cayhamlem1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cayhamlem1		Structured version Visualization version GIF version

Theorem cayhamlem1 23239

Description: Lemma 1 for cayleyhamilton 23263. (Contributed by AV, 11-Nov-2019.)

**Hypotheses**
Ref	Expression
cayhamlem1.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
cayhamlem1.b	⊢ 𝐵 = (Base‘𝐴)
cayhamlem1.p	⊢ 𝑃 = (Poly₁‘𝑅)
cayhamlem1.y	⊢ 𝑌 = (𝑁 Mat 𝑃)
cayhamlem1.r	⊢ × = (._r‘𝑌)
cayhamlem1.s	⊢ − = (-_g‘𝑌)
cayhamlem1.0	⊢ 0 = (0_g‘𝑌)
cayhamlem1.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
cayhamlem1.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))))))))
cayhamlem1.e	⊢ ↑ = (._g‘(mulGrp‘𝑌))

**Assertion**
Ref	Expression
cayhamlem1	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = 0 )

Distinct variable groups: 𝐵,𝑛 𝑛,𝑀 𝑛,𝑁 𝑅,𝑛 𝑛,𝑌 𝑛,𝑏 𝑛,𝑠 0 ,𝑛 𝐵,𝑖 𝑖,𝐺 𝑖,𝑀 𝑖,𝑁 𝑅,𝑖 𝑇,𝑖 × ,𝑖 ↑ ,𝑖 𝑖,𝑠 𝑖,𝑏 𝑇,𝑛,𝑖 𝑖,𝑌 × ,𝑛 − ,𝑛,𝑖 Allowed substitution hints: 𝐴(𝑖, 𝑛, 𝑠, 𝑏) 𝐵(𝑠, 𝑏) 𝑃(𝑖, 𝑛, 𝑠, 𝑏) 𝑅(𝑠, 𝑏) 𝑇(𝑠, 𝑏) × (𝑠, 𝑏) ↑ (𝑛, 𝑠, 𝑏) 𝐺(𝑛, 𝑠, 𝑏) 𝑀(𝑠, 𝑏) − (𝑠, 𝑏) 𝑁(𝑠, 𝑏) 𝑌(𝑠, 𝑏) 0 (𝑖, 𝑠, 𝑏)

Proof of Theorem cayhamlem1 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cayhamlem1.a	. . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		cayhamlem1.b	. . 3 ⊢ 𝐵 = (Base‘𝐴)
3		cayhamlem1.p	. . 3 ⊢ 𝑃 = (Poly₁‘𝑅)
4		cayhamlem1.y	. . 3 ⊢ 𝑌 = (𝑁 Mat 𝑃)
5		cayhamlem1.r	. . 3 ⊢ × = (._r‘𝑌)
6		cayhamlem1.s	. . 3 ⊢ − = (-_g‘𝑌)
7		cayhamlem1.0	. . 3 ⊢ 0 = (0_g‘𝑌)
8		cayhamlem1.t	. . 3 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
9		cayhamlem1.g	. . 3 ⊢ 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))))))))
10		cayhamlem1.e	. . 3 ⊢ ↑ = (._g‘(mulGrp‘𝑌))
11		eqid 2763	. . 3 ⊢ (+_g‘𝑌) = (+_g‘𝑌)
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	chfacfpmmulgsum2 23238	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖))))))(+_g‘𝑌)((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
13		elfzelz 13696	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℤ)
14	13	zcnd 12844	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℂ)
15		pncan1 11779	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ℂ → ((𝑖 + 1) − 1) = 𝑖)
16	14, 15	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (1...𝑠) → ((𝑖 + 1) − 1) = 𝑖)
17	16	eqcomd 2769	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 = ((𝑖 + 1) − 1))
18	17	adantl 487	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑠)) → 𝑖 = ((𝑖 + 1) − 1))
19	18	fveq2d 6920	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑠)) → (𝑏‘𝑖) = (𝑏‘((𝑖 + 1) − 1)))
20	19	fveq2d 6920	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑠)) → (𝑇‘(𝑏‘𝑖)) = (𝑇‘(𝑏‘((𝑖 + 1) − 1))))
21	20	oveq2d 7467	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑠)) → (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖))) = (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1)))))
22	21	oveq2d 7467	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑠)) → (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖)))) = (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1))))))
23	22	mpteq2dva 5204	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖))))) = (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1)))))))
24	23	oveq2d 7467	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖)))))) = (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1))))))))
25	24	adantr 486	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖)))))) = (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1))))))))
26		eqid 2763	. . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌)
27		crngring 20523	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
28	27	anim2i 629	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
29	28	3adant3 1150	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
30	3, 4	pmatring 23065	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
31	29, 30	syl 18	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
32		ringabl 20561	. . . . . . 7 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Abel)
33	31, 32	syl 18	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Abel)
34	33	adantr 486	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ Abel)
35		elnnuz 13045	. . . . . . 7 ⊢ (𝑠 ∈ ℕ ↔ 𝑠 ∈ (ℤ_≥‘1))
36	35	biimpi 219	. . . . . 6 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ (ℤ_≥‘1))
37	36	ad2antrl 741	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑠 ∈ (ℤ_≥‘1))
38	31	adantr 486	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ Ring)
39	38	adantr 486	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑌 ∈ Ring)
40	28, 30	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
41	40	3adant3 1150	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
42		eqid 2763	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌)
43	42	ringmgp 20516	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ Ring → (mulGrp‘𝑌) ∈ Mnd)
44	41, 43	syl 18	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝑌) ∈ Mnd)
45	44	adantr 486	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (mulGrp‘𝑌) ∈ Mnd)
46	45	adantr 486	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (mulGrp‘𝑌) ∈ Mnd)
47		mndmgm 18980	. . . . . . . . 9 ⊢ ((mulGrp‘𝑌) ∈ Mnd → (mulGrp‘𝑌) ∈ Mgm)
48	46, 47	syl 18	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (mulGrp‘𝑌) ∈ Mgm)
49		elfznn 13727	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...(𝑠 + 1)) → 𝑘 ∈ ℕ)
50	49	adantl 487	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑘 ∈ ℕ)
51	8, 1, 2, 3, 4	mat2pmatbas 23099	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
52	27, 51	syl3an2 1182	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
53	52	adantr 486	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑇‘𝑀) ∈ (Base‘𝑌))
54	53	adantr 486	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (𝑇‘𝑀) ∈ (Base‘𝑌))
55	42, 26	mgpbas 20416	. . . . . . . . 9 ⊢ (Base‘𝑌) = (Base‘(mulGrp‘𝑌))
56	55, 10	mulgnncl 19349	. . . . . . . 8 ⊢ (((mulGrp‘𝑌) ∈ Mgm ∧ 𝑘 ∈ ℕ ∧ (𝑇‘𝑀) ∈ (Base‘𝑌)) → (𝑘 ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌))
57	48, 50, 54, 56	syl3anc 1399	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (𝑘 ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌))
58		simpl1 1210	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑁 ∈ Fin)
59	58	adantr 486	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑁 ∈ Fin)
60	27	3ad2ant2 1152	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
61	60	adantr 486	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑅 ∈ Ring)
62	61	adantr 486	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑅 ∈ Ring)
63		elmapi 8907	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ (𝐵 ↑_m (0...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
64	63	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝑏:(0...𝑠)⟶𝐵)
65	64	adantl 487	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑏:(0...𝑠)⟶𝐵)
66	65	adantr 486	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑏:(0...𝑠)⟶𝐵)
67		nnz 12754	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
68		peano2nn 12387	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℕ)
69	68	nnzd 12759	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℤ)
70		elfzm1b 13776	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℤ ∧ (𝑠 + 1) ∈ ℤ) → (𝑘 ∈ (1...(𝑠 + 1)) ↔ (𝑘 − 1) ∈ (0...((𝑠 + 1) − 1))))
71	67, 69, 70	syl2an 608	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (𝑘 ∈ (1...(𝑠 + 1)) ↔ (𝑘 − 1) ∈ (0...((𝑠 + 1) − 1))))
72		nncn 12383	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℂ)
73		pncan1 11779	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 ∈ ℂ → ((𝑠 + 1) − 1) = 𝑠)
74	72, 73	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 ∈ ℕ → ((𝑠 + 1) − 1) = 𝑠)
75	74	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → ((𝑠 + 1) − 1) = 𝑠)
76	75	oveq2d 7467	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (0...((𝑠 + 1) − 1)) = (0...𝑠))
77	76	eleq2d 2850	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → ((𝑘 − 1) ∈ (0...((𝑠 + 1) − 1)) ↔ (𝑘 − 1) ∈ (0...𝑠)))
78	77	biimpd 232	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → ((𝑘 − 1) ∈ (0...((𝑠 + 1) − 1)) → (𝑘 − 1) ∈ (0...𝑠)))
79	71, 78	sylbid 243	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (𝑘 ∈ (1...(𝑠 + 1)) → (𝑘 − 1) ∈ (0...𝑠)))
80	79	expcom 419	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ ℕ → (𝑘 ∈ ℕ → (𝑘 ∈ (1...(𝑠 + 1)) → (𝑘 − 1) ∈ (0...𝑠))))
81	80	com13 89	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...(𝑠 + 1)) → (𝑘 ∈ ℕ → (𝑠 ∈ ℕ → (𝑘 − 1) ∈ (0...𝑠))))
82	49, 81	mpd 16	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...(𝑠 + 1)) → (𝑠 ∈ ℕ → (𝑘 − 1) ∈ (0...𝑠)))
83	82	com12 33	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ℕ → (𝑘 ∈ (1...(𝑠 + 1)) → (𝑘 − 1) ∈ (0...𝑠)))
84	83	ad2antrl 741	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑘 ∈ (1...(𝑠 + 1)) → (𝑘 − 1) ∈ (0...𝑠)))
85	84	imp 412	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (𝑘 − 1) ∈ (0...𝑠))
86	66, 85	ffvelcdmd 7117	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (𝑏‘(𝑘 − 1)) ∈ 𝐵)
87	8, 1, 2, 3, 4	mat2pmatbas 23099	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑘 − 1)) ∈ 𝐵) → (𝑇‘(𝑏‘(𝑘 − 1))) ∈ (Base‘𝑌))
88	59, 62, 86, 87	syl3anc 1399	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (𝑇‘(𝑏‘(𝑘 − 1))) ∈ (Base‘𝑌))
89	26, 5	ringcl 20528	. . . . . . 7 ⊢ ((𝑌 ∈ Ring ∧ (𝑘 ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘(𝑘 − 1))) ∈ (Base‘𝑌)) → ((𝑘 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) ∈ (Base‘𝑌))
90	39, 57, 88, 89	syl3anc 1399	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → ((𝑘 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) ∈ (Base‘𝑌))
91	90	ralrimiva 3158	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ∀𝑘 ∈ (1...(𝑠 + 1))((𝑘 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) ∈ (Base‘𝑌))
92		oveq1 7458	. . . . . 6 ⊢ (𝑘 = 𝑖 → (𝑘 ↑ (𝑇‘𝑀)) = (𝑖 ↑ (𝑇‘𝑀)))
93		fvoveq1 7474	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (𝑏‘(𝑘 − 1)) = (𝑏‘(𝑖 − 1)))
94	93	fveq2d 6920	. . . . . 6 ⊢ (𝑘 = 𝑖 → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘(𝑖 − 1))))
95	92, 94	oveq12d 7469	. . . . 5 ⊢ (𝑘 = 𝑖 → ((𝑘 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) = ((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))))
96		oveq1 7458	. . . . . 6 ⊢ (𝑘 = (𝑖 + 1) → (𝑘 ↑ (𝑇‘𝑀)) = ((𝑖 + 1) ↑ (𝑇‘𝑀)))
97		fvoveq1 7474	. . . . . . 7 ⊢ (𝑘 = (𝑖 + 1) → (𝑏‘(𝑘 − 1)) = (𝑏‘((𝑖 + 1) − 1)))
98	97	fveq2d 6920	. . . . . 6 ⊢ (𝑘 = (𝑖 + 1) → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘((𝑖 + 1) − 1))))
99	96, 98	oveq12d 7469	. . . . 5 ⊢ (𝑘 = (𝑖 + 1) → ((𝑘 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) = (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1)))))
100		oveq1 7458	. . . . . 6 ⊢ (𝑘 = 1 → (𝑘 ↑ (𝑇‘𝑀)) = (1 ↑ (𝑇‘𝑀)))
101		fvoveq1 7474	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑏‘(𝑘 − 1)) = (𝑏‘(1 − 1)))
102	101	fveq2d 6920	. . . . . 6 ⊢ (𝑘 = 1 → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘(1 − 1))))
103	100, 102	oveq12d 7469	. . . . 5 ⊢ (𝑘 = 1 → ((𝑘 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) = ((1 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(1 − 1)))))
104		oveq1 7458	. . . . . 6 ⊢ (𝑘 = (𝑠 + 1) → (𝑘 ↑ (𝑇‘𝑀)) = ((𝑠 + 1) ↑ (𝑇‘𝑀)))
105		fvoveq1 7474	. . . . . . 7 ⊢ (𝑘 = (𝑠 + 1) → (𝑏‘(𝑘 − 1)) = (𝑏‘((𝑠 + 1) − 1)))
106	105	fveq2d 6920	. . . . . 6 ⊢ (𝑘 = (𝑠 + 1) → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘((𝑠 + 1) − 1))))
107	104, 106	oveq12d 7469	. . . . 5 ⊢ (𝑘 = (𝑠 + 1) → ((𝑘 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) = (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) − 1)))))
108	26, 34, 6, 37, 91, 95, 99, 103, 107	telgsumfz 20255	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1))))))) = (((1 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(1 − 1)))) − (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) − 1))))))
109	25, 108	eqtrd 2798	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖)))))) = (((1 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(1 − 1)))) − (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) − 1))))))
110	109	oveq1d 7466	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖))))))(+_g‘𝑌)((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))) = ((((1 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(1 − 1)))) − (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) − 1)))))(+_g‘𝑌)((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
111	55, 10	mulg1 19341	. . . . . . . 8 ⊢ ((𝑇‘𝑀) ∈ (Base‘𝑌) → (1 ↑ (𝑇‘𝑀)) = (𝑇‘𝑀))
112	52, 111	syl 18	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (1 ↑ (𝑇‘𝑀)) = (𝑇‘𝑀))
113	112	adantr 486	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (1 ↑ (𝑇‘𝑀)) = (𝑇‘𝑀))
114		1cnd 11341	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 1 ∈ ℂ)
115	114	subidd 11696	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (1 − 1) = 0)
116	115	fveq2d 6920	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑏‘(1 − 1)) = (𝑏‘0))
117	116	fveq2d 6920	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑇‘(𝑏‘(1 − 1))) = (𝑇‘(𝑏‘0)))
118	113, 117	oveq12d 7469	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((1 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(1 − 1)))) = ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))
119	72	ad2antrl 741	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑠 ∈ ℂ)
120	119, 114	pncand 11709	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑠 + 1) − 1) = 𝑠)
121	120	fveq2d 6920	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑏‘((𝑠 + 1) − 1)) = (𝑏‘𝑠))
122	121	fveq2d 6920	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑇‘(𝑏‘((𝑠 + 1) − 1))) = (𝑇‘(𝑏‘𝑠)))
123	122	oveq2d 7467	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) − 1)))) = (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))))
124	118, 123	oveq12d 7469	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((1 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(1 − 1)))) − (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) − 1))))) = (((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) − (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠)))))
125	124	oveq1d 7466	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((((1 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(1 − 1)))) − (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) − 1)))))(+_g‘𝑌)((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))) = ((((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) − (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))))(+_g‘𝑌)((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
126		ringgrp 20515	. . . . . 6 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp)
127	31, 126	syl 18	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Grp)
128	127	adantr 486	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ Grp)
129		nnnn0 12653	. . . . . . . . 9 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℕ₀)
130		0elfz 13798	. . . . . . . . 9 ⊢ (𝑠 ∈ ℕ₀ → 0 ∈ (0...𝑠))
131	129, 130	syl 18	. . . . . . . 8 ⊢ (𝑠 ∈ ℕ → 0 ∈ (0...𝑠))
132	131	ad2antrl 741	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 0 ∈ (0...𝑠))
133	65, 132	ffvelcdmd 7117	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑏‘0) ∈ 𝐵)
134	8, 1, 2, 3, 4	mat2pmatbas 23099	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
135	58, 61, 133, 134	syl3anc 1399	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
136	26, 5	ringcl 20528	. . . . 5 ⊢ ((𝑌 ∈ Ring ∧ (𝑇‘𝑀) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌)) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
137	38, 53, 135, 136	syl3anc 1399	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
138	45, 47	syl 18	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (mulGrp‘𝑌) ∈ Mgm)
139		simprl 783	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑠 ∈ ℕ)
140	139	peano2nnd 12392	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑠 + 1) ∈ ℕ)
141	55, 10	mulgnncl 19349	. . . . . 6 ⊢ (((mulGrp‘𝑌) ∈ Mgm ∧ (𝑠 + 1) ∈ ℕ ∧ (𝑇‘𝑀) ∈ (Base‘𝑌)) → ((𝑠 + 1) ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌))
142	138, 140, 53, 141	syl3anc 1399	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑠 + 1) ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌))
143		nn0fz0 13799	. . . . . . . . 9 ⊢ (𝑠 ∈ ℕ₀ ↔ 𝑠 ∈ (0...𝑠))
144	129, 143	sylib 221	. . . . . . . 8 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ (0...𝑠))
145	144	ad2antrl 741	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑠 ∈ (0...𝑠))
146	65, 145	ffvelcdmd 7117	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑏‘𝑠) ∈ 𝐵)
147	8, 1, 2, 3, 4	mat2pmatbas 23099	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘𝑠) ∈ 𝐵) → (𝑇‘(𝑏‘𝑠)) ∈ (Base‘𝑌))
148	58, 61, 146, 147	syl3anc 1399	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑇‘(𝑏‘𝑠)) ∈ (Base‘𝑌))
149	26, 5	ringcl 20528	. . . . 5 ⊢ ((𝑌 ∈ Ring ∧ ((𝑠 + 1) ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘𝑠)) ∈ (Base‘𝑌)) → (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) ∈ (Base‘𝑌))
150	38, 142, 148, 149	syl3anc 1399	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) ∈ (Base‘𝑌))
151	26, 11, 6, 7	grpnpncan0 19296	. . . 4 ⊢ ((𝑌 ∈ Grp ∧ (((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌) ∧ (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) ∈ (Base‘𝑌))) → ((((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) − (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))))(+_g‘𝑌)((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))) = 0 )
152	128, 137, 150, 151	syl12anc 850	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) − (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))))(+_g‘𝑌)((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))) = 0 )
153	125, 152	eqtrd 2798	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((((1 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(1 − 1)))) − (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) − 1)))))(+_g‘𝑌)((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))) = 0 )
154	12, 110, 153	3eqtrd 2802	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = 0 )

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ifcif 4485 class class class wbr 5108 ↦ cmpt 5192 ⟶wf 6562 ‘cfv 6566 (class class class)co 7451 ↑_m cmap 8885 Fincfn 9011 ℂcc 11237 0cc0 11239 1c1 11240 + caddc 11242 < clt 11382 − cmin 11580 ℕcn 12375 ℕ₀cn0 12646 ℤcz 12733 ℤ_≥cuz 13005 ...cfz 13679 Basecbs 17431 +_gcplusg 17472 ._rcmulr 17473 0_gc0g 17660 Σ_g cgsu 17661 Mgmcmgm 18864 Mndcmnd 18973 Grpcgrp 19194 -_gcsg 19196 ._gcmg 19327 Abelcabl 20046 mulGrpcmgp 20411 Ringcrg 20510 CRingccrg 20511 Poly₁cpl1 22550 Mat cmat 22777 matToPolyMat cmat2pmat 23077

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-ot 4864 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-ofr 7726 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-er 8755 df-map 8887 df-pm 8888 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-sup 9471 df-oi 9541 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-rp 13161 df-fz 13680 df-fzo 13829 df-seq 14185 df-hash 14516 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-hom 17497 df-cco 17498 df-0g 17662 df-gsum 17663 df-prds 17668 df-pws 17670 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-mhm 19028 df-submnd 19029 df-grp 19197 df-minusg 19198 df-sbg 19199 df-mulg 19328 df-subg 19383 df-ghm 19478 df-cntz 19582 df-cmn 20047 df-abl 20048 df-mgp 20412 df-rng 20426 df-ur 20459 df-ring 20512 df-cring 20513 df-subrng 20849 df-subrg 20873 df-lmod 21188 df-lss 21258 df-sra 21499 df-rgmod 21500 df-dsmm 22093 df-frlm 22108 df-ascl 22218 df-psr 22272 df-mpl 22274 df-opsr 22276 df-psr1 22553 df-ply1 22555 df-mamu 22761 df-mat 22778 df-mat2pmat 23080

This theorem is used by: cayleyhamilton0 23262 cayleyhamiltonALT 23264

W3C validator