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Theorem cpmadugsumlemF 22808

Description: Lemma F for cpmadugsum 22810. (Contributed by AV, 7-Nov-2019.)

**Hypotheses**
Ref	Expression
cpmadugsum.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
cpmadugsum.b	⊢ 𝐵 = (Base‘𝐴)
cpmadugsum.p	⊢ 𝑃 = (Poly₁‘𝑅)
cpmadugsum.y	⊢ 𝑌 = (𝑁 Mat 𝑃)
cpmadugsum.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
cpmadugsum.x	⊢ 𝑋 = (var₁‘𝑅)
cpmadugsum.e	⊢ ↑ = (._g‘(mulGrp‘𝑃))
cpmadugsum.m	⊢ · = ( ·_𝑠 ‘𝑌)
cpmadugsum.r	⊢ × = (._r‘𝑌)
cpmadugsum.1	⊢ 1 = (1_r‘𝑌)
cpmadugsum.g	⊢ + = (+_g‘𝑌)
cpmadugsum.s	⊢ − = (-_g‘𝑌)

**Assertion**
Ref	Expression
cpmadugsumlemF	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑋 · 1 ) × (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) − ((𝑇‘𝑀) × (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))

Distinct variable groups: 𝐵,𝑖 𝑖,𝑀 𝑖,𝑁 𝑅,𝑖 𝑖,𝑋 𝑖,𝑌 × ,𝑖 · ,𝑖 1 ,𝑖 𝑖,𝑏 𝑖,𝑠 𝑇,𝑖 ↑ ,𝑖 − ,𝑖 Allowed substitution hints: 𝐴(𝑖,𝑠,𝑏) 𝐵(𝑠,𝑏) 𝑃(𝑖,𝑠,𝑏) + (𝑖,𝑠,𝑏) 𝑅(𝑠,𝑏) 𝑇(𝑠,𝑏) · (𝑠,𝑏) × (𝑠,𝑏) 1 (𝑠,𝑏) ↑ (𝑠,𝑏) 𝑀(𝑠,𝑏) − (𝑠,𝑏) 𝑁(𝑠,𝑏) 𝑋(𝑠,𝑏) 𝑌(𝑠,𝑏)

Proof of Theorem cpmadugsumlemF Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnnn0 12509	. . . 4 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℕ₀)
2		cpmadugsum.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
3		cpmadugsum.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
4		cpmadugsum.p	. . . . 5 ⊢ 𝑃 = (Poly₁‘𝑅)
5		cpmadugsum.y	. . . . 5 ⊢ 𝑌 = (𝑁 Mat 𝑃)
6		cpmadugsum.t	. . . . 5 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
7		cpmadugsum.x	. . . . 5 ⊢ 𝑋 = (var₁‘𝑅)
8		cpmadugsum.e	. . . . 5 ⊢ ↑ = (._g‘(mulGrp‘𝑃))
9		cpmadugsum.m	. . . . 5 ⊢ · = ( ·_𝑠 ‘𝑌)
10		cpmadugsum.r	. . . . 5 ⊢ × = (._r‘𝑌)
11		cpmadugsum.1	. . . . 5 ⊢ 1 = (1_r‘𝑌)
12	2, 3, 4, 5, 6, 7, 8, 9, 10, 11	cpmadugsumlemB 22806	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑋 · 1 ) × (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) = (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))))
13	1, 12	sylanr1 680	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑋 · 1 ) × (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) = (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))))
14	2, 3, 4, 5, 6, 7, 8, 9, 10, 11	cpmadugsumlemC 22807	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑇‘𝑀) × (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) = (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))))
15	1, 14	sylanr1 680	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑇‘𝑀) × (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) = (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))))
16	13, 15	oveq12d 7435	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑋 · 1 ) × (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) − ((𝑇‘𝑀) × (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))))) = ((𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) − (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))))
17		nncn 12250	. . . . . . . . 9 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℂ)
18		npcan1 11669	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℂ → ((𝑠 − 1) + 1) = 𝑠)
19	18	eqcomd 2731	. . . . . . . . 9 ⊢ (𝑠 ∈ ℂ → 𝑠 = ((𝑠 − 1) + 1))
20	17, 19	syl 17	. . . . . . . 8 ⊢ (𝑠 ∈ ℕ → 𝑠 = ((𝑠 − 1) + 1))
21	20	oveq2d 7433	. . . . . . 7 ⊢ (𝑠 ∈ ℕ → (0...𝑠) = (0...((𝑠 − 1) + 1)))
22	21	mpteq1d 5243	. . . . . 6 ⊢ (𝑠 ∈ ℕ → (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) = (𝑖 ∈ (0...((𝑠 − 1) + 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))
23	22	oveq2d 7433	. . . . 5 ⊢ (𝑠 ∈ ℕ → (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) = (𝑌 Σ_g (𝑖 ∈ (0...((𝑠 − 1) + 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))))
24	23	ad2antrl 726	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) = (𝑌 Σ_g (𝑖 ∈ (0...((𝑠 − 1) + 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))))
25		eqid 2725	. . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌)
26		cpmadugsum.g	. . . . 5 ⊢ + = (+_g‘𝑌)
27		crngring 20189	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
28	27	anim2i 615	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
29	28	3adant3 1129	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
30	4, 5	pmatring 22624	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
31	29, 30	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
32		ringcmn 20222	. . . . . . 7 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ CMnd)
33	31, 32	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ CMnd)
34	33	adantr 479	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ CMnd)
35		nnm1nn0 12543	. . . . . 6 ⊢ (𝑠 ∈ ℕ → (𝑠 − 1) ∈ ℕ₀)
36	35	ad2antrl 726	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑠 − 1) ∈ ℕ₀)
37		simpll1 1209	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...((𝑠 − 1) + 1))) → 𝑁 ∈ Fin)
38	27	3ad2ant2 1131	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
39	38	adantr 479	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑅 ∈ Ring)
40	39	adantr 479	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...((𝑠 − 1) + 1))) → 𝑅 ∈ Ring)
41		elmapi 8866	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝐵 ↑_m (0...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
42	21	feq2d 6707	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℕ → (𝑏:(0...𝑠)⟶𝐵 ↔ 𝑏:(0...((𝑠 − 1) + 1))⟶𝐵))
43	41, 42	syl5ibcom 244	. . . . . . . . 9 ⊢ (𝑏 ∈ (𝐵 ↑_m (0...𝑠)) → (𝑠 ∈ ℕ → 𝑏:(0...((𝑠 − 1) + 1))⟶𝐵))
44	43	impcom 406	. . . . . . . 8 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝑏:(0...((𝑠 − 1) + 1))⟶𝐵)
45	44	adantl 480	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑏:(0...((𝑠 − 1) + 1))⟶𝐵)
46	45	ffvelcdmda 7091	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...((𝑠 − 1) + 1))) → (𝑏‘𝑖) ∈ 𝐵)
47		elfznn0 13626	. . . . . . . 8 ⊢ (𝑖 ∈ (0...((𝑠 − 1) + 1)) → 𝑖 ∈ ℕ₀)
48	47	adantl 480	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...((𝑠 − 1) + 1))) → 𝑖 ∈ ℕ₀)
49		1nn0 12518	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
50	49	a1i 11	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...((𝑠 − 1) + 1))) → 1 ∈ ℕ₀)
51	48, 50	nn0addcld 12566	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...((𝑠 − 1) + 1))) → (𝑖 + 1) ∈ ℕ₀)
52	2, 3, 6, 4, 5, 25, 9, 8, 7	mat2pmatscmxcl 22672	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝑏‘𝑖) ∈ 𝐵 ∧ (𝑖 + 1) ∈ ℕ₀)) → (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) ∈ (Base‘𝑌))
53	37, 40, 46, 51, 52	syl22anc 837	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...((𝑠 − 1) + 1))) → (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) ∈ (Base‘𝑌))
54	25, 26, 34, 36, 53	gsummptfzsplit 19891	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...((𝑠 − 1) + 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) = ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) + (𝑌 Σ_g (𝑖 ∈ {((𝑠 − 1) + 1)} ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))))
55		ringmnd 20187	. . . . . . . 8 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Mnd)
56	31, 55	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Mnd)
57	56	adantr 479	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ Mnd)
58		ovexd 7452	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑠 − 1) + 1) ∈ V)
59		simpl1 1188	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑁 ∈ Fin)
60		nn0fz0 13631	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ℕ₀ ↔ 𝑠 ∈ (0...𝑠))
61	1, 60	sylib 217	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ (0...𝑠))
62		ffvelcdm 7088	. . . . . . . . . 10 ⊢ ((𝑏:(0...𝑠)⟶𝐵 ∧ 𝑠 ∈ (0...𝑠)) → (𝑏‘𝑠) ∈ 𝐵)
63	41, 61, 62	syl2anr 595	. . . . . . . . 9 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑏‘𝑠) ∈ 𝐵)
64	1	adantr 479	. . . . . . . . . 10 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝑠 ∈ ℕ₀)
65	49	a1i 11	. . . . . . . . . 10 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 1 ∈ ℕ₀)
66	64, 65	nn0addcld 12566	. . . . . . . . 9 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑠 + 1) ∈ ℕ₀)
67	63, 66	jca 510	. . . . . . . 8 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → ((𝑏‘𝑠) ∈ 𝐵 ∧ (𝑠 + 1) ∈ ℕ₀))
68	67	adantl 480	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑏‘𝑠) ∈ 𝐵 ∧ (𝑠 + 1) ∈ ℕ₀))
69	2, 3, 6, 4, 5, 25, 9, 8, 7	mat2pmatscmxcl 22672	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝑏‘𝑠) ∈ 𝐵 ∧ (𝑠 + 1) ∈ ℕ₀)) → (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) ∈ (Base‘𝑌))
70	59, 39, 68, 69	syl21anc 836	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) ∈ (Base‘𝑌))
71		oveq1 7424	. . . . . . . . 9 ⊢ (𝑖 = ((𝑠 − 1) + 1) → (𝑖 + 1) = (((𝑠 − 1) + 1) + 1))
72	71	oveq1d 7432	. . . . . . . 8 ⊢ (𝑖 = ((𝑠 − 1) + 1) → ((𝑖 + 1) ↑ 𝑋) = ((((𝑠 − 1) + 1) + 1) ↑ 𝑋))
73		2fveq3 6899	. . . . . . . 8 ⊢ (𝑖 = ((𝑠 − 1) + 1) → (𝑇‘(𝑏‘𝑖)) = (𝑇‘(𝑏‘((𝑠 − 1) + 1))))
74	72, 73	oveq12d 7435	. . . . . . 7 ⊢ (𝑖 = ((𝑠 − 1) + 1) → (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) = (((((𝑠 − 1) + 1) + 1) ↑ 𝑋) · (𝑇‘(𝑏‘((𝑠 − 1) + 1)))))
75	17, 18	syl 17	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ℕ → ((𝑠 − 1) + 1) = 𝑠)
76	75	oveq1d 7432	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℕ → (((𝑠 − 1) + 1) + 1) = (𝑠 + 1))
77	76	oveq1d 7432	. . . . . . . . 9 ⊢ (𝑠 ∈ ℕ → ((((𝑠 − 1) + 1) + 1) ↑ 𝑋) = ((𝑠 + 1) ↑ 𝑋))
78	75	fveq2d 6898	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℕ → (𝑏‘((𝑠 − 1) + 1)) = (𝑏‘𝑠))
79	78	fveq2d 6898	. . . . . . . . 9 ⊢ (𝑠 ∈ ℕ → (𝑇‘(𝑏‘((𝑠 − 1) + 1))) = (𝑇‘(𝑏‘𝑠)))
80	77, 79	oveq12d 7435	. . . . . . . 8 ⊢ (𝑠 ∈ ℕ → (((((𝑠 − 1) + 1) + 1) ↑ 𝑋) · (𝑇‘(𝑏‘((𝑠 − 1) + 1)))) = (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))))
81	80	ad2antrl 726	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((((𝑠 − 1) + 1) + 1) ↑ 𝑋) · (𝑇‘(𝑏‘((𝑠 − 1) + 1)))) = (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))))
82	74, 81	sylan9eqr 2787	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 = ((𝑠 − 1) + 1)) → (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) = (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))))
83	25, 57, 58, 70, 82	gsumsnd 19911	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ {((𝑠 − 1) + 1)} ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) = (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))))
84	83	oveq2d 7433	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) + (𝑌 Σ_g (𝑖 ∈ {((𝑠 − 1) + 1)} ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) = ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) + (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠)))))
85	24, 54, 84	3eqtrd 2769	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) = ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) + (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠)))))
86	1	ad2antrl 726	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑠 ∈ ℕ₀)
87	4, 5	pmatlmod 22625	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ LMod)
88	28, 87	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ LMod)
89	88	3adant3 1129	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ LMod)
90	89	adantr 479	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ LMod)
91	90	adantr 479	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑌 ∈ LMod)
92		eqid 2725	. . . . . . . . 9 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
93		eqid 2725	. . . . . . . . 9 ⊢ (Base‘𝑃) = (Base‘𝑃)
94	92, 93	mgpbas 20084	. . . . . . . 8 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
95	4	ply1ring 22175	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
96	27, 95	syl 17	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
97	96	3ad2ant2 1131	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ Ring)
98	92	ringmgp 20183	. . . . . . . . . . 11 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
99	97, 98	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝑃) ∈ Mnd)
100	99	adantr 479	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (mulGrp‘𝑃) ∈ Mnd)
101	100	adantr 479	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (mulGrp‘𝑃) ∈ Mnd)
102		elfznn0 13626	. . . . . . . . 9 ⊢ (𝑖 ∈ (0...𝑠) → 𝑖 ∈ ℕ₀)
103	102	adantl 480	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑖 ∈ ℕ₀)
104	7, 4, 93	vr1cl 22145	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
105	27, 104	syl 17	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑋 ∈ (Base‘𝑃))
106	105	3ad2ant2 1131	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑃))
107	106	adantr 479	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑋 ∈ (Base‘𝑃))
108	107	adantr 479	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑋 ∈ (Base‘𝑃))
109	94, 8, 101, 103, 108	mulgnn0cld 19054	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘𝑃))
110	4	ply1crng 22126	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing)
111	110	anim2i 615	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
112	111	3adant3 1129	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
113	5	matsca2 22352	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑃 = (Scalar‘𝑌))
114	112, 113	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 = (Scalar‘𝑌))
115	114	eqcomd 2731	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑌) = 𝑃)
116	115	fveq2d 6898	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃))
117	116	eleq2d 2811	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)) ↔ (𝑖 ↑ 𝑋) ∈ (Base‘𝑃)))
118	117	adantr 479	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)) ↔ (𝑖 ↑ 𝑋) ∈ (Base‘𝑃)))
119	118	adantr 479	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)) ↔ (𝑖 ↑ 𝑋) ∈ (Base‘𝑃)))
120	109, 119	mpbird 256	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)))
121	31	adantr 479	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ Ring)
122	121	adantr 479	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑌 ∈ Ring)
123		simpll1 1209	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑁 ∈ Fin)
124	39	adantr 479	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑅 ∈ Ring)
125		simpll3 1211	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑀 ∈ 𝐵)
126	6, 2, 3, 4, 5	mat2pmatbas 22658	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
127	123, 124, 125, 126	syl3anc 1368	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑇‘𝑀) ∈ (Base‘𝑌))
128	86	adantr 479	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑠 ∈ ℕ₀)
129		simprr 771	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))
130	129	anim1i 613	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑏 ∈ (𝐵 ↑_m (0...𝑠)) ∧ 𝑖 ∈ (0...𝑠)))
131	2, 3, 4, 5, 6	m2pmfzmap 22679	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ₀) ∧ (𝑏 ∈ (𝐵 ↑_m (0...𝑠)) ∧ 𝑖 ∈ (0...𝑠))) → (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌))
132	123, 124, 128, 130, 131	syl31anc 1370	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌))
133	25, 10	ringcl 20194	. . . . . . 7 ⊢ ((𝑌 ∈ Ring ∧ (𝑇‘𝑀) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌)) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))) ∈ (Base‘𝑌))
134	122, 127, 132, 133	syl3anc 1368	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))) ∈ (Base‘𝑌))
135		eqid 2725	. . . . . . 7 ⊢ (Scalar‘𝑌) = (Scalar‘𝑌)
136		eqid 2725	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌))
137	25, 135, 9, 136	lmodvscl 20765	. . . . . 6 ⊢ ((𝑌 ∈ LMod ∧ (𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)) ∧ ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))) ∈ (Base‘𝑌)) → ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))) ∈ (Base‘𝑌))
138	91, 120, 134, 137	syl3anc 1368	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))) ∈ (Base‘𝑌))
139	25, 26, 34, 86, 138	gsummptfzsplitl 19892	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))) + (𝑌 Σ_g (𝑖 ∈ {0} ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))))
140		0nn0 12517	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
141	140	a1i 11	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 0 ∈ ℕ₀)
142		eqid 2725	. . . . . . . . . . . . 13 ⊢ (0_g‘(mulGrp‘𝑃)) = (0_g‘(mulGrp‘𝑃))
143	94, 142, 8	mulg0 19034	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ (Base‘𝑃) → (0 ↑ 𝑋) = (0_g‘(mulGrp‘𝑃)))
144	106, 143	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (0 ↑ 𝑋) = (0_g‘(mulGrp‘𝑃)))
145	144	adantr 479	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (0 ↑ 𝑋) = (0_g‘(mulGrp‘𝑃)))
146	145	oveq1d 7432	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((0 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = ((0_g‘(mulGrp‘𝑃)) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
147		eqid 2725	. . . . . . . . . . . . 13 ⊢ (1_r‘𝑃) = (1_r‘𝑃)
148	92, 147	ringidval 20127	. . . . . . . . . . . 12 ⊢ (1_r‘𝑃) = (0_g‘(mulGrp‘𝑃))
149	148	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (1_r‘𝑃) = (0_g‘(mulGrp‘𝑃)))
150	149	eqcomd 2731	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (0_g‘(mulGrp‘𝑃)) = (1_r‘𝑃))
151	150	oveq1d 7432	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((0_g‘(mulGrp‘𝑃)) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = ((1_r‘𝑃) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
152	114	adantr 479	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑃 = (Scalar‘𝑌))
153	152	fveq2d 6898	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (1_r‘𝑃) = (1_r‘(Scalar‘𝑌)))
154	153	oveq1d 7432	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((1_r‘𝑃) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = ((1_r‘(Scalar‘𝑌)) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
155	27, 126	syl3an2 1161	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
156	155	adantr 479	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑇‘𝑀) ∈ (Base‘𝑌))
157		simpl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑏:(0...𝑠)⟶𝐵 ∧ 𝑠 ∈ ℕ) → 𝑏:(0...𝑠)⟶𝐵)
158		elnn0uz 12897	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 ∈ ℕ₀ ↔ 𝑠 ∈ (ℤ_≥‘0))
159	1, 158	sylib 217	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ (ℤ_≥‘0))
160		eluzfz1 13540	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑠))
161	159, 160	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℕ → 0 ∈ (0...𝑠))
162	161	adantl 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑏:(0...𝑠)⟶𝐵 ∧ 𝑠 ∈ ℕ) → 0 ∈ (0...𝑠))
163	157, 162	ffvelcdmd 7092	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏:(0...𝑠)⟶𝐵 ∧ 𝑠 ∈ ℕ) → (𝑏‘0) ∈ 𝐵)
164	163	ex 411	. . . . . . . . . . . . . . . 16 ⊢ (𝑏:(0...𝑠)⟶𝐵 → (𝑠 ∈ ℕ → (𝑏‘0) ∈ 𝐵))
165	41, 164	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ (𝐵 ↑_m (0...𝑠)) → (𝑠 ∈ ℕ → (𝑏‘0) ∈ 𝐵))
166	165	impcom 406	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑏‘0) ∈ 𝐵)
167	166	adantl 480	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑏‘0) ∈ 𝐵)
168	6, 2, 3, 4, 5	mat2pmatbas 22658	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
169	59, 39, 167, 168	syl3anc 1368	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
170	25, 10	ringcl 20194	. . . . . . . . . . . 12 ⊢ ((𝑌 ∈ Ring ∧ (𝑇‘𝑀) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌)) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
171	121, 156, 169, 170	syl3anc 1368	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
172		eqid 2725	. . . . . . . . . . . 12 ⊢ (1_r‘(Scalar‘𝑌)) = (1_r‘(Scalar‘𝑌))
173	25, 135, 9, 172	lmodvs1 20777	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ LMod ∧ ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌)) → ((1_r‘(Scalar‘𝑌)) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))
174	90, 171, 173	syl2anc 582	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((1_r‘(Scalar‘𝑌)) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))
175	154, 174	eqtrd 2765	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((1_r‘𝑃) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))
176	146, 151, 175	3eqtrd 2769	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((0 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))
177	176, 171	eqeltrd 2825	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((0 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) ∈ (Base‘𝑌))
178		oveq1 7424	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝑖 ↑ 𝑋) = (0 ↑ 𝑋))
179		2fveq3 6899	. . . . . . . . . 10 ⊢ (𝑖 = 0 → (𝑇‘(𝑏‘𝑖)) = (𝑇‘(𝑏‘0)))
180	179	oveq2d 7433	. . . . . . . . 9 ⊢ (𝑖 = 0 → ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))) = ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))
181	178, 180	oveq12d 7435	. . . . . . . 8 ⊢ (𝑖 = 0 → ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))) = ((0 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
182	181	adantl 480	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 = 0) → ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))) = ((0 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
183	25, 57, 141, 177, 182	gsumsnd 19911	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ {0} ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))) = ((0 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
184	94, 148, 8	mulg0 19034	. . . . . . . . 9 ⊢ (𝑋 ∈ (Base‘𝑃) → (0 ↑ 𝑋) = (1_r‘𝑃))
185	106, 184	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (0 ↑ 𝑋) = (1_r‘𝑃))
186	185	adantr 479	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (0 ↑ 𝑋) = (1_r‘𝑃))
187	186	oveq1d 7432	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((0 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = ((1_r‘𝑃) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
188	183, 187, 175	3eqtrd 2769	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ {0} ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))) = ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))
189	188	oveq2d 7433	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))) + (𝑌 Σ_g (𝑖 ∈ {0} ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))) + ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
190	139, 189	eqtrd 2765	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))) + ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
191	85, 190	oveq12d 7435	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) − (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) = (((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) + (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠)))) − ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))) + ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
192		fzfid 13970	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (0...(𝑠 − 1)) ∈ Fin)
193		simpll1 1209	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...(𝑠 − 1))) → 𝑁 ∈ Fin)
194	39	adantr 479	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...(𝑠 − 1))) → 𝑅 ∈ Ring)
195	41	adantl 480	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝑏:(0...𝑠)⟶𝐵)
196	195	adantr 479	. . . . . . . . . 10 ⊢ (((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑖 ∈ (0...(𝑠 − 1))) → 𝑏:(0...𝑠)⟶𝐵)
197		nnz 12609	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℤ)
198		fzoval 13665	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℤ → (0..^𝑠) = (0...(𝑠 − 1)))
199	197, 198	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ ℕ → (0..^𝑠) = (0...(𝑠 − 1)))
200	199	eqcomd 2731	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ ℕ → (0...(𝑠 − 1)) = (0..^𝑠))
201	200	eleq2d 2811	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℕ → (𝑖 ∈ (0...(𝑠 − 1)) ↔ 𝑖 ∈ (0..^𝑠)))
202		elfzofz 13680	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0..^𝑠) → 𝑖 ∈ (0...𝑠))
203	201, 202	biimtrdi 252	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℕ → (𝑖 ∈ (0...(𝑠 − 1)) → 𝑖 ∈ (0...𝑠)))
204	203	adantr 479	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑖 ∈ (0...(𝑠 − 1)) → 𝑖 ∈ (0...𝑠)))
205	204	imp 405	. . . . . . . . . 10 ⊢ (((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑖 ∈ (0...(𝑠 − 1))) → 𝑖 ∈ (0...𝑠))
206	196, 205	ffvelcdmd 7092	. . . . . . . . 9 ⊢ (((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑖 ∈ (0...(𝑠 − 1))) → (𝑏‘𝑖) ∈ 𝐵)
207	206	adantll 712	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...(𝑠 − 1))) → (𝑏‘𝑖) ∈ 𝐵)
208		elfznn0 13626	. . . . . . . . . 10 ⊢ (𝑖 ∈ (0...(𝑠 − 1)) → 𝑖 ∈ ℕ₀)
209	208	adantl 480	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...(𝑠 − 1))) → 𝑖 ∈ ℕ₀)
210	49	a1i 11	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...(𝑠 − 1))) → 1 ∈ ℕ₀)
211	209, 210	nn0addcld 12566	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...(𝑠 − 1))) → (𝑖 + 1) ∈ ℕ₀)
212	193, 194, 207, 211, 52	syl22anc 837	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...(𝑠 − 1))) → (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) ∈ (Base‘𝑌))
213	212	ralrimiva 3136	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ∀𝑖 ∈ (0...(𝑠 − 1))(((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) ∈ (Base‘𝑌))
214	25, 34, 192, 213	gsummptcl 19926	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) ∈ (Base‘𝑌))
215	25, 26	cmncom 19757	. . . . 5 ⊢ ((𝑌 ∈ CMnd ∧ (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) ∈ (Base‘𝑌) ∧ (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) ∈ (Base‘𝑌)) → ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) + (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠)))) = ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))))
216	34, 214, 70, 215	syl3anc 1368	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) + (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠)))) = ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))))
217	216	oveq1d 7432	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) + (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠)))) − ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))) + ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))) = (((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) − ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))) + ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
218		ringgrp 20182	. . . . . . . . 9 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp)
219	31, 218	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Grp)
220	219	adantr 479	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ Grp)
221		fzfid 13970	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (1...𝑠) ∈ Fin)
222	90	adantr 479	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑌 ∈ LMod)
223	100	adantr 479	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (mulGrp‘𝑃) ∈ Mnd)
224		elfznn 13562	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℕ)
225	224	nnnn0d 12562	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℕ₀)
226	225	adantl 480	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑖 ∈ ℕ₀)
227	107	adantr 479	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑋 ∈ (Base‘𝑃))
228	94, 8, 223, 226, 227	mulgnn0cld 19054	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘𝑃))
229	114	fveq2d 6898	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘𝑃) = (Base‘(Scalar‘𝑌)))
230	229	adantr 479	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (Base‘𝑃) = (Base‘(Scalar‘𝑌)))
231	230	adantr 479	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (Base‘𝑃) = (Base‘(Scalar‘𝑌)))
232	228, 231	eleqtrd 2827	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)))
233	121	adantr 479	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑌 ∈ Ring)
234	156	adantr 479	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑇‘𝑀) ∈ (Base‘𝑌))
235		simpll1 1209	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑁 ∈ Fin)
236	39	adantr 479	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑅 ∈ Ring)
237	195	adantl 480	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑏:(0...𝑠)⟶𝐵)
238	237	adantr 479	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
239		1eluzge0 12906	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ (ℤ_≥‘0)
240		fzss1 13572	. . . . . . . . . . . . . . . . 17 ⊢ (1 ∈ (ℤ_≥‘0) → (1...𝑠) ⊆ (0...𝑠))
241	239, 240	mp1i 13	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℕ → (1...𝑠) ⊆ (0...𝑠))
242	241	sseld 3976	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ ℕ → (𝑖 ∈ (1...𝑠) → 𝑖 ∈ (0...𝑠)))
243	242	ad2antrl 726	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑖 ∈ (1...𝑠) → 𝑖 ∈ (0...𝑠)))
244	243	imp 405	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑖 ∈ (0...𝑠))
245	238, 244	ffvelcdmd 7092	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑏‘𝑖) ∈ 𝐵)
246	6, 2, 3, 4, 5	mat2pmatbas 22658	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘𝑖) ∈ 𝐵) → (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌))
247	235, 236, 245, 246	syl3anc 1368	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌))
248	233, 234, 247, 133	syl3anc 1368	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))) ∈ (Base‘𝑌))
249	222, 232, 248, 137	syl3anc 1368	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))) ∈ (Base‘𝑌))
250	249	ralrimiva 3136	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ∀𝑖 ∈ (1...𝑠)((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))) ∈ (Base‘𝑌))
251	25, 34, 221, 250	gsummptcl 19926	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))) ∈ (Base‘𝑌))
252		cpmadugsum.s	. . . . . . . 8 ⊢ − = (-_g‘𝑌)
253	25, 26, 252	grpaddsubass 18990	. . . . . . 7 ⊢ ((𝑌 ∈ Grp ∧ ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) ∈ (Base‘𝑌) ∧ (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) ∈ (Base‘𝑌) ∧ (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))) ∈ (Base‘𝑌))) → (((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) − (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) = ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) − (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))))))
254	220, 70, 214, 251, 253	syl13anc 1369	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) − (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) = ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) − (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))))))
255		oveq1 7424	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑖 → (𝑥 − 1) = (𝑖 − 1))
256	255	oveq1d 7432	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑖 → ((𝑥 − 1) + 1) = ((𝑖 − 1) + 1))
257	256	oveq1d 7432	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑖 → (((𝑥 − 1) + 1) ↑ 𝑋) = (((𝑖 − 1) + 1) ↑ 𝑋))
258	255	fveq2d 6898	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑖 → (𝑏‘(𝑥 − 1)) = (𝑏‘(𝑖 − 1)))
259	258	fveq2d 6898	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑖 → (𝑇‘(𝑏‘(𝑥 − 1))) = (𝑇‘(𝑏‘(𝑖 − 1))))
260	257, 259	oveq12d 7435	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑖 → ((((𝑥 − 1) + 1) ↑ 𝑋) · (𝑇‘(𝑏‘(𝑥 − 1)))) = ((((𝑖 − 1) + 1) ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))))
261	260	cbvmptv 5261	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1...𝑠) ↦ ((((𝑥 − 1) + 1) ↑ 𝑋) · (𝑇‘(𝑏‘(𝑥 − 1))))) = (𝑖 ∈ (1...𝑠) ↦ ((((𝑖 − 1) + 1) ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))))
262	224	nncnd 12258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℂ)
263	262	adantl 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → 𝑖 ∈ ℂ)
264		npcan1 11669	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℂ → ((𝑖 − 1) + 1) = 𝑖)
265	263, 264	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 − 1) + 1) = 𝑖)
266	265	oveq1d 7432	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → (((𝑖 − 1) + 1) ↑ 𝑋) = (𝑖 ↑ 𝑋))
267	266	oveq1d 7432	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → ((((𝑖 − 1) + 1) ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) = ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))))
268	267	mpteq2dva 5248	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℕ → (𝑖 ∈ (1...𝑠) ↦ ((((𝑖 − 1) + 1) ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1))))) = (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1))))))
269	261, 268	eqtrid 2777	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ℕ → (𝑥 ∈ (1...𝑠) ↦ ((((𝑥 − 1) + 1) ↑ 𝑋) · (𝑇‘(𝑏‘(𝑥 − 1))))) = (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1))))))
270	269	oveq2d 7433	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℕ → (𝑌 Σ_g (𝑥 ∈ (1...𝑠) ↦ ((((𝑥 − 1) + 1) ↑ 𝑋) · (𝑇‘(𝑏‘(𝑥 − 1)))))) = (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))))))
271	270	ad2antrl 726	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑥 ∈ (1...𝑠) ↦ ((((𝑥 − 1) + 1) ↑ 𝑋) · (𝑇‘(𝑏‘(𝑥 − 1)))))) = (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))))))
272	271	oveq1d 7432	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑥 ∈ (1...𝑠) ↦ ((((𝑥 − 1) + 1) ↑ 𝑋) · (𝑇‘(𝑏‘(𝑥 − 1)))))) − (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))))) − (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))))
273		eqid 2725	. . . . . . . . . . 11 ⊢ (0_g‘𝑌) = (0_g‘𝑌)
274		1zzd 12623	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 1 ∈ ℤ)
275		0zd 12600	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 0 ∈ ℤ)
276	36	nn0zd 12614	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑠 − 1) ∈ ℤ)
277		oveq1 7424	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑥 − 1) → (𝑖 + 1) = ((𝑥 − 1) + 1))
278	277	oveq1d 7432	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑥 − 1) → ((𝑖 + 1) ↑ 𝑋) = (((𝑥 − 1) + 1) ↑ 𝑋))
279		2fveq3 6899	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑥 − 1) → (𝑇‘(𝑏‘𝑖)) = (𝑇‘(𝑏‘(𝑥 − 1))))
280	278, 279	oveq12d 7435	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑥 − 1) → (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) = ((((𝑥 − 1) + 1) ↑ 𝑋) · (𝑇‘(𝑏‘(𝑥 − 1)))))
281	25, 273, 34, 274, 275, 276, 212, 280	gsummptshft 19895	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) = (𝑌 Σ_g (𝑥 ∈ ((0 + 1)...((𝑠 − 1) + 1)) ↦ ((((𝑥 − 1) + 1) ↑ 𝑋) · (𝑇‘(𝑏‘(𝑥 − 1)))))))
282		0p1e1 12364	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
283	282	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (0 + 1) = 1)
284	75	ad2antrl 726	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑠 − 1) + 1) = 𝑠)
285	283, 284	oveq12d 7435	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((0 + 1)...((𝑠 − 1) + 1)) = (1...𝑠))
286	285	mpteq1d 5243	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑥 ∈ ((0 + 1)...((𝑠 − 1) + 1)) ↦ ((((𝑥 − 1) + 1) ↑ 𝑋) · (𝑇‘(𝑏‘(𝑥 − 1))))) = (𝑥 ∈ (1...𝑠) ↦ ((((𝑥 − 1) + 1) ↑ 𝑋) · (𝑇‘(𝑏‘(𝑥 − 1))))))
287	286	oveq2d 7433	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑥 ∈ ((0 + 1)...((𝑠 − 1) + 1)) ↦ ((((𝑥 − 1) + 1) ↑ 𝑋) · (𝑇‘(𝑏‘(𝑥 − 1)))))) = (𝑌 Σ_g (𝑥 ∈ (1...𝑠) ↦ ((((𝑥 − 1) + 1) ↑ 𝑋) · (𝑇‘(𝑏‘(𝑥 − 1)))))))
288	281, 287	eqtrd 2765	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) = (𝑌 Σ_g (𝑥 ∈ (1...𝑠) ↦ ((((𝑥 − 1) + 1) ↑ 𝑋) · (𝑇‘(𝑏‘(𝑥 − 1)))))))
289	288	oveq1d 7432	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) − (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) = ((𝑌 Σ_g (𝑥 ∈ (1...𝑠) ↦ ((((𝑥 − 1) + 1) ↑ 𝑋) · (𝑇‘(𝑏‘(𝑥 − 1)))))) − (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))))
290		ringabl 20221	. . . . . . . . . . 11 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Abel)
291	31, 290	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Abel)
292	291	adantr 479	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ Abel)
293	224	adantl 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → 𝑖 ∈ ℕ)
294		nnz 12609	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℕ → 𝑖 ∈ ℤ)
295		elfzm1b 13611	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ ℤ ∧ 𝑠 ∈ ℤ) → (𝑖 ∈ (1...𝑠) ↔ (𝑖 − 1) ∈ (0...(𝑠 − 1))))
296	294, 197, 295	syl2an 594	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (𝑖 ∈ (1...𝑠) ↔ (𝑖 − 1) ∈ (0...(𝑠 − 1))))
297	199	adantl 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (0..^𝑠) = (0...(𝑠 − 1)))
298	297	eqcomd 2731	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (0...(𝑠 − 1)) = (0..^𝑠))
299	298	eleq2d 2811	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ ℕ ∧ 𝑠 ∈ ℕ) → ((𝑖 − 1) ∈ (0...(𝑠 − 1)) ↔ (𝑖 − 1) ∈ (0..^𝑠)))
300		elfzofz 13680	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 − 1) ∈ (0..^𝑠) → (𝑖 − 1) ∈ (0...𝑠))
301	299, 300	biimtrdi 252	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ ℕ ∧ 𝑠 ∈ ℕ) → ((𝑖 − 1) ∈ (0...(𝑠 − 1)) → (𝑖 − 1) ∈ (0...𝑠)))
302	296, 301	sylbid 239	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (𝑖 ∈ (1...𝑠) → (𝑖 − 1) ∈ (0...𝑠)))
303	302	expimpd 452	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℕ → ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → (𝑖 − 1) ∈ (0...𝑠)))
304	293, 303	mpcom 38	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → (𝑖 − 1) ∈ (0...𝑠))
305	304	ex 411	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℕ → (𝑖 ∈ (1...𝑠) → (𝑖 − 1) ∈ (0...𝑠)))
306	305	ad2antrl 726	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑖 ∈ (1...𝑠) → (𝑖 − 1) ∈ (0...𝑠)))
307	306	imp 405	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑖 − 1) ∈ (0...𝑠))
308	238, 307	ffvelcdmd 7092	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑏‘(𝑖 − 1)) ∈ 𝐵)
309	2, 3, 6, 4, 5, 25, 9, 8, 7	mat2pmatscmxcl 22672	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝑏‘(𝑖 − 1)) ∈ 𝐵 ∧ 𝑖 ∈ ℕ₀)) → ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) ∈ (Base‘𝑌))
310	235, 236, 308, 226, 309	syl22anc 837	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) ∈ (Base‘𝑌))
311		eqid 2725	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1))))) = (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))))
312		eqid 2725	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))) = (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
313	25, 252, 292, 221, 310, 249, 311, 312	gsummptfidmsub 19909	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))))) − (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))))
314	272, 289, 313	3eqtr4d 2775	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) − (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) = (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))))
315	314	oveq2d 7433	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) − (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))))) = ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))))))
316	220	adantr 479	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑌 ∈ Grp)
317	25, 252	grpsubcl 18980	. . . . . . . . . 10 ⊢ ((𝑌 ∈ Grp ∧ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) ∈ (Base‘𝑌) ∧ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))) ∈ (Base‘𝑌)) → (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))) ∈ (Base‘𝑌))
318	316, 310, 249, 317	syl3anc 1368	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))) ∈ (Base‘𝑌))
319	318	ralrimiva 3136	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ∀𝑖 ∈ (1...𝑠)(((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))) ∈ (Base‘𝑌))
320	25, 34, 221, 319	gsummptcl 19926	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) ∈ (Base‘𝑌))
321	25, 26	cmncom 19757	. . . . . . 7 ⊢ ((𝑌 ∈ CMnd ∧ (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) ∈ (Base‘𝑌) ∧ (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) ∈ (Base‘𝑌)) → ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠)))))
322	34, 70, 320, 321	syl3anc 1368	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠)))))
323	254, 315, 322	3eqtrd 2769	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) − (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠)))))
324	323	oveq1d 7432	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) − (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = (((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠)))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
325	25, 26	mndcl 18701	. . . . . 6 ⊢ ((𝑌 ∈ Mnd ∧ (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) ∈ (Base‘𝑌) ∧ (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) ∈ (Base‘𝑌)) → ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) ∈ (Base‘𝑌))
326	57, 70, 214, 325	syl3anc 1368	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) ∈ (Base‘𝑌))
327	25, 26, 252, 292, 326, 251, 171	ablsubsub4 19777	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) − (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = (((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) − ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))) + ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
328	25, 26, 252	grpaddsubass 18990	. . . . 5 ⊢ ((𝑌 ∈ Grp ∧ ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) ∈ (Base‘𝑌) ∧ (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) ∈ (Base‘𝑌) ∧ ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))) → (((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠)))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
329	220, 320, 70, 171, 328	syl13anc 1369	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠)))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
330	324, 327, 329	3eqtr3d 2773	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) − ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))) + ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
331	6, 2, 3, 4, 5	mat2pmatbas 22658	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑖 − 1)) ∈ 𝐵) → (𝑇‘(𝑏‘(𝑖 − 1))) ∈ (Base‘𝑌))
332	235, 236, 308, 331	syl3anc 1368	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑇‘(𝑏‘(𝑖 − 1))) ∈ (Base‘𝑌))
333	25, 9, 135, 136, 252, 222, 232, 332, 248	lmodsubdi 20806	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))) = (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))
334	333	eqcomd 2731	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))) = ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))
335	334	mpteq2dva 5248	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))) = (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))))
336	335	oveq2d 7433	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) = (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))))
337	336	oveq1d 7432	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
338	217, 330, 337	3eqtrd 2769	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 − 1)) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) + (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠)))) − ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))) + ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
339	16, 191, 338	3eqtrd 2769	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑋 · 1 ) × (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) − ((𝑇‘𝑀) × (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ⊆ wss 3945 {csn 4629 ↦ cmpt 5231 ⟶wf 6543 ‘cfv 6547 (class class class)co 7417 ↑_m cmap 8843 Fincfn 8962 ℂcc 11136 0cc0 11138 1c1 11139 + caddc 11141 − cmin 11474 ℕcn 12242 ℕ₀cn0 12502 ℤcz 12588 ℤ_≥cuz 12852 ...cfz 13516 ..^cfzo 13659 Basecbs 17179 +_gcplusg 17232 ._rcmulr 17233 Scalarcsca 17235 ·_𝑠 cvsca 17236 0_gc0g 17420 Σ_g cgsu 17421 Mndcmnd 18693 Grpcgrp 18894 -_gcsg 18896 ._gcmg 19027 CMndccmn 19739 Abelcabl 19740 mulGrpcmgp 20078 1_rcur 20125 Ringcrg 20177 CRingccrg 20178 LModclmod 20747 var₁cv1 22103 Poly₁cpl1 22104 Mat cmat 22337 matToPolyMat cmat2pmat 22636

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-ot 4638 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-isom 6556 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7683 df-ofr 7684 df-om 7870 df-1st 7992 df-2nd 7993 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-fzo 13660 df-seq 13999 df-hash 14322 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-hom 17256 df-cco 17257 df-0g 17422 df-gsum 17423 df-prds 17428 df-pws 17430 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-grp 18897 df-minusg 18898 df-sbg 18899 df-mulg 19028 df-subg 19082 df-ghm 19172 df-cntz 19272 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-cring 20180 df-subrng 20487 df-subrg 20512 df-lmod 20749 df-lss 20820 df-sra 21062 df-rgmod 21063 df-dsmm 21670 df-frlm 21685 df-assa 21791 df-ascl 21793 df-psr 21846 df-mvr 21847 df-mpl 21848 df-opsr 21850 df-psr1 22107 df-vr1 22108 df-ply1 22109 df-mamu 22321 df-mat 22338 df-mat2pmat 22639

This theorem is referenced by: cpmadugsumfi 22809

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