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Theorem cayhamlem1 21735

Description: Lemma 1 for cayleyhamilton 21759. (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 2734	. . 3 ⊢ (+_g‘𝑌) = (+_g‘𝑌)
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	chfacfpmmulgsum2 21734	. 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 13095	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℤ)
14	13	zcnd 12266	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℂ)
15		pncan1 11239	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ℂ → ((𝑖 + 1) − 1) = 𝑖)
16	14, 15	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (1...𝑠) → ((𝑖 + 1) − 1) = 𝑖)
17	16	eqcomd 2740	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 = ((𝑖 + 1) − 1))
18	17	adantl 485	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑠)) → 𝑖 = ((𝑖 + 1) − 1))
19	18	fveq2d 6710	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑠)) → (𝑏‘𝑖) = (𝑏‘((𝑖 + 1) − 1)))
20	19	fveq2d 6710	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑠)) → (𝑇‘(𝑏‘𝑖)) = (𝑇‘(𝑏‘((𝑖 + 1) − 1))))
21	20	oveq2d 7218	. . . . . . . 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 7218	. . . . . . 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 5139	. . . . . 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 7218	. . . . 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 484	. . . 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 2734	. . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌)
27		crngring 19546	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
28	27	anim2i 620	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
29	28	3adant3 1134	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
30	3, 4	pmatring 21561	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
31	29, 30	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
32		ringabl 19570	. . . . . . 7 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Abel)
33	31, 32	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Abel)
34	33	adantr 484	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ Abel)
35		elnnuz 12461	. . . . . . 7 ⊢ (𝑠 ∈ ℕ ↔ 𝑠 ∈ (ℤ_≥‘1))
36	35	biimpi 219	. . . . . 6 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ (ℤ_≥‘1))
37	36	ad2antrl 728	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑠 ∈ (ℤ_≥‘1))
38	31	adantr 484	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ Ring)
39	38	adantr 484	. . . . . . 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 1134	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
42		eqid 2734	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌)
43	42	ringmgp 19540	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ Ring → (mulGrp‘𝑌) ∈ Mnd)
44	41, 43	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝑌) ∈ Mnd)
45	44	adantr 484	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (mulGrp‘𝑌) ∈ Mnd)
46	45	adantr 484	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (mulGrp‘𝑌) ∈ Mnd)
47		mndmgm 18152	. . . . . . . . 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 13124	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...(𝑠 + 1)) → 𝑘 ∈ ℕ)
50	49	adantl 485	. . . . . . . 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 21595	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
52	27, 51	syl3an2 1166	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
53	52	adantr 484	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑇‘𝑀) ∈ (Base‘𝑌))
54	53	adantr 484	. . . . . . . 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 19482	. . . . . . . . 9 ⊢ (Base‘𝑌) = (Base‘(mulGrp‘𝑌))
56	55, 10	mulgnncl 18479	. . . . . . . 8 ⊢ (((mulGrp‘𝑌) ∈ Mgm ∧ 𝑘 ∈ ℕ ∧ (𝑇‘𝑀) ∈ (Base‘𝑌)) → (𝑘 ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌))
57	48, 50, 54, 56	syl3anc 1373	. . . . . . 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 1193	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑁 ∈ Fin)
59	58	adantr 484	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑁 ∈ Fin)
60	27	3ad2ant2 1136	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
61	60	adantr 484	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑅 ∈ Ring)
62	61	adantr 484	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑅 ∈ Ring)
63		elmapi 8519	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ (𝐵 ↑_m (0...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
64	63	adantl 485	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝑏:(0...𝑠)⟶𝐵)
65	64	adantl 485	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑏:(0...𝑠)⟶𝐵)
66	65	adantr 484	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑏:(0...𝑠)⟶𝐵)
67		nnz 12182	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
68		peano2nn 11825	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℕ)
69	68	nnzd 12264	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℤ)
70		elfzm1b 13173	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℤ ∧ (𝑠 + 1) ∈ ℤ) → (𝑘 ∈ (1...(𝑠 + 1)) ↔ (𝑘 − 1) ∈ (0...((𝑠 + 1) − 1))))
71	67, 69, 70	syl2an 599	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (𝑘 ∈ (1...(𝑠 + 1)) ↔ (𝑘 − 1) ∈ (0...((𝑠 + 1) − 1))))
72		nncn 11821	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℂ)
73		pncan1 11239	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 ∈ ℂ → ((𝑠 + 1) − 1) = 𝑠)
74	72, 73	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 ∈ ℕ → ((𝑠 + 1) − 1) = 𝑠)
75	74	adantl 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → ((𝑠 + 1) − 1) = 𝑠)
76	75	oveq2d 7218	. . . . . . . . . . . . . . . . . 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 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 417	. . . . . . . . . . . . . 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 728	. . . . . . . . . 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 410	. . . . . . . . 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	ffvelrnd 6894	. . . . . . . 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 21595	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑘 − 1)) ∈ 𝐵) → (𝑇‘(𝑏‘(𝑘 − 1))) ∈ (Base‘𝑌))
88	59, 62, 86, 87	syl3anc 1373	. . . . . . 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 19551	. . . . . . 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 1373	. . . . . 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 3098	. . . . 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 7209	. . . . . 6 ⊢ (𝑘 = 𝑖 → (𝑘 ↑ (𝑇‘𝑀)) = (𝑖 ↑ (𝑇‘𝑀)))
93		fvoveq1 7225	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (𝑏‘(𝑘 − 1)) = (𝑏‘(𝑖 − 1)))
94	93	fveq2d 6710	. . . . . 6 ⊢ (𝑘 = 𝑖 → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘(𝑖 − 1))))
95	92, 94	oveq12d 7220	. . . . 5 ⊢ (𝑘 = 𝑖 → ((𝑘 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) = ((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))))
96		oveq1 7209	. . . . . 6 ⊢ (𝑘 = (𝑖 + 1) → (𝑘 ↑ (𝑇‘𝑀)) = ((𝑖 + 1) ↑ (𝑇‘𝑀)))
97		fvoveq1 7225	. . . . . . 7 ⊢ (𝑘 = (𝑖 + 1) → (𝑏‘(𝑘 − 1)) = (𝑏‘((𝑖 + 1) − 1)))
98	97	fveq2d 6710	. . . . . 6 ⊢ (𝑘 = (𝑖 + 1) → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘((𝑖 + 1) − 1))))
99	96, 98	oveq12d 7220	. . . . 5 ⊢ (𝑘 = (𝑖 + 1) → ((𝑘 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) = (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1)))))
100		oveq1 7209	. . . . . 6 ⊢ (𝑘 = 1 → (𝑘 ↑ (𝑇‘𝑀)) = (1 ↑ (𝑇‘𝑀)))
101		fvoveq1 7225	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑏‘(𝑘 − 1)) = (𝑏‘(1 − 1)))
102	101	fveq2d 6710	. . . . . 6 ⊢ (𝑘 = 1 → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘(1 − 1))))
103	100, 102	oveq12d 7220	. . . . 5 ⊢ (𝑘 = 1 → ((𝑘 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) = ((1 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(1 − 1)))))
104		oveq1 7209	. . . . . 6 ⊢ (𝑘 = (𝑠 + 1) → (𝑘 ↑ (𝑇‘𝑀)) = ((𝑠 + 1) ↑ (𝑇‘𝑀)))
105		fvoveq1 7225	. . . . . . 7 ⊢ (𝑘 = (𝑠 + 1) → (𝑏‘(𝑘 − 1)) = (𝑏‘((𝑠 + 1) − 1)))
106	105	fveq2d 6710	. . . . . 6 ⊢ (𝑘 = (𝑠 + 1) → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘((𝑠 + 1) − 1))))
107	104, 106	oveq12d 7220	. . . . 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 19347	. . . 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 2774	. . 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 7217	. 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 18471	. . . . . . . 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 484	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (1 ↑ (𝑇‘𝑀)) = (𝑇‘𝑀))
114		1cnd 10811	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 1 ∈ ℂ)
115	114	subidd 11160	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (1 − 1) = 0)
116	115	fveq2d 6710	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑏‘(1 − 1)) = (𝑏‘0))
117	116	fveq2d 6710	. . . . . 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 7220	. . . . 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 728	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑠 ∈ ℂ)
120	119, 114	pncand 11173	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑠 + 1) − 1) = 𝑠)
121	120	fveq2d 6710	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑏‘((𝑠 + 1) − 1)) = (𝑏‘𝑠))
122	121	fveq2d 6710	. . . . . 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 7218	. . . . 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 7220	. . . 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 7217	. . 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 19539	. . . . . 6 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp)
127	31, 126	syl 17	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Grp)
128	127	adantr 484	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ Grp)
129		nnnn0 12080	. . . . . . . . 9 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℕ₀)
130		0elfz 13192	. . . . . . . . 9 ⊢ (𝑠 ∈ ℕ₀ → 0 ∈ (0...𝑠))
131	129, 130	syl 17	. . . . . . . 8 ⊢ (𝑠 ∈ ℕ → 0 ∈ (0...𝑠))
132	131	ad2antrl 728	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 0 ∈ (0...𝑠))
133	65, 132	ffvelrnd 6894	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑏‘0) ∈ 𝐵)
134	8, 1, 2, 3, 4	mat2pmatbas 21595	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
135	58, 61, 133, 134	syl3anc 1373	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
136	26, 5	ringcl 19551	. . . . 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 1373	. . . 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 771	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑠 ∈ ℕ)
140	139	peano2nnd 11830	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑠 + 1) ∈ ℕ)
141	55, 10	mulgnncl 18479	. . . . . 6 ⊢ (((mulGrp‘𝑌) ∈ Mgm ∧ (𝑠 + 1) ∈ ℕ ∧ (𝑇‘𝑀) ∈ (Base‘𝑌)) → ((𝑠 + 1) ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌))
142	138, 140, 53, 141	syl3anc 1373	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑠 + 1) ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌))
143		nn0fz0 13193	. . . . . . . . 9 ⊢ (𝑠 ∈ ℕ₀ ↔ 𝑠 ∈ (0...𝑠))
144	129, 143	sylib 221	. . . . . . . 8 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ (0...𝑠))
145	144	ad2antrl 728	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑠 ∈ (0...𝑠))
146	65, 145	ffvelrnd 6894	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑏‘𝑠) ∈ 𝐵)
147	8, 1, 2, 3, 4	mat2pmatbas 21595	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘𝑠) ∈ 𝐵) → (𝑇‘(𝑏‘𝑠)) ∈ (Base‘𝑌))
148	58, 61, 146, 147	syl3anc 1373	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑇‘(𝑏‘𝑠)) ∈ (Base‘𝑌))
149	26, 5	ringcl 19551	. . . . 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 1373	. . . 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 18431	. . . 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 837	. . 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 2774	. 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 2778	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 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ifcif 4429 class class class wbr 5043 ↦ cmpt 5124 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 ↑_m cmap 8497 Fincfn 8615 ℂcc 10710 0cc0 10712 1c1 10713 + caddc 10715 < clt 10850 − cmin 11045 ℕcn 11813 ℕ₀cn0 12073 ℤcz 12159 ℤ_≥cuz 12421 ...cfz 13078 Basecbs 16684 +_gcplusg 16767 ._rcmulr 16768 0_gc0g 16916 Σ_g cgsu 16917 Mgmcmgm 18084 Mndcmnd 18145 Grpcgrp 18337 -_gcsg 18339 ._gcmg 18460 Abelcabl 19143 mulGrpcmgp 19476 Ringcrg 19534 CRingccrg 19535 Poly₁cpl1 21070 Mat cmat 21276 matToPolyMat cmat2pmat 21573

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-ot 4540 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-ofr 7459 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-pm 8500 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-sup 9047 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-rp 12570 df-fz 13079 df-fzo 13222 df-seq 13558 df-hash 13880 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-hom 16791 df-cco 16792 df-0g 16918 df-gsum 16919 df-prds 16924 df-pws 16926 df-mre 17061 df-mrc 17062 df-acs 17064 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-mhm 18190 df-submnd 18191 df-grp 18340 df-minusg 18341 df-sbg 18342 df-mulg 18461 df-subg 18512 df-ghm 18592 df-cntz 18683 df-cmn 19144 df-abl 19145 df-mgp 19477 df-ur 19489 df-ring 19536 df-cring 19537 df-subrg 19770 df-lmod 19873 df-lss 19941 df-sra 20181 df-rgmod 20182 df-dsmm 20666 df-frlm 20681 df-ascl 20789 df-psr 20840 df-mpl 20842 df-opsr 20844 df-psr1 21073 df-ply1 21075 df-mamu 21255 df-mat 21277 df-mat2pmat 21576

This theorem is referenced by: cayleyhamilton0 21758 cayleyhamiltonALT 21760

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