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Theorem decpmatmul 22690

Description: The matrix consisting of the coefficients in the polynomial entries of the product of two polynomial matrices is a sum of products of the matrices consisting of the coefficients in the polynomial entries of the polynomial matrices for the same power. (Contributed by AV, 21-Oct-2019.) (Revised by AV, 3-Dec-2019.)

**Hypotheses**
Ref	Expression
decpmatmul.p	⊢ 𝑃 = (Poly₁‘𝑅)
decpmatmul.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
decpmatmul.b	⊢ 𝐵 = (Base‘𝐶)
decpmatmul.a	⊢ 𝐴 = (𝑁 Mat 𝑅)

**Assertion**
Ref	Expression
decpmatmul	⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) → ((𝑈(._r‘𝐶)𝑊) decompPMat 𝐾) = (𝐴 Σ_g (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(._r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))))))

Distinct variable groups: 𝐵,𝑘 𝑘,𝐾 𝑘,𝑁 𝑃,𝑘 𝑅,𝑘 𝑈,𝑘 𝑘,𝑊 𝐴,𝑘 Allowed substitution hint: 𝐶(𝑘)

Proof of Theorem decpmatmul Dummy variables 𝑡 𝑖 𝑗 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2726	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))))) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))))))
2		oveq1 7422	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑖 → (𝑥(𝑈 decompPMat 𝑘)𝑡) = (𝑖(𝑈 decompPMat 𝑘)𝑡))
3		oveq2 7423	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑗 → (𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦) = (𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗))
4	2, 3	oveqan12d 7434	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)) = ((𝑖(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)))
5	4	mpteq2dv 5245	. . . . . . . . 9 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))) = (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗))))
6	5	oveq2d 7431	. . . . . . . 8 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))) = (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)))))
7	6	mpteq2dv 5245	. . . . . . 7 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))) = (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗))))))
8	7	oveq2d 7431	. . . . . 6 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑅 Σ_g (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))) = (𝑅 Σ_g (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)))))))
9	8	adantl 480	. . . . 5 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑥 = 𝑖 ∧ 𝑦 = 𝑗)) → (𝑅 Σ_g (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))) = (𝑅 Σ_g (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)))))))
10		simprl 769	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
11		simprr 771	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁)
12		ovexd 7450	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑅 Σ_g (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)))))) ∈ V)
13	1, 9, 10, 11, 12	ovmpod 7569	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))))𝑗) = (𝑅 Σ_g (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)))))))
14		decpmatmul.c	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐶 = (𝑁 Mat 𝑃)
15		decpmatmul.b	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐵 = (Base‘𝐶)
16	14, 15	matrcl 22328	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑃 ∈ V))
17	16	simpld 493	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ 𝐵 → 𝑁 ∈ Fin)
18	17	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → 𝑁 ∈ Fin)
19	18	anim2i 615	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
20	19	ancomd 460	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
21	20	3adant3 1129	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
22		decpmatmul.a	. . . . . . . . . . . . . . 15 ⊢ 𝐴 = (𝑁 Mat 𝑅)
23		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
24	22, 23	matmulr 22356	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (._r‘𝐴))
25	21, 24	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (._r‘𝐴))
26	25	adantr 479	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (._r‘𝐴))
27	26	adantr 479	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (._r‘𝐴))
28	27	eqcomd 2731	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (._r‘𝐴) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩))
29	28	oveqd 7432	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → ((𝑈 decompPMat 𝑘)(._r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))) = ((𝑈 decompPMat 𝑘)(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)(𝑊 decompPMat (𝐾 − 𝑘))))
30		eqid 2725	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
31		eqid 2725	. . . . . . . . . 10 ⊢ (._r‘𝑅) = (._r‘𝑅)
32		simp1 1133	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) → 𝑅 ∈ Ring)
33	32	adantr 479	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Ring)
34	33	adantr 479	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑅 ∈ Ring)
35	21	simpld 493	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) → 𝑁 ∈ Fin)
36	35	adantr 479	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑁 ∈ Fin)
37	36	adantr 479	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑁 ∈ Fin)
38		simpl2l 1223	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑈 ∈ 𝐵)
39	38	adantr 479	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑈 ∈ 𝐵)
40		elfznn0 13624	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝐾) → 𝑘 ∈ ℕ₀)
41	40	adantl 480	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑘 ∈ ℕ₀)
42	34, 39, 41	3jca 1125	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐵 ∧ 𝑘 ∈ ℕ₀))
43		decpmatmul.p	. . . . . . . . . . . . 13 ⊢ 𝑃 = (Poly₁‘𝑅)
44		eqid 2725	. . . . . . . . . . . . 13 ⊢ (Base‘𝐴) = (Base‘𝐴)
45	43, 14, 15, 22, 44	decpmatcl 22685	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐵 ∧ 𝑘 ∈ ℕ₀) → (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴))
46	42, 45	syl 17	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴))
47	22, 30, 44	matbas2i 22340	. . . . . . . . . . 11 ⊢ ((𝑈 decompPMat 𝑘) ∈ (Base‘𝐴) → (𝑈 decompPMat 𝑘) ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
48	46, 47	syl 17	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑈 decompPMat 𝑘) ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
49		simpl2r 1224	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑊 ∈ 𝐵)
50	49	adantr 479	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑊 ∈ 𝐵)
51		fznn0sub 13563	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝐾) → (𝐾 − 𝑘) ∈ ℕ₀)
52	51	adantl 480	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝐾 − 𝑘) ∈ ℕ₀)
53	34, 50, 52	3jca 1125	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 ∈ Ring ∧ 𝑊 ∈ 𝐵 ∧ (𝐾 − 𝑘) ∈ ℕ₀))
54	43, 14, 15, 22, 44	decpmatcl 22685	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑊 ∈ 𝐵 ∧ (𝐾 − 𝑘) ∈ ℕ₀) → (𝑊 decompPMat (𝐾 − 𝑘)) ∈ (Base‘𝐴))
55	53, 54	syl 17	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑊 decompPMat (𝐾 − 𝑘)) ∈ (Base‘𝐴))
56	22, 30, 44	matbas2i 22340	. . . . . . . . . . 11 ⊢ ((𝑊 decompPMat (𝐾 − 𝑘)) ∈ (Base‘𝐴) → (𝑊 decompPMat (𝐾 − 𝑘)) ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
57	55, 56	syl 17	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑊 decompPMat (𝐾 − 𝑘)) ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
58	23, 30, 31, 34, 37, 37, 37, 48, 57	mamuval 22309	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → ((𝑈 decompPMat 𝑘)(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)(𝑊 decompPMat (𝐾 − 𝑘))) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))))
59	29, 58	eqtrd 2765	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → ((𝑈 decompPMat 𝑘)(._r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))))
60	59	mpteq2dva 5243	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(._r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘)))) = (𝑘 ∈ (0...𝐾) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))))
61	60	oveq2d 7431	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝐴 Σ_g (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(._r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))))) = (𝐴 Σ_g (𝑘 ∈ (0...𝐾) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))))))
62		eqid 2725	. . . . . . 7 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
63		ovexd 7450	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (0...𝐾) ∈ V)
64		ringcmn 20220	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
65	32, 64	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) → 𝑅 ∈ CMnd)
66	65	adantr 479	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ CMnd)
67	66	adantr 479	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑅 ∈ CMnd)
68	67	3ad2ant1 1130	. . . . . . . . 9 ⊢ (((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑅 ∈ CMnd)
69	37	3ad2ant1 1130	. . . . . . . . 9 ⊢ (((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑁 ∈ Fin)
70	34	3ad2ant1 1130	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑅 ∈ Ring)
71	70	adantr 479	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) ∧ 𝑡 ∈ 𝑁) → 𝑅 ∈ Ring)
72		simpl2 1189	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) ∧ 𝑡 ∈ 𝑁) → 𝑥 ∈ 𝑁)
73		simpr 483	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) ∧ 𝑡 ∈ 𝑁) → 𝑡 ∈ 𝑁)
74	42	3ad2ant1 1130	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐵 ∧ 𝑘 ∈ ℕ₀))
75	74	adantr 479	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) ∧ 𝑡 ∈ 𝑁) → (𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐵 ∧ 𝑘 ∈ ℕ₀))
76	75, 45	syl 17	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) ∧ 𝑡 ∈ 𝑁) → (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴))
77	22, 30, 44, 72, 73, 76	matecld 22344	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) ∧ 𝑡 ∈ 𝑁) → (𝑥(𝑈 decompPMat 𝑘)𝑡) ∈ (Base‘𝑅))
78		simpl3 1190	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) ∧ 𝑡 ∈ 𝑁) → 𝑦 ∈ 𝑁)
79	55	3ad2ant1 1130	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑊 decompPMat (𝐾 − 𝑘)) ∈ (Base‘𝐴))
80	79	adantr 479	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) ∧ 𝑡 ∈ 𝑁) → (𝑊 decompPMat (𝐾 − 𝑘)) ∈ (Base‘𝐴))
81	22, 30, 44, 73, 78, 80	matecld 22344	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) ∧ 𝑡 ∈ 𝑁) → (𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦) ∈ (Base‘𝑅))
82	30, 31	ringcl 20192	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ (𝑥(𝑈 decompPMat 𝑘)𝑡) ∈ (Base‘𝑅) ∧ (𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦) ∈ (Base‘𝑅)) → ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)) ∈ (Base‘𝑅))
83	71, 77, 81, 82	syl3anc 1368	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) ∧ 𝑡 ∈ 𝑁) → ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)) ∈ (Base‘𝑅))
84	83	ralrimiva 3136	. . . . . . . . 9 ⊢ (((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → ∀𝑡 ∈ 𝑁 ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)) ∈ (Base‘𝑅))
85	30, 68, 69, 84	gsummptcl 19924	. . . . . . . 8 ⊢ (((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))) ∈ (Base‘𝑅))
86	22, 30, 44, 37, 34, 85	matbas2d 22341	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))) ∈ (Base‘𝐴))
87		eqid 2725	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝐾) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))) = (𝑘 ∈ (0...𝐾) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))))
88		fzfid 13968	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (0...𝐾) ∈ Fin)
89		simpl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ V) → 𝑁 ∈ Fin)
90	89, 89	jca 510	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ V) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
91	16, 90	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
92	91	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
93	92	3ad2ant2 1131	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
94	93	adantr 479	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
95	94	adantr 479	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
96		mpoexga 8078	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))) ∈ V)
97	95, 96	syl 17	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))) ∈ V)
98		fvexd 6906	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (0_g‘𝐴) ∈ V)
99	87, 88, 97, 98	fsuppmptdm 9397	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑘 ∈ (0...𝐾) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))) finSupp (0_g‘𝐴))
100	22, 44, 62, 36, 63, 33, 86, 99	matgsum 22355	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝐴 Σ_g (𝑘 ∈ (0...𝐾) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))))) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))))))
101	61, 100	eqtrd 2765	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝐴 Σ_g (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(._r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))))) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))))))
102	101	oveqd 7432	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝐴 Σ_g (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(._r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘)))))𝑗) = (𝑖(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))))𝑗))
103		simpl2 1189	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵))
104		simpl3 1190	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝐾 ∈ ℕ₀)
105	43, 14, 15	decpmatmullem 22689	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ∧ 𝐾 ∈ ℕ₀)) → (𝑖((𝑈(._r‘𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑘 ∈ (0...𝐾) ↦ (((coe₁‘(𝑖𝑈𝑡))‘𝑘)(._r‘𝑅)((coe₁‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))))))))
106	36, 33, 103, 10, 11, 104, 105	syl213anc 1386	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖((𝑈(._r‘𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑘 ∈ (0...𝐾) ↦ (((coe₁‘(𝑖𝑈𝑡))‘𝑘)(._r‘𝑅)((coe₁‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))))))))
107		simpll1 1209	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑅 ∈ Ring)
108		simplrl 775	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑖 ∈ 𝑁)
109		simprl 769	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑡 ∈ 𝑁)
110	15	eleq2i 2817	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ 𝐵 ↔ 𝑈 ∈ (Base‘𝐶))
111	110	biimpi 215	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ 𝐵 → 𝑈 ∈ (Base‘𝐶))
112	111	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → 𝑈 ∈ (Base‘𝐶))
113	112	3ad2ant2 1131	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) → 𝑈 ∈ (Base‘𝐶))
114	113	adantr 479	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑈 ∈ (Base‘𝐶))
115	114	adantr 479	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑈 ∈ (Base‘𝐶))
116		eqid 2725	. . . . . . . . . 10 ⊢ (Base‘𝑃) = (Base‘𝑃)
117	14, 116	matecl 22343	. . . . . . . . 9 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑡 ∈ 𝑁 ∧ 𝑈 ∈ (Base‘𝐶)) → (𝑖𝑈𝑡) ∈ (Base‘𝑃))
118	108, 109, 115, 117	syl3anc 1368	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → (𝑖𝑈𝑡) ∈ (Base‘𝑃))
119	40	ad2antll 727	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑘 ∈ ℕ₀)
120		eqid 2725	. . . . . . . . 9 ⊢ (coe₁‘(𝑖𝑈𝑡)) = (coe₁‘(𝑖𝑈𝑡))
121	120, 116, 43, 30	coe1fvalcl 22138	. . . . . . . 8 ⊢ (((𝑖𝑈𝑡) ∈ (Base‘𝑃) ∧ 𝑘 ∈ ℕ₀) → ((coe₁‘(𝑖𝑈𝑡))‘𝑘) ∈ (Base‘𝑅))
122	118, 119, 121	syl2anc 582	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → ((coe₁‘(𝑖𝑈𝑡))‘𝑘) ∈ (Base‘𝑅))
123		simplrr 776	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑗 ∈ 𝑁)
124	49	adantr 479	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑊 ∈ 𝐵)
125	14, 116, 15, 109, 123, 124	matecld 22344	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → (𝑡𝑊𝑗) ∈ (Base‘𝑃))
126	51	ad2antll 727	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → (𝐾 − 𝑘) ∈ ℕ₀)
127		eqid 2725	. . . . . . . . 9 ⊢ (coe₁‘(𝑡𝑊𝑗)) = (coe₁‘(𝑡𝑊𝑗))
128	127, 116, 43, 30	coe1fvalcl 22138	. . . . . . . 8 ⊢ (((𝑡𝑊𝑗) ∈ (Base‘𝑃) ∧ (𝐾 − 𝑘) ∈ ℕ₀) → ((coe₁‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘)) ∈ (Base‘𝑅))
129	125, 126, 128	syl2anc 582	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → ((coe₁‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘)) ∈ (Base‘𝑅))
130	30, 31	ringcl 20192	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ ((coe₁‘(𝑖𝑈𝑡))‘𝑘) ∈ (Base‘𝑅) ∧ ((coe₁‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘)) ∈ (Base‘𝑅)) → (((coe₁‘(𝑖𝑈𝑡))‘𝑘)(._r‘𝑅)((coe₁‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))) ∈ (Base‘𝑅))
131	107, 122, 129, 130	syl3anc 1368	. . . . . 6 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → (((coe₁‘(𝑖𝑈𝑡))‘𝑘)(._r‘𝑅)((coe₁‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))) ∈ (Base‘𝑅))
132	30, 66, 36, 88, 131	gsumcom3fi 19936	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑘 ∈ (0...𝐾) ↦ (((coe₁‘(𝑖𝑈𝑡))‘𝑘)(._r‘𝑅)((coe₁‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))))))) = (𝑅 Σ_g (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ (((coe₁‘(𝑖𝑈𝑡))‘𝑘)(._r‘𝑅)((coe₁‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))))))))
133	10	adantr 479	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑖 ∈ 𝑁)
134	133	anim1i 613	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡 ∈ 𝑁) → (𝑖 ∈ 𝑁 ∧ 𝑡 ∈ 𝑁))
135	43, 14, 15	decpmate 22684	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐵 ∧ 𝑘 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑡 ∈ 𝑁)) → (𝑖(𝑈 decompPMat 𝑘)𝑡) = ((coe₁‘(𝑖𝑈𝑡))‘𝑘))
136	42, 134, 135	syl2an2r 683	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡 ∈ 𝑁) → (𝑖(𝑈 decompPMat 𝑘)𝑡) = ((coe₁‘(𝑖𝑈𝑡))‘𝑘))
137		simplrr 776	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑗 ∈ 𝑁)
138	137	anim1ci 614	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡 ∈ 𝑁) → (𝑡 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
139	43, 14, 15	decpmate 22684	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑊 ∈ 𝐵 ∧ (𝐾 − 𝑘) ∈ ℕ₀) ∧ (𝑡 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗) = ((coe₁‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘)))
140	53, 138, 139	syl2an2r 683	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡 ∈ 𝑁) → (𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗) = ((coe₁‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘)))
141	136, 140	oveq12d 7433	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡 ∈ 𝑁) → ((𝑖(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)) = (((coe₁‘(𝑖𝑈𝑡))‘𝑘)(._r‘𝑅)((coe₁‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))))
142	141	eqcomd 2731	. . . . . . . . 9 ⊢ (((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡 ∈ 𝑁) → (((coe₁‘(𝑖𝑈𝑡))‘𝑘)(._r‘𝑅)((coe₁‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))) = ((𝑖(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)))
143	142	mpteq2dva 5243	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑡 ∈ 𝑁 ↦ (((coe₁‘(𝑖𝑈𝑡))‘𝑘)(._r‘𝑅)((coe₁‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘)))) = (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗))))
144	143	oveq2d 7431	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ (((coe₁‘(𝑖𝑈𝑡))‘𝑘)(._r‘𝑅)((coe₁‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))))) = (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)))))
145	144	mpteq2dva 5243	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ (((coe₁‘(𝑖𝑈𝑡))‘𝑘)(._r‘𝑅)((coe₁‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘)))))) = (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗))))))
146	145	oveq2d 7431	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑅 Σ_g (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ (((coe₁‘(𝑖𝑈𝑡))‘𝑘)(._r‘𝑅)((coe₁‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))))))) = (𝑅 Σ_g (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)))))))
147	106, 132, 146	3eqtrd 2769	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖((𝑈(._r‘𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑅 Σ_g (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σ_g (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(._r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)))))))
148	13, 102, 147	3eqtr4rd 2776	. . 3 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖((𝑈(._r‘𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑖(𝐴 Σ_g (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(._r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘)))))𝑗))
149	148	ralrimivva 3191	. 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖((𝑈(._r‘𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑖(𝐴 Σ_g (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(._r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘)))))𝑗))
150	43, 14	pmatring 22610	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
151	20, 150	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝐶 ∈ Ring)
152		simprl 769	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑈 ∈ 𝐵)
153		simprr 771	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑊 ∈ 𝐵)
154		eqid 2725	. . . . . . 7 ⊢ (._r‘𝐶) = (._r‘𝐶)
155	15, 154	ringcl 20192	. . . . . 6 ⊢ ((𝐶 ∈ Ring ∧ 𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑈(._r‘𝐶)𝑊) ∈ 𝐵)
156	151, 152, 153, 155	syl3anc 1368	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (𝑈(._r‘𝐶)𝑊) ∈ 𝐵)
157	156	3adant3 1129	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) → (𝑈(._r‘𝐶)𝑊) ∈ 𝐵)
158	43, 14, 15, 22, 44	decpmatcl 22685	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑈(._r‘𝐶)𝑊) ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀) → ((𝑈(._r‘𝐶)𝑊) decompPMat 𝐾) ∈ (Base‘𝐴))
159	157, 158	syld3an2 1408	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) → ((𝑈(._r‘𝐶)𝑊) decompPMat 𝐾) ∈ (Base‘𝐴))
160	22	matring 22361	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
161	21, 160	syl 17	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) → 𝐴 ∈ Ring)
162		ringcmn 20220	. . . . 5 ⊢ (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
163	161, 162	syl 17	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) → 𝐴 ∈ CMnd)
164		fzfid 13968	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) → (0...𝐾) ∈ Fin)
165	161	adantr 479	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐾)) → 𝐴 ∈ Ring)
166	32	adantr 479	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐾)) → 𝑅 ∈ Ring)
167		simpl2l 1223	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐾)) → 𝑈 ∈ 𝐵)
168	40	adantl 480	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐾)) → 𝑘 ∈ ℕ₀)
169	166, 167, 168	3jca 1125	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐵 ∧ 𝑘 ∈ ℕ₀))
170	169, 45	syl 17	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐾)) → (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴))
171		simpl2r 1224	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐾)) → 𝑊 ∈ 𝐵)
172	51	adantl 480	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐾)) → (𝐾 − 𝑘) ∈ ℕ₀)
173	166, 171, 172	3jca 1125	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 ∈ Ring ∧ 𝑊 ∈ 𝐵 ∧ (𝐾 − 𝑘) ∈ ℕ₀))
174	173, 54	syl 17	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐾)) → (𝑊 decompPMat (𝐾 − 𝑘)) ∈ (Base‘𝐴))
175		eqid 2725	. . . . . . 7 ⊢ (._r‘𝐴) = (._r‘𝐴)
176	44, 175	ringcl 20192	. . . . . 6 ⊢ ((𝐴 ∈ Ring ∧ (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴) ∧ (𝑊 decompPMat (𝐾 − 𝑘)) ∈ (Base‘𝐴)) → ((𝑈 decompPMat 𝑘)(._r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))) ∈ (Base‘𝐴))
177	165, 170, 174, 176	syl3anc 1368	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐾)) → ((𝑈 decompPMat 𝑘)(._r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))) ∈ (Base‘𝐴))
178	177	ralrimiva 3136	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) → ∀𝑘 ∈ (0...𝐾)((𝑈 decompPMat 𝑘)(._r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))) ∈ (Base‘𝐴))
179	44, 163, 164, 178	gsummptcl 19924	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) → (𝐴 Σ_g (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(._r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))))) ∈ (Base‘𝐴))
180	22, 44	eqmat 22342	. . 3 ⊢ ((((𝑈(._r‘𝐶)𝑊) decompPMat 𝐾) ∈ (Base‘𝐴) ∧ (𝐴 Σ_g (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(._r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))))) ∈ (Base‘𝐴)) → (((𝑈(._r‘𝐶)𝑊) decompPMat 𝐾) = (𝐴 Σ_g (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(._r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))))) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖((𝑈(._r‘𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑖(𝐴 Σ_g (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(._r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘)))))𝑗)))
181	159, 179, 180	syl2anc 582	. 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) → (((𝑈(._r‘𝐶)𝑊) decompPMat 𝐾) = (𝐴 Σ_g (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(._r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))))) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖((𝑈(._r‘𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑖(𝐴 Σ_g (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(._r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘)))))𝑗)))
182	149, 181	mpbird 256	1 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) → ((𝑈(._r‘𝐶)𝑊) decompPMat 𝐾) = (𝐴 Σ_g (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(._r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3051 Vcvv 3463 ⟨cotp 4632 ↦ cmpt 5226 × cxp 5670 ‘cfv 6542 (class class class)co 7415 ∈ cmpo 7417 ↑_m cmap 8841 Fincfn 8960 0cc0 11136 − cmin 11472 ℕ₀cn0 12500 ...cfz 13514 Basecbs 17177 ._rcmulr 17231 0_gc0g 17418 Σ_g cgsu 17419 CMndccmn 19737 Ringcrg 20175 Poly₁cpl1 22102 coe₁cco1 22103 maMul cmmul 22306 Mat cmat 22323 decompPMat cdecpmat 22680

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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213

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 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-ofr 7682 df-om 7868 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-pm 8844 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-sup 9463 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-fzo 13658 df-seq 13997 df-hash 14320 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-hom 17254 df-cco 17255 df-0g 17420 df-gsum 17421 df-prds 17426 df-pws 17428 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18737 df-submnd 18738 df-grp 18895 df-minusg 18896 df-sbg 18897 df-mulg 19026 df-subg 19080 df-ghm 19170 df-cntz 19270 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-subrng 20485 df-subrg 20510 df-lmod 20747 df-lss 20818 df-sra 21060 df-rgmod 21061 df-dsmm 21668 df-frlm 21683 df-psr 21844 df-mpl 21846 df-opsr 21848 df-psr1 22105 df-ply1 22107 df-coe1 22108 df-mamu 22307 df-mat 22324 df-decpmat 22681

This theorem is referenced by: decpmatmulsumfsupp 22691 pm2mpmhmlem1 22736 pm2mpmhmlem2 22737

W3C validator