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 22367

Description: Lemma 1 for cayleyhamilton 22391. (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 2732	. . 3 ⊢ (+_g‘𝑌) = (+_g‘𝑌)
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	chfacfpmmulgsum2 22366	. 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 13500	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℤ)
14	13	zcnd 12666	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℂ)
15		pncan1 11637	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ℂ → ((𝑖 + 1) − 1) = 𝑖)
16	14, 15	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (1...𝑠) → ((𝑖 + 1) − 1) = 𝑖)
17	16	eqcomd 2738	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 = ((𝑖 + 1) − 1))
18	17	adantl 482	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑠)) → 𝑖 = ((𝑖 + 1) − 1))
19	18	fveq2d 6895	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑠)) → (𝑏‘𝑖) = (𝑏‘((𝑖 + 1) − 1)))
20	19	fveq2d 6895	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑠)) → (𝑇‘(𝑏‘𝑖)) = (𝑇‘(𝑏‘((𝑖 + 1) − 1))))
21	20	oveq2d 7424	. . . . . . . 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 7424	. . . . . . 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 5248	. . . . . 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 7424	. . . . 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 481	. . . 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 2732	. . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌)
27		crngring 20067	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
28	27	anim2i 617	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
29	28	3adant3 1132	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
30	3, 4	pmatring 22193	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
31	29, 30	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
32		ringabl 20097	. . . . . . 7 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Abel)
33	31, 32	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Abel)
34	33	adantr 481	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ Abel)
35		elnnuz 12865	. . . . . . 7 ⊢ (𝑠 ∈ ℕ ↔ 𝑠 ∈ (ℤ_≥‘1))
36	35	biimpi 215	. . . . . 6 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ (ℤ_≥‘1))
37	36	ad2antrl 726	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑠 ∈ (ℤ_≥‘1))
38	31	adantr 481	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ Ring)
39	38	adantr 481	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑌 ∈ Ring)
40	28, 30	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
41	40	3adant3 1132	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
42		eqid 2732	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌)
43	42	ringmgp 20061	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ Ring → (mulGrp‘𝑌) ∈ Mnd)
44	41, 43	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝑌) ∈ Mnd)
45	44	adantr 481	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (mulGrp‘𝑌) ∈ Mnd)
46	45	adantr 481	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (mulGrp‘𝑌) ∈ Mnd)
47		mndmgm 18631	. . . . . . . . 9 ⊢ ((mulGrp‘𝑌) ∈ Mnd → (mulGrp‘𝑌) ∈ Mgm)
48	46, 47	syl 17	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (mulGrp‘𝑌) ∈ Mgm)
49		elfznn 13529	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...(𝑠 + 1)) → 𝑘 ∈ ℕ)
50	49	adantl 482	. . . . . . . 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 22227	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
52	27, 51	syl3an2 1164	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
53	52	adantr 481	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑇‘𝑀) ∈ (Base‘𝑌))
54	53	adantr 481	. . . . . . . 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 19992	. . . . . . . . 9 ⊢ (Base‘𝑌) = (Base‘(mulGrp‘𝑌))
56	55, 10	mulgnncl 18968	. . . . . . . 8 ⊢ (((mulGrp‘𝑌) ∈ Mgm ∧ 𝑘 ∈ ℕ ∧ (𝑇‘𝑀) ∈ (Base‘𝑌)) → (𝑘 ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌))
57	48, 50, 54, 56	syl3anc 1371	. . . . . . 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 1191	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑁 ∈ Fin)
59	58	adantr 481	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑁 ∈ Fin)
60	27	3ad2ant2 1134	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
61	60	adantr 481	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑅 ∈ Ring)
62	61	adantr 481	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑅 ∈ Ring)
63		elmapi 8842	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ (𝐵 ↑_m (0...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
64	63	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝑏:(0...𝑠)⟶𝐵)
65	64	adantl 482	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑏:(0...𝑠)⟶𝐵)
66	65	adantr 481	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑏:(0...𝑠)⟶𝐵)
67		nnz 12578	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
68		peano2nn 12223	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℕ)
69	68	nnzd 12584	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℤ)
70		elfzm1b 13578	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℤ ∧ (𝑠 + 1) ∈ ℤ) → (𝑘 ∈ (1...(𝑠 + 1)) ↔ (𝑘 − 1) ∈ (0...((𝑠 + 1) − 1))))
71	67, 69, 70	syl2an 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (𝑘 ∈ (1...(𝑠 + 1)) ↔ (𝑘 − 1) ∈ (0...((𝑠 + 1) − 1))))
72		nncn 12219	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℂ)
73		pncan1 11637	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 ∈ ℂ → ((𝑠 + 1) − 1) = 𝑠)
74	72, 73	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 ∈ ℕ → ((𝑠 + 1) − 1) = 𝑠)
75	74	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → ((𝑠 + 1) − 1) = 𝑠)
76	75	oveq2d 7424	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (0...((𝑠 + 1) − 1)) = (0...𝑠))
77	76	eleq2d 2819	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → ((𝑘 − 1) ∈ (0...((𝑠 + 1) − 1)) ↔ (𝑘 − 1) ∈ (0...𝑠)))
78	77	biimpd 228	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → ((𝑘 − 1) ∈ (0...((𝑠 + 1) − 1)) → (𝑘 − 1) ∈ (0...𝑠)))
79	71, 78	sylbid 239	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (𝑘 ∈ (1...(𝑠 + 1)) → (𝑘 − 1) ∈ (0...𝑠)))
80	79	expcom 414	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ ℕ → (𝑘 ∈ ℕ → (𝑘 ∈ (1...(𝑠 + 1)) → (𝑘 − 1) ∈ (0...𝑠))))
81	80	com13 88	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...(𝑠 + 1)) → (𝑘 ∈ ℕ → (𝑠 ∈ ℕ → (𝑘 − 1) ∈ (0...𝑠))))
82	49, 81	mpd 15	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...(𝑠 + 1)) → (𝑠 ∈ ℕ → (𝑘 − 1) ∈ (0...𝑠)))
83	82	com12 32	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ℕ → (𝑘 ∈ (1...(𝑠 + 1)) → (𝑘 − 1) ∈ (0...𝑠)))
84	83	ad2antrl 726	. . . . . . . . . 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 407	. . . . . . . . 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 7087	. . . . . . . 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 22227	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑘 − 1)) ∈ 𝐵) → (𝑇‘(𝑏‘(𝑘 − 1))) ∈ (Base‘𝑌))
88	59, 62, 86, 87	syl3anc 1371	. . . . . . 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 20072	. . . . . . 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 1371	. . . . . 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 3146	. . . . 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 7415	. . . . . 6 ⊢ (𝑘 = 𝑖 → (𝑘 ↑ (𝑇‘𝑀)) = (𝑖 ↑ (𝑇‘𝑀)))
93		fvoveq1 7431	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (𝑏‘(𝑘 − 1)) = (𝑏‘(𝑖 − 1)))
94	93	fveq2d 6895	. . . . . 6 ⊢ (𝑘 = 𝑖 → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘(𝑖 − 1))))
95	92, 94	oveq12d 7426	. . . . 5 ⊢ (𝑘 = 𝑖 → ((𝑘 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) = ((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))))
96		oveq1 7415	. . . . . 6 ⊢ (𝑘 = (𝑖 + 1) → (𝑘 ↑ (𝑇‘𝑀)) = ((𝑖 + 1) ↑ (𝑇‘𝑀)))
97		fvoveq1 7431	. . . . . . 7 ⊢ (𝑘 = (𝑖 + 1) → (𝑏‘(𝑘 − 1)) = (𝑏‘((𝑖 + 1) − 1)))
98	97	fveq2d 6895	. . . . . 6 ⊢ (𝑘 = (𝑖 + 1) → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘((𝑖 + 1) − 1))))
99	96, 98	oveq12d 7426	. . . . 5 ⊢ (𝑘 = (𝑖 + 1) → ((𝑘 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) = (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1)))))
100		oveq1 7415	. . . . . 6 ⊢ (𝑘 = 1 → (𝑘 ↑ (𝑇‘𝑀)) = (1 ↑ (𝑇‘𝑀)))
101		fvoveq1 7431	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑏‘(𝑘 − 1)) = (𝑏‘(1 − 1)))
102	101	fveq2d 6895	. . . . . 6 ⊢ (𝑘 = 1 → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘(1 − 1))))
103	100, 102	oveq12d 7426	. . . . 5 ⊢ (𝑘 = 1 → ((𝑘 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) = ((1 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(1 − 1)))))
104		oveq1 7415	. . . . . 6 ⊢ (𝑘 = (𝑠 + 1) → (𝑘 ↑ (𝑇‘𝑀)) = ((𝑠 + 1) ↑ (𝑇‘𝑀)))
105		fvoveq1 7431	. . . . . . 7 ⊢ (𝑘 = (𝑠 + 1) → (𝑏‘(𝑘 − 1)) = (𝑏‘((𝑠 + 1) − 1)))
106	105	fveq2d 6895	. . . . . 6 ⊢ (𝑘 = (𝑠 + 1) → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘((𝑠 + 1) − 1))))
107	104, 106	oveq12d 7426	. . . . 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 19857	. . . 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 2772	. . 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 7423	. 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 18960	. . . . . . . 8 ⊢ ((𝑇‘𝑀) ∈ (Base‘𝑌) → (1 ↑ (𝑇‘𝑀)) = (𝑇‘𝑀))
112	52, 111	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (1 ↑ (𝑇‘𝑀)) = (𝑇‘𝑀))
113	112	adantr 481	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (1 ↑ (𝑇‘𝑀)) = (𝑇‘𝑀))
114		1cnd 11208	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 1 ∈ ℂ)
115	114	subidd 11558	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (1 − 1) = 0)
116	115	fveq2d 6895	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑏‘(1 − 1)) = (𝑏‘0))
117	116	fveq2d 6895	. . . . . 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 7426	. . . . 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 726	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑠 ∈ ℂ)
120	119, 114	pncand 11571	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑠 + 1) − 1) = 𝑠)
121	120	fveq2d 6895	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑏‘((𝑠 + 1) − 1)) = (𝑏‘𝑠))
122	121	fveq2d 6895	. . . . . 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 7424	. . . . 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 7426	. . . 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 7423	. . 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 20060	. . . . . 6 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp)
127	31, 126	syl 17	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Grp)
128	127	adantr 481	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ Grp)
129		nnnn0 12478	. . . . . . . . 9 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℕ₀)
130		0elfz 13597	. . . . . . . . 9 ⊢ (𝑠 ∈ ℕ₀ → 0 ∈ (0...𝑠))
131	129, 130	syl 17	. . . . . . . 8 ⊢ (𝑠 ∈ ℕ → 0 ∈ (0...𝑠))
132	131	ad2antrl 726	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 0 ∈ (0...𝑠))
133	65, 132	ffvelcdmd 7087	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑏‘0) ∈ 𝐵)
134	8, 1, 2, 3, 4	mat2pmatbas 22227	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
135	58, 61, 133, 134	syl3anc 1371	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
136	26, 5	ringcl 20072	. . . . 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 1371	. . . 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 17	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (mulGrp‘𝑌) ∈ Mgm)
139		simprl 769	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑠 ∈ ℕ)
140	139	peano2nnd 12228	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑠 + 1) ∈ ℕ)
141	55, 10	mulgnncl 18968	. . . . . 6 ⊢ (((mulGrp‘𝑌) ∈ Mgm ∧ (𝑠 + 1) ∈ ℕ ∧ (𝑇‘𝑀) ∈ (Base‘𝑌)) → ((𝑠 + 1) ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌))
142	138, 140, 53, 141	syl3anc 1371	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑠 + 1) ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌))
143		nn0fz0 13598	. . . . . . . . 9 ⊢ (𝑠 ∈ ℕ₀ ↔ 𝑠 ∈ (0...𝑠))
144	129, 143	sylib 217	. . . . . . . 8 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ (0...𝑠))
145	144	ad2antrl 726	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑠 ∈ (0...𝑠))
146	65, 145	ffvelcdmd 7087	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑏‘𝑠) ∈ 𝐵)
147	8, 1, 2, 3, 4	mat2pmatbas 22227	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘𝑠) ∈ 𝐵) → (𝑇‘(𝑏‘𝑠)) ∈ (Base‘𝑌))
148	58, 61, 146, 147	syl3anc 1371	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑇‘(𝑏‘𝑠)) ∈ (Base‘𝑌))
149	26, 5	ringcl 20072	. . . . 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 1371	. . . 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 18918	. . . 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 835	. . 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 2772	. 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 2776	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

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ifcif 4528 class class class wbr 5148 ↦ cmpt 5231 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 ↑_m cmap 8819 Fincfn 8938 ℂcc 11107 0cc0 11109 1c1 11110 + caddc 11112 < clt 11247 − cmin 11443 ℕcn 12211 ℕ₀cn0 12471 ℤcz 12557 ℤ_≥cuz 12821 ...cfz 13483 Basecbs 17143 +_gcplusg 17196 ._rcmulr 17197 0_gc0g 17384 Σ_g cgsu 17385 Mgmcmgm 18558 Mndcmnd 18624 Grpcgrp 18818 -_gcsg 18820 ._gcmg 18949 Abelcabl 19648 mulGrpcmgp 19986 Ringcrg 20055 CRingccrg 20056 Poly₁cpl1 21700 Mat cmat 21906 matToPolyMat cmat2pmat 22205

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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-ofr 7670 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-rp 12974 df-fz 13484 df-fzo 13627 df-seq 13966 df-hash 14290 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-0g 17386 df-gsum 17387 df-prds 17392 df-pws 17394 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-mhm 18670 df-submnd 18671 df-grp 18821 df-minusg 18822 df-sbg 18823 df-mulg 18950 df-subg 19002 df-ghm 19089 df-cntz 19180 df-cmn 19649 df-abl 19650 df-mgp 19987 df-ur 20004 df-ring 20057 df-cring 20058 df-subrg 20316 df-lmod 20472 df-lss 20542 df-sra 20784 df-rgmod 20785 df-dsmm 21286 df-frlm 21301 df-ascl 21409 df-psr 21461 df-mpl 21463 df-opsr 21465 df-psr1 21703 df-ply1 21705 df-mamu 21885 df-mat 21907 df-mat2pmat 22208

This theorem is referenced by: cayleyhamilton0 22390 cayleyhamiltonALT 22392

W3C validator