) ≤ 𝑠)))
94	93	imp 406	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (𝑛 < (𝑠 + 1) → (𝑛 − 1) ≤ 𝑠))
95	94	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → (𝑛 < (𝑠 + 1) → (𝑛 − 1) ≤ 𝑠))
96	95	imp 406	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → (𝑛 − 1) ≤ 𝑠)
97		elfz2nn0 13675	. . . . . . . . . . . . 13 ⊢ ((𝑛 − 1) ∈ (0...𝑠) ↔ ((𝑛 − 1) ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ (𝑛 − 1) ≤ 𝑠))
98	82, 84, 96, 97	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))
99	75, 98	ffvelcdmd 7119	. . . . . . . . . . 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)
100		df-3an 1089	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑛 − 1)) ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑏‘(𝑛 − 1)) ∈ 𝐵))
101	74, 99, 100	sylanbrc 582	. . . . . . . . . 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))
102	1, 15, 16, 17	m2cpm 22768	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑛 − 1)) ∈ 𝐵) → (𝑇‘(𝑏‘(𝑛 − 1))) ∈ 𝑆)
103	101, 102	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))) ∈ 𝑆)
104	14	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑆 ∈ (SubRing‘𝑌))
105	19	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → (𝑇‘𝑀) ∈ 𝑆)
106	20, 83	anim12i 612	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑠 ∈ ℕ0))
107		df-3an 1089	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑠 ∈ ℕ0))
108	106, 107	sylibr 234	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0))
109	108	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0))
110	109	simp1d 1142	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑁 ∈ Fin)
111	109	simp2d 1143	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑅 ∈ Ring)
112	44	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑏:(0...𝑠)⟶𝐵)
113		simplr 768	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑛 ∈ ℕ0)
114	23	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑠 ∈ ℕ0)
115		nn0z 12664	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℤ)
116		nnz 12660	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℤ)
117		zleltp1 12694	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℤ ∧ 𝑠 ∈ ℤ) → (𝑛 ≤ 𝑠 ↔ 𝑛 < (𝑠 + 1)))
118	115, 116, 117	syl2anr 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (𝑛 ≤ 𝑠 ↔ 𝑛 < (𝑠 + 1)))
119	118	biimpar 477	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑛 ≤ 𝑠)
120		elfz2nn0 13675	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (0...𝑠) ↔ (𝑛 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝑛 ≤ 𝑠))
121	113, 114, 119, 120	syl3anbrc 1343	. . . . . . . . . . . . . . . 16 ⊢ (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑛 ∈ (0...𝑠))
122	121	exp31 419	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ ℕ → (𝑛 ∈ ℕ0 → (𝑛 < (𝑠 + 1) → 𝑛 ∈ (0...𝑠))))
123	122	ad2antrl 727	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑛 ∈ ℕ0 → (𝑛 < (𝑠 + 1) → 𝑛 ∈ (0...𝑠))))
124	123	imp31 417	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑛 ∈ (0...𝑠))
125	112, 124	ffvelcdmd 7119	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → (𝑏‘𝑛) ∈ 𝐵)
126	1, 15, 16, 17	m2cpm 22768	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘𝑛) ∈ 𝐵) → (𝑇‘(𝑏‘𝑛)) ∈ 𝑆)
127	110, 111, 125, 126	syl3anc 1371	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → (𝑇‘(𝑏‘𝑛)) ∈ 𝑆)
128	35	subrgmcl 20612	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ (SubRing‘𝑌) ∧ (𝑇‘𝑀) ∈ 𝑆 ∧ (𝑇‘(𝑏‘𝑛)) ∈ 𝑆) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))) ∈ 𝑆)
129	104, 105, 127, 128	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))) ∈ 𝑆)
130	129	adantlr 714	. . . . . . . . 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))) ∈ 𝑆)
131	38	subgsubcl 19177	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubGrp‘𝑌) ∧ (𝑇‘(𝑏‘(𝑛 − 1))) ∈ 𝑆 ∧ ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))) ∈ 𝑆) → ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))) ∈ 𝑆)
132	73, 103, 130, 131	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)))) ∈ 𝑆)
133	132	ex 412	. . . . . . 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)))) ∈ 𝑆))
134	72, 133	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)))) ∈ 𝑆))
135	134	impl 455	. . . . 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)))) ∈ 𝑆)
136	53, 135	ifclda 4583	. . . 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))))) ∈ 𝑆)
137	52, 136	ifclda 4583	. . 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)))))) ∈ 𝑆)
138	41, 137	ifclda 4583	. 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))))))) ∈ 𝑆)
139		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))))))))
140	138, 139	fmptd 7148	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ifcif 4548   class class class wbr 5166   ↦ cmpt 5249  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ↑_m cmap 8884  Fincfn 9003  ℝcr 11183  0cc0 11184  1c1 11185   + caddc 11187   < clt 11324   ≤ cle 11325   − cmin 11520  ℕcn 12293  ℕ₀cn0 12553  ℤcz 12639  ℤ_≥cuz 12903  ...cfz 13567  Basecbs 17258  ._rcmulr 17312  0_gc0g 17499  SubMndcsubmnd 18817  -_gcsg 18975  SubGrpcsubg 19160  Ringcrg 20260  SubRingcsubrg 20595  Poly₁cpl1 22199   Mat cmat 22432   ConstPolyMat ccpmat 22730   matToPolyMat cmat2pmat 22731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-ofr 7715  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-pm 8887  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-sup 9511  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-fzo 13712  df-seq 14053  df-hash 14380  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-hom 17335  df-cco 17336  df-0g 17501  df-gsum 17502  df-prds 17507  df-pws 17509  df-mre 17644  df-mrc 17645  df-acs 17647  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-mhm 18818  df-submnd 18819  df-grp 18976  df-minusg 18977  df-sbg 18978  df-mulg 19108  df-subg 19163  df-ghm 19253  df-cntz 19357  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-srg 20214  df-ring 20262  df-subrng 20572  df-subrg 20597  df-lmod 20882  df-lss 20953  df-sra 21195  df-rgmod 21196  df-dsmm 21775  df-frlm 21790  df-ascl 21898  df-psr 21952  df-mvr 21953  df-mpl 21954  df-opsr 21956  df-psr1 22202  df-vr1 22203  df-ply1 22204  df-coe1 22205  df-mamu 22416  df-mat 22433  df-cpmat 22733  df-mat2pmat 22734

This theorem is referenced by:  cpmadumatpolylem1  22908  cpmadumatpolylem2  22909  cpmadumatpoly  22910  chcoeffeqlem  22912  cayhamlem4  22915