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Theorem chfacfpmmulgsum 22335

Description: Breaking up a sum of values of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-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))
chfacfpmmulgsum.p	⊢ + = (+_g‘𝑌)

**Assertion**
Ref	Expression
chfacfpmmulgsum	⊢ (((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))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘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 Allowed substitution hints: 𝐴(𝑖,𝑛,𝑠,𝑏) 𝐵(𝑠,𝑏) 𝑃(𝑖,𝑛,𝑠,𝑏) + (𝑖,𝑛,𝑠,𝑏) 𝑅(𝑠,𝑏) 𝑇(𝑠,𝑏) × (𝑠,𝑏) ↑ (𝑛,𝑠,𝑏) 𝐺(𝑛,𝑠,𝑏) 𝑀(𝑠,𝑏) − (𝑖,𝑠,𝑏) 𝑁(𝑠,𝑏) 𝑌(𝑠,𝑏) 0 (𝑖,𝑠,𝑏)

**Proof of Theorem chfacfpmmulgsum**
Step	Hyp	Ref	Expression
1		eqid 2733	. . 3 ⊢ (Base‘𝑌) = (Base‘𝑌)
2		cayhamlem1.0	. . 3 ⊢ 0 = (0_g‘𝑌)
3		chfacfpmmulgsum.p	. . 3 ⊢ + = (+_g‘𝑌)
4		crngring 20050	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5	4	anim2i 618	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
6	5	3adant3 1133	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
7		cayhamlem1.p	. . . . . . 7 ⊢ 𝑃 = (Poly₁‘𝑅)
8		cayhamlem1.y	. . . . . . 7 ⊢ 𝑌 = (𝑁 Mat 𝑃)
9	7, 8	pmatring 22163	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
10	6, 9	syl 17	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
11		ringcmn 20080	. . . . 5 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ CMnd)
12	10, 11	syl 17	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ CMnd)
13	12	adantr 482	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ CMnd)
14		nn0ex 12465	. . . 4 ⊢ ℕ₀ ∈ V
15	14	a1i 11	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ℕ₀ ∈ V)
16		simpll 766	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ ℕ₀) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵))
17		simplr 768	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ ℕ₀) → (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))))
18		simpr 486	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ ℕ₀) → 𝑖 ∈ ℕ₀)
19	16, 17, 18	3jca 1129	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ ℕ₀) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑖 ∈ ℕ₀))
20		cayhamlem1.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
21		cayhamlem1.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
22		cayhamlem1.r	. . . . 5 ⊢ × = (._r‘𝑌)
23		cayhamlem1.s	. . . . 5 ⊢ − = (-_g‘𝑌)
24		cayhamlem1.t	. . . . 5 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
25		cayhamlem1.g	. . . . 5 ⊢ 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))))))))
26		cayhamlem1.e	. . . . 5 ⊢ ↑ = (._g‘(mulGrp‘𝑌))
27	20, 21, 7, 8, 22, 23, 2, 24, 25, 26	chfacfpmmulcl 22332	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑖 ∈ ℕ₀) → ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)) ∈ (Base‘𝑌))
28	19, 27	syl 17	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ ℕ₀) → ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)) ∈ (Base‘𝑌))
29	20, 21, 7, 8, 22, 23, 2, 24, 25, 26	chfacfpmmulfsupp 22334	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖))) finSupp 0 )
30		nn0disj 13604	. . . 4 ⊢ ((0...(𝑠 + 1)) ∩ (ℤ_≥‘((𝑠 + 1) + 1))) = ∅
31	30	a1i 11	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((0...(𝑠 + 1)) ∩ (ℤ_≥‘((𝑠 + 1) + 1))) = ∅)
32		nnnn0 12466	. . . . . 6 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℕ₀)
33		peano2nn0 12499	. . . . . 6 ⊢ (𝑠 ∈ ℕ₀ → (𝑠 + 1) ∈ ℕ₀)
34	32, 33	syl 17	. . . . 5 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℕ₀)
35		nn0split 13603	. . . . 5 ⊢ ((𝑠 + 1) ∈ ℕ₀ → ℕ₀ = ((0...(𝑠 + 1)) ∪ (ℤ_≥‘((𝑠 + 1) + 1))))
36	34, 35	syl 17	. . . 4 ⊢ (𝑠 ∈ ℕ → ℕ₀ = ((0...(𝑠 + 1)) ∪ (ℤ_≥‘((𝑠 + 1) + 1))))
37	36	ad2antrl 727	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ℕ₀ = ((0...(𝑠 + 1)) ∪ (ℤ_≥‘((𝑠 + 1) + 1))))
38	1, 2, 3, 13, 15, 28, 29, 31, 37	gsumsplit2 19780	. 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 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) + (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖))))))
39		simpll 766	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵))
40		simplr 768	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1))) → (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))))
41		nncn 12207	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℂ)
42		add1p1 12450	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℂ → ((𝑠 + 1) + 1) = (𝑠 + 2))
43	41, 42	syl 17	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℕ → ((𝑠 + 1) + 1) = (𝑠 + 2))
44	43	ad2antrl 727	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑠 + 1) + 1) = (𝑠 + 2))
45	44	fveq2d 6885	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (ℤ_≥‘((𝑠 + 1) + 1)) = (ℤ_≥‘(𝑠 + 2)))
46	45	eleq2d 2820	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↔ 𝑖 ∈ (ℤ_≥‘(𝑠 + 2))))
47	46	biimpa 478	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1))) → 𝑖 ∈ (ℤ_≥‘(𝑠 + 2)))
48	20, 21, 7, 8, 22, 23, 2, 24, 25, 26	chfacfpmmul0 22333	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑖 ∈ (ℤ_≥‘(𝑠 + 2))) → ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)) = 0 )
49	39, 40, 47, 48	syl3anc 1372	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1))) → ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)) = 0 )
50	49	mpteq2dva 5244	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖))) = (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ 0 ))
51	50	oveq2d 7412	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ 0 )))
52	4, 9	sylan2 594	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
53		ringmnd 20048	. . . . . . . . . 10 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Mnd)
54	52, 53	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Mnd)
55	54	3adant3 1133	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Mnd)
56		fvex 6894	. . . . . . . 8 ⊢ (ℤ_≥‘((𝑠 + 1) + 1)) ∈ V
57	55, 56	jctir 522	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑌 ∈ Mnd ∧ (ℤ_≥‘((𝑠 + 1) + 1)) ∈ V))
58	57	adantr 482	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 ∈ Mnd ∧ (ℤ_≥‘((𝑠 + 1) + 1)) ∈ V))
59	2	gsumz 18704	. . . . . 6 ⊢ ((𝑌 ∈ Mnd ∧ (ℤ_≥‘((𝑠 + 1) + 1)) ∈ V) → (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ 0 )) = 0 )
60	58, 59	syl 17	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ 0 )) = 0 )
61	51, 60	eqtrd 2773	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = 0 )
62	61	oveq2d 7412	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) + (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖))))) = ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) + 0 ))
63		fzfid 13925	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (0...(𝑠 + 1)) ∈ Fin)
64		elfznn0 13581	. . . . . . . 8 ⊢ (𝑖 ∈ (0...(𝑠 + 1)) → 𝑖 ∈ ℕ₀)
65	64, 19	sylan2 594	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...(𝑠 + 1))) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑖 ∈ ℕ₀))
66	65, 27	syl 17	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...(𝑠 + 1))) → ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)) ∈ (Base‘𝑌))
67	66	ralrimiva 3147	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ∀𝑖 ∈ (0...(𝑠 + 1))((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)) ∈ (Base‘𝑌))
68	1, 13, 63, 67	gsummptcl 19818	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) ∈ (Base‘𝑌))
69	1, 3, 2	mndrid 18633	. . . 4 ⊢ ((𝑌 ∈ Mnd ∧ (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) ∈ (Base‘𝑌)) → ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) + 0 ) = (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))))
70	55, 68, 69	syl2an2r 684	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) + 0 ) = (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))))
71	62, 70	eqtrd 2773	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) + (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖))))) = (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))))
72	32	ad2antrl 727	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑠 ∈ ℕ₀)
73	1, 3, 13, 72, 66	gsummptfzsplit 19783	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) + (𝑌 Σ_g (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖))))))
74		elfznn0 13581	. . . . . . 7 ⊢ (𝑖 ∈ (0...𝑠) → 𝑖 ∈ ℕ₀)
75	74, 28	sylan2 594	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)) ∈ (Base‘𝑌))
76	1, 3, 13, 72, 75	gsummptfzsplitl 19784	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) + (𝑌 Σ_g (𝑖 ∈ {0} ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖))))))
77	55	adantr 482	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ Mnd)
78		0nn0 12474	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
79	78	a1i 11	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 0 ∈ ℕ₀)
80	20, 21, 7, 8, 22, 23, 2, 24, 25, 26	chfacfpmmulcl 22332	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 0 ∈ ℕ₀) → ((0 ↑ (𝑇‘𝑀)) × (𝐺‘0)) ∈ (Base‘𝑌))
81	79, 80	mpd3an3 1463	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((0 ↑ (𝑇‘𝑀)) × (𝐺‘0)) ∈ (Base‘𝑌))
82		oveq1 7403	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝑖 ↑ (𝑇‘𝑀)) = (0 ↑ (𝑇‘𝑀)))
83		fveq2 6881	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝐺‘𝑖) = (𝐺‘0))
84	82, 83	oveq12d 7414	. . . . . . . 8 ⊢ (𝑖 = 0 → ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)) = ((0 ↑ (𝑇‘𝑀)) × (𝐺‘0)))
85	1, 84	gsumsn 19805	. . . . . . 7 ⊢ ((𝑌 ∈ Mnd ∧ 0 ∈ ℕ₀ ∧ ((0 ↑ (𝑇‘𝑀)) × (𝐺‘0)) ∈ (Base‘𝑌)) → (𝑌 Σ_g (𝑖 ∈ {0} ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = ((0 ↑ (𝑇‘𝑀)) × (𝐺‘0)))
86	77, 79, 81, 85	syl3anc 1372	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ {0} ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = ((0 ↑ (𝑇‘𝑀)) × (𝐺‘0)))
87	86	oveq2d 7412	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) + (𝑌 Σ_g (𝑖 ∈ {0} ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖))))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) + ((0 ↑ (𝑇‘𝑀)) × (𝐺‘0))))
88	76, 87	eqtrd 2773	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) + ((0 ↑ (𝑇‘𝑀)) × (𝐺‘0))))
89		ovexd 7431	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑠 + 1) ∈ V)
90		1nn0 12475	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
91	90	a1i 11	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 1 ∈ ℕ₀)
92	72, 91	nn0addcld 12523	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑠 + 1) ∈ ℕ₀)
93	20, 21, 7, 8, 22, 23, 2, 24, 25, 26	chfacfpmmulcl 22332	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ (𝑠 + 1) ∈ ℕ₀) → (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌))
94	92, 93	mpd3an3 1463	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌))
95		oveq1 7403	. . . . . . 7 ⊢ (𝑖 = (𝑠 + 1) → (𝑖 ↑ (𝑇‘𝑀)) = ((𝑠 + 1) ↑ (𝑇‘𝑀)))
96		fveq2 6881	. . . . . . 7 ⊢ (𝑖 = (𝑠 + 1) → (𝐺‘𝑖) = (𝐺‘(𝑠 + 1)))
97	95, 96	oveq12d 7414	. . . . . 6 ⊢ (𝑖 = (𝑠 + 1) → ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)) = (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝐺‘(𝑠 + 1))))
98	1, 97	gsumsn 19805	. . . . 5 ⊢ ((𝑌 ∈ Mnd ∧ (𝑠 + 1) ∈ V ∧ (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌)) → (𝑌 Σ_g (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝐺‘(𝑠 + 1))))
99	77, 89, 94, 98	syl3anc 1372	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝐺‘(𝑠 + 1))))
100	88, 99	oveq12d 7414	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) + (𝑌 Σ_g (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖))))) = (((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) + ((0 ↑ (𝑇‘𝑀)) × (𝐺‘0))) + (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝐺‘(𝑠 + 1)))))
101		fzfid 13925	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (1...𝑠) ∈ Fin)
102		simpll 766	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵))
103		simplr 768	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))))
104		elfznn 13517	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℕ)
105	104	nnnn0d 12519	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℕ₀)
106	105	adantl 483	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑖 ∈ ℕ₀)
107	102, 103, 106, 27	syl3anc 1372	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)) ∈ (Base‘𝑌))
108	107	ralrimiva 3147	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ∀𝑖 ∈ (1...𝑠)((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)) ∈ (Base‘𝑌))
109	1, 13, 101, 108	gsummptcl 19818	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) ∈ (Base‘𝑌))
110	1, 3	mndass 18621	. . . . 5 ⊢ ((𝑌 ∈ Mnd ∧ ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) ∈ (Base‘𝑌) ∧ ((0 ↑ (𝑇‘𝑀)) × (𝐺‘0)) ∈ (Base‘𝑌) ∧ (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌))) → (((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) + ((0 ↑ (𝑇‘𝑀)) × (𝐺‘0))) + (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝐺‘(𝑠 + 1)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) + (((0 ↑ (𝑇‘𝑀)) × (𝐺‘0)) + (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝐺‘(𝑠 + 1))))))
111	77, 109, 81, 94, 110	syl13anc 1373	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) + ((0 ↑ (𝑇‘𝑀)) × (𝐺‘0))) + (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝐺‘(𝑠 + 1)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) + (((0 ↑ (𝑇‘𝑀)) × (𝐺‘0)) + (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝐺‘(𝑠 + 1))))))
112	104	nnne0d 12249	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ≠ 0)
113	112	ad2antlr 726	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑖 ≠ 0)
114		neeq1 3004	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑖 → (𝑛 ≠ 0 ↔ 𝑖 ≠ 0))
115	114	adantl 483	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑛 ≠ 0 ↔ 𝑖 ≠ 0))
116	113, 115	mpbird 257	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 ≠ 0)
117		eqneqall 2952	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → (𝑛 ≠ 0 → 0 = (𝑇‘(𝑏‘(𝑖 − 1)))))
118	116, 117	mpan9 508	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → 0 = (𝑇‘(𝑏‘(𝑖 − 1))))
119		simplr 768	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → 𝑛 = 𝑖)
120		eqeq1 2737	. . . . . . . . . . . . . . . . 17 ⊢ (0 = 𝑛 → (0 = 𝑖 ↔ 𝑛 = 𝑖))
121	120	eqcoms 2741	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 0 → (0 = 𝑖 ↔ 𝑛 = 𝑖))
122	121	adantl 483	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → (0 = 𝑖 ↔ 𝑛 = 𝑖))
123	119, 122	mpbird 257	. . . . . . . . . . . . . 14 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → 0 = 𝑖)
124	123	fveq2d 6885	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → (𝑏‘0) = (𝑏‘𝑖))
125	124	fveq2d 6885	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → (𝑇‘(𝑏‘0)) = (𝑇‘(𝑏‘𝑖)))
126	125	oveq2d 7412	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) = ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))
127	118, 126	oveq12d 7414	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
128		elfz2 13478	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (1...𝑠) ↔ ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠)))
129		zleltp1 12600	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ ℤ ∧ 𝑠 ∈ ℤ) → (𝑖 ≤ 𝑠 ↔ 𝑖 < (𝑠 + 1)))
130	129	ancoms 460	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ≤ 𝑠 ↔ 𝑖 < (𝑠 + 1)))
131	130	3adant1 1131	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ≤ 𝑠 ↔ 𝑖 < (𝑠 + 1)))
132	131	biimpcd 248	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ≤ 𝑠 → ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → 𝑖 < (𝑠 + 1)))
133	132	adantl 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠) → ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → 𝑖 < (𝑠 + 1)))
134	133	impcom 409	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠)) → 𝑖 < (𝑠 + 1))
135	134	orcd 872	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠)) → (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖))
136		zre 12549	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 ∈ ℤ → 𝑠 ∈ ℝ)
137		1red 11202	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 ∈ ℤ → 1 ∈ ℝ)
138	136, 137	readdcld 11230	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑠 ∈ ℤ → (𝑠 + 1) ∈ ℝ)
139		zre 12549	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℝ)
140	138, 139	anim12ci 615	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ))
141	140	3adant1 1131	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ))
142		lttri2 11283	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ) → (𝑖 ≠ (𝑠 + 1) ↔ (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖)))
143	141, 142	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ≠ (𝑠 + 1) ↔ (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖)))
144	143	adantr 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠)) → (𝑖 ≠ (𝑠 + 1) ↔ (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖)))
145	135, 144	mpbird 257	. . . . . . . . . . . . . . . . . 18 ⊢ (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠)) → 𝑖 ≠ (𝑠 + 1))
146	128, 145	sylbi 216	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ≠ (𝑠 + 1))
147	146	ad2antlr 726	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑖 ≠ (𝑠 + 1))
148		neeq1 3004	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑖 → (𝑛 ≠ (𝑠 + 1) ↔ 𝑖 ≠ (𝑠 + 1)))
149	148	adantl 483	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑛 ≠ (𝑠 + 1) ↔ 𝑖 ≠ (𝑠 + 1)))
150	147, 149	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 ≠ (𝑠 + 1))
151	150	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → 𝑛 ≠ (𝑠 + 1))
152	151	neneqd 2946	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → ¬ 𝑛 = (𝑠 + 1))
153	152	pm2.21d 121	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → (𝑛 = (𝑠 + 1) → (𝑇‘(𝑏‘𝑠)) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))
154	153	imp 408	. . . . . . . . . . 11 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ 𝑛 = (𝑠 + 1)) → (𝑇‘(𝑏‘𝑠)) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
155	104	nnred 12214	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℝ)
156		eleq1w 2817	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑖 → (𝑛 ∈ ℝ ↔ 𝑖 ∈ ℝ))
157	155, 156	syl5ibrcom 246	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (1...𝑠) → (𝑛 = 𝑖 → 𝑛 ∈ ℝ))
158	157	adantl 483	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑛 = 𝑖 → 𝑛 ∈ ℝ))
159	158	imp 408	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 ∈ ℝ)
160	72	nn0red 12520	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑠 ∈ ℝ)
161	160	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑠 ∈ ℝ)
162		1red 11202	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 1 ∈ ℝ)
163	161, 162	readdcld 11230	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑠 + 1) ∈ ℝ)
164	128, 134	sylbi 216	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 < (𝑠 + 1))
165	164	ad2antlr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑖 < (𝑠 + 1))
166		breq1 5147	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑖 → (𝑛 < (𝑠 + 1) ↔ 𝑖 < (𝑠 + 1)))
167	166	adantl 483	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑛 < (𝑠 + 1) ↔ 𝑖 < (𝑠 + 1)))
168	165, 167	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 < (𝑠 + 1))
169	159, 163, 168	ltnsymd 11350	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → ¬ (𝑠 + 1) < 𝑛)
170	169	pm2.21d 121	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → ((𝑠 + 1) < 𝑛 → 0 = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))
171	170	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) → ((𝑠 + 1) < 𝑛 → 0 = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))
172	171	imp 408	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ (𝑠 + 1) < 𝑛) → 0 = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
173		simp-4r 783	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 = 𝑖)
174	173	fvoveq1d 7418	. . . . . . . . . . . . . 14 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑏‘(𝑛 − 1)) = (𝑏‘(𝑖 − 1)))
175	174	fveq2d 6885	. . . . . . . . . . . . 13 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑇‘(𝑏‘(𝑛 − 1))) = (𝑇‘(𝑏‘(𝑖 − 1))))
176	173	fveq2d 6885	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑏‘𝑛) = (𝑏‘𝑖))
177	176	fveq2d 6885	. . . . . . . . . . . . . 14 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑇‘(𝑏‘𝑛)) = (𝑇‘(𝑏‘𝑖)))
178	177	oveq2d 7412	. . . . . . . . . . . . 13 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))) = ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))
179	175, 178	oveq12d 7414	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
180	172, 179	ifeqda 4560	. . . . . . . . . . 11 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) → if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
181	154, 180	ifeqda 4560	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
182	127, 181	ifeqda 4560	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 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))))))) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
183		ovexd 7431	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))) ∈ V)
184	25, 182, 106, 183	fvmptd2 6995	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝐺‘𝑖) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
185	184	oveq2d 7412	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)) = ((𝑖 ↑ (𝑇‘𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))
186	185	mpteq2dva 5244	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖))) = (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))))
187	186	oveq2d 7412	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))))
188		nn0p1gt0 12488	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ ℕ₀ → 0 < (𝑠 + 1))
189		0red 11204	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℕ₀ → 0 ∈ ℝ)
190		ltne 11298	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 0 < (𝑠 + 1)) → (𝑠 + 1) ≠ 0)
191	189, 190	sylan 581	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℕ₀ ∧ 0 < (𝑠 + 1)) → (𝑠 + 1) ≠ 0)
192		neeq1 3004	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑠 + 1) → (𝑛 ≠ 0 ↔ (𝑠 + 1) ≠ 0))
193	191, 192	syl5ibrcom 246	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℕ₀ ∧ 0 < (𝑠 + 1)) → (𝑛 = (𝑠 + 1) → 𝑛 ≠ 0))
194	32, 188, 193	syl2anc2 586	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℕ → (𝑛 = (𝑠 + 1) → 𝑛 ≠ 0))
195	194	ad2antrl 727	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑛 = (𝑠 + 1) → 𝑛 ≠ 0))
196	195	imp 408	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) → 𝑛 ≠ 0)
197		eqneqall 2952	. . . . . . . . . . 11 ⊢ (𝑛 = 0 → (𝑛 ≠ 0 → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = (𝑇‘(𝑏‘𝑠))))
198	196, 197	mpan9 508	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) ∧ 𝑛 = 0) → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = (𝑇‘(𝑏‘𝑠)))
199		iftrue 4530	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑠 + 1) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))) = (𝑇‘(𝑏‘𝑠)))
200	199	ad2antlr 726	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) ∧ ¬ 𝑛 = 0) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))) = (𝑇‘(𝑏‘𝑠)))
201	198, 200	ifeqda 4560	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) → 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))))))) = (𝑇‘(𝑏‘𝑠)))
202	72, 33	syl 17	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑠 + 1) ∈ ℕ₀)
203		fvexd 6896	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑇‘(𝑏‘𝑠)) ∈ V)
204	25, 201, 202, 203	fvmptd2 6995	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝐺‘(𝑠 + 1)) = (𝑇‘(𝑏‘𝑠)))
205	204	oveq2d 7412	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝐺‘(𝑠 + 1))) = (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))))
206	24, 20, 21, 7, 8	mat2pmatbas 22197	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
207	4, 206	syl3an2 1165	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
208		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌)
209	208, 1	mgpbas 19976	. . . . . . . . . . . . 13 ⊢ (Base‘𝑌) = (Base‘(mulGrp‘𝑌))
210		eqid 2733	. . . . . . . . . . . . 13 ⊢ (0_g‘(mulGrp‘𝑌)) = (0_g‘(mulGrp‘𝑌))
211	209, 210, 26	mulg0 18942	. . . . . . . . . . . 12 ⊢ ((𝑇‘𝑀) ∈ (Base‘𝑌) → (0 ↑ (𝑇‘𝑀)) = (0_g‘(mulGrp‘𝑌)))
212	207, 211	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (0 ↑ (𝑇‘𝑀)) = (0_g‘(mulGrp‘𝑌)))
213		eqid 2733	. . . . . . . . . . . 12 ⊢ (1_r‘𝑌) = (1_r‘𝑌)
214	208, 213	ringidval 19989	. . . . . . . . . . 11 ⊢ (1_r‘𝑌) = (0_g‘(mulGrp‘𝑌))
215	212, 214	eqtr4di 2791	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (0 ↑ (𝑇‘𝑀)) = (1_r‘𝑌))
216	215	adantr 482	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (0 ↑ (𝑇‘𝑀)) = (1_r‘𝑌))
217	216	oveq1d 7411	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((0 ↑ (𝑇‘𝑀)) × (𝐺‘0)) = ((1_r‘𝑌) × (𝐺‘0)))
218	52	3adant3 1133	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
219	20, 21, 7, 8, 22, 23, 2, 24, 25	chfacfisf 22325	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺:ℕ₀⟶(Base‘𝑌))
220	4, 219	syl3anl2 1414	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺:ℕ₀⟶(Base‘𝑌))
221	220, 79	ffvelcdmd 7075	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝐺‘0) ∈ (Base‘𝑌))
222	1, 22, 213	ringlidm 20067	. . . . . . . . 9 ⊢ ((𝑌 ∈ Ring ∧ (𝐺‘0) ∈ (Base‘𝑌)) → ((1_r‘𝑌) × (𝐺‘0)) = (𝐺‘0))
223	218, 221, 222	syl2an2r 684	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((1_r‘𝑌) × (𝐺‘0)) = (𝐺‘0))
224		iftrue 4530	. . . . . . . . 9 ⊢ (𝑛 = 0 → 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))))))) = ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
225		ovexd 7431	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) ∈ V)
226	25, 224, 79, 225	fvmptd3 7010	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝐺‘0) = ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
227	217, 223, 226	3eqtrd 2777	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((0 ↑ (𝑇‘𝑀)) × (𝐺‘0)) = ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
228	205, 227	oveq12d 7414	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝐺‘(𝑠 + 1))) + ((0 ↑ (𝑇‘𝑀)) × (𝐺‘0))) = ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) + ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
229	1, 3	cmncom 19650	. . . . . . 7 ⊢ ((𝑌 ∈ CMnd ∧ ((0 ↑ (𝑇‘𝑀)) × (𝐺‘0)) ∈ (Base‘𝑌) ∧ (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌)) → (((0 ↑ (𝑇‘𝑀)) × (𝐺‘0)) + (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝐺‘(𝑠 + 1)))) = ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝐺‘(𝑠 + 1))) + ((0 ↑ (𝑇‘𝑀)) × (𝐺‘0))))
230	13, 81, 94, 229	syl3anc 1372	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((0 ↑ (𝑇‘𝑀)) × (𝐺‘0)) + (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝐺‘(𝑠 + 1)))) = ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝐺‘(𝑠 + 1))) + ((0 ↑ (𝑇‘𝑀)) × (𝐺‘0))))
231		ringgrp 20043	. . . . . . . . 9 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp)
232	10, 231	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Grp)
233	232	adantr 482	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ Grp)
234	205, 94	eqeltrrd 2835	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) ∈ (Base‘𝑌))
235	10	adantr 482	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ Ring)
236	207	adantr 482	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑇‘𝑀) ∈ (Base‘𝑌))
237		simpl1 1192	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑁 ∈ Fin)
238	4	3ad2ant2 1135	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
239	238	adantr 482	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑅 ∈ Ring)
240		elmapi 8831	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ (𝐵 ↑_m (0...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
241	240	adantl 483	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝑏:(0...𝑠)⟶𝐵)
242	241	adantl 483	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑏:(0...𝑠)⟶𝐵)
243		0elfz 13585	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℕ₀ → 0 ∈ (0...𝑠))
244	32, 243	syl 17	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ℕ → 0 ∈ (0...𝑠))
245	244	ad2antrl 727	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 0 ∈ (0...𝑠))
246	242, 245	ffvelcdmd 7075	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑏‘0) ∈ 𝐵)
247	24, 20, 21, 7, 8	mat2pmatbas 22197	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
248	237, 239, 246, 247	syl3anc 1372	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
249	1, 22	ringcl 20055	. . . . . . . 8 ⊢ ((𝑌 ∈ Ring ∧ (𝑇‘𝑀) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌)) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
250	235, 236, 248, 249	syl3anc 1372	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
251	1, 2, 23, 3	grpsubadd0sub 18897	. . . . . . 7 ⊢ ((𝑌 ∈ Grp ∧ (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) ∈ (Base‘𝑌) ∧ ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌)) → ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) + ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
252	233, 234, 250, 251	syl3anc 1372	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) + ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
253	228, 230, 252	3eqtr4d 2783	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((0 ↑ (𝑇‘𝑀)) × (𝐺‘0)) + (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝐺‘(𝑠 + 1)))) = ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
254	187, 253	oveq12d 7414	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) + (((0 ↑ (𝑇‘𝑀)) × (𝐺‘0)) + (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝐺‘(𝑠 + 1))))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
255	111, 254	eqtrd 2773	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) + ((0 ↑ (𝑇‘𝑀)) × (𝐺‘0))) + (((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝐺‘(𝑠 + 1)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
256	73, 100, 255	3eqtrd 2777	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
257	38, 71, 256	3eqtrd 2777	1 ⊢ (((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))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ∪ cun 3944 ∩ cin 3945 ∅c0 4320 ifcif 4524 {csn 4624 class class class wbr 5144 ↦ cmpt 5227 ⟶wf 6531 ‘cfv 6535 (class class class)co 7396 ↑_m cmap 8808 Fincfn 8927 ℂcc 11095 ℝcr 11096 0cc0 11097 1c1 11098 + caddc 11100 < clt 11235 ≤ cle 11236 − cmin 11431 ℕcn 12199 2c2 12254 ℕ₀cn0 12459 ℤcz 12545 ℤ_≥cuz 12809 ...cfz 13471 Basecbs 17131 +_gcplusg 17184 ._rcmulr 17185 0_gc0g 17372 Σ_g cgsu 17373 Mndcmnd 18612 Grpcgrp 18806 -_gcsg 18808 ._gcmg 18935 CMndccmn 19632 mulGrpcmgp 19970 1_rcur 19987 Ringcrg 20038 CRingccrg 20039 Poly₁cpl1 21670 Mat cmat 21876 matToPolyMat cmat2pmat 22175

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4905 df-int 4947 df-iun 4995 df-iin 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7657 df-ofr 7658 df-om 7843 df-1st 7962 df-2nd 7963 df-supp 8134 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8691 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9350 df-sup 9424 df-oi 9492 df-card 9921 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-z 12546 df-dec 12665 df-uz 12810 df-rp 12962 df-fz 13472 df-fzo 13615 df-seq 13954 df-hash 14278 df-struct 17067 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-plusg 17197 df-mulr 17198 df-sca 17200 df-vsca 17201 df-ip 17202 df-tset 17203 df-ple 17204 df-ds 17206 df-hom 17208 df-cco 17209 df-0g 17374 df-gsum 17375 df-prds 17380 df-pws 17382 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-mhm 18658 df-submnd 18659 df-grp 18809 df-minusg 18810 df-sbg 18811 df-mulg 18936 df-subg 18988 df-ghm 19075 df-cntz 19166 df-cmn 19634 df-abl 19635 df-mgp 19971 df-ur 19988 df-ring 20040 df-cring 20041 df-subrg 20338 df-lmod 20450 df-lss 20520 df-sra 20762 df-rgmod 20763 df-dsmm 21260 df-frlm 21275 df-ascl 21383 df-psr 21433 df-mpl 21435 df-opsr 21437 df-psr1 21673 df-ply1 21675 df-mamu 21855 df-mat 21877 df-mat2pmat 22178

This theorem is referenced by: chfacfpmmulgsum2 22336

W3C validator