Theorem decpmatmul 23120

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 Σg (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 2764	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))))) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))))))
2		oveq1 7450	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑖 → (𝑥(𝑈 decompPMat 𝑘)𝑡) = (𝑖(𝑈 decompPMat 𝑘)𝑡))
3		oveq2 7451	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑗 → (𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦) = (𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗))
4	2, 3	oveqan12d 7462	. . . . . . . . . 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 5202	. . . . . . . . 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 7459	. . . . . . . 8 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))) = (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)))))
7	6	mpteq2dv 5202	. . . . . . 7 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))) = (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗))))))
8	7	oveq2d 7459	. . . . . 6 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))) = (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)))))))
9	8	adantl 487	. . . . 5 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑥 = 𝑖 ∧ 𝑦 = 𝑗)) → (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))) = (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)))))))
10		simprl 783	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
11		simprr 785	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁)
12		ovexd 7478	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)))))) ∈ V)
13	1, 9, 10, 11, 12	ovmpod 7596	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))))𝑗) = (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (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 22757	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑃 ∈ V))
17	16	simpld 500	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ 𝐵 → 𝑁 ∈ Fin)
18	17	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → 𝑁 ∈ Fin)
19	18	anim2i 629	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
20	19	ancomd 467	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
21	20	3adant3 1150	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
22		decpmatmul.a	. . . . . . . . . . . . . . 15 ⊢ 𝐴 = (𝑁 Mat 𝑅)
23		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
24	22, 23	matmulr 22783	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r‘𝐴))
25	21, 24	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r‘𝐴))
26	25	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r‘𝐴))
27	26	adantr 486	. . . . . . . . . . 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 2769	. . . . . . . . . 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 7460	. . . . . . . . 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 2763	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
31		eqid 2763	. . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅)
32		simp1 1154	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ Ring)
33	32	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Ring)
34	33	adantr 486	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑅 ∈ Ring)
35	21	simpld 500	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑁 ∈ Fin)
36	35	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑁 ∈ Fin)
37	36	adantr 486	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑁 ∈ Fin)
38		simpl2l 1245	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑈 ∈ 𝐵)
39	38	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑈 ∈ 𝐵)
40		elfznn0 13781	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝐾) → 𝑘 ∈ ℕ0)
41	40	adantl 487	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑘 ∈ ℕ0)
42	34, 39, 41	3jca 1146	. . . . . . . . . . . 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 2763	. . . . . . . . . . . . 13 ⊢ (Base‘𝐴) = (Base‘𝐴)
45	43, 14, 15, 22, 44	decpmatcl 23115	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐵 ∧ 𝑘 ∈ ℕ0) → (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴))
46	42, 45	syl 18	. . . . . . . . . . 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 22767	. . . . . . . . . . 11 ⊢ ((𝑈 decompPMat 𝑘) ∈ (Base‘𝐴) → (𝑈 decompPMat 𝑘) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
48	46, 47	syl 18	. . . . . . . . . 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 1246	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑊 ∈ 𝐵)
50	49	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑊 ∈ 𝐵)
51		fznn0sub 13717	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝐾) → (𝐾 − 𝑘) ∈ ℕ0)
52	51	adantl 487	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝐾 − 𝑘) ∈ ℕ0)
53	34, 50, 52	3jca 1146	. . . . . . . . . . . 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 23115	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑊 ∈ 𝐵 ∧ (𝐾 − 𝑘) ∈ ℕ0) → (𝑊 decompPMat (𝐾 − 𝑘)) ∈ (Base‘𝐴))
55	53, 54	syl 18	. . . . . . . . . . 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 22767	. . . . . . . . . . 11 ⊢ ((𝑊 decompPMat (𝐾 − 𝑘)) ∈ (Base‘𝐴) → (𝑊 decompPMat (𝐾 − 𝑘)) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
57	55, 56	syl 18	. . . . . . . . . 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 22738	. . . . . . . . 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 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))))
59	29, 58	eqtrd 2798	. . . . . . . 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 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))))
60	59	mpteq2dva 5201	. . . . . . 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 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))))
61	60	oveq2d 7459	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))))) = (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))))))
62		eqid 2763	. . . . . . 7 ⊢ (0g‘𝐴) = (0g‘𝐴)
63		ovexd 7478	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (0...𝐾) ∈ V)
64		ringcmn 20541	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
65	32, 64	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ CMnd)
66	65	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ CMnd)
67	66	adantr 486	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑅 ∈ CMnd)
68	67	3ad2ant1 1151	. . . . . . . . 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 1151	. . . . . . . . 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 1151	. . . . . . . . . . . 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 486	. . . . . . . . . . 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 1211	. . . . . . . . . . . 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 490	. . . . . . . . . . . 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 1151	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 22771	. . . . . . . . . . 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 1212	. . . . . . . . . . . 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 1151	. . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 22771	. . . . . . . . . . 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 20507	. . . . . . . . . . 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 1399	. . . . . . . . . 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 3158	. . . . . . . . 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 20211	. . . . . . . 8 ⊢ (((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑅 Σg (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 22768	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))) ∈ (Base‘𝐴))
87		eqid 2763	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝐾) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))) = (𝑘 ∈ (0...𝐾) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))))
88		fzfid 14143	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (0...𝐾) ∈ Fin)
89		simpl 488	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ V) → 𝑁 ∈ Fin)
90	89, 89	jca 521	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ V) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
91	16, 90	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
92	91	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
93	92	3ad2ant2 1152	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
94	93	adantr 486	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
95	94	adantr 486	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
96		mpoexga 8115	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))) ∈ V)
97	95, 96	syl 18	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))) ∈ V)
98		fvexd 6923	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (0g‘𝐴) ∈ V)
99	87, 88, 97, 98	fsuppmptdm 9395	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑘 ∈ (0...𝐾) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))) finSupp (0g‘𝐴))
100	22, 44, 62, 36, 63, 33, 86, 99	matgsum 22782	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))))) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))))))
101	61, 100	eqtrd 2798	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))))) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))))))
102	101	oveqd 7460	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘)))))𝑗) = (𝑖(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))))𝑗))
103		simpl2 1211	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵))
104		simpl3 1212	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝐾 ∈ ℕ0)
105	43, 14, 15	decpmatmullem 23119	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ∧ 𝐾 ∈ ℕ0)) → (𝑖((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑅 Σg (𝑡 ∈ 𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r‘𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))))))))
106	36, 33, 103, 10, 11, 104, 105	syl213anc 1417	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑅 Σg (𝑡 ∈ 𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r‘𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))))))))
107		simpll1 1231	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑅 ∈ Ring)
108		simplrl 789	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑖 ∈ 𝑁)
109		simprl 783	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑡 ∈ 𝑁)
110	15	eleq2i 2856	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ 𝐵 ↔ 𝑈 ∈ (Base‘𝐶))
111	110	birani 509	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → 𝑈 ∈ (Base‘𝐶))
112	111	3ad2ant2 1152	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑈 ∈ (Base‘𝐶))
113	112	adantr 486	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑈 ∈ (Base‘𝐶))
114	113	adantr 486	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑈 ∈ (Base‘𝐶))
115		eqid 2763	. . . . . . . . . 10 ⊢ (Base‘𝑃) = (Base‘𝑃)
116	14, 115	matecl 22770	. . . . . . . . 9 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑡 ∈ 𝑁 ∧ 𝑈 ∈ (Base‘𝐶)) → (𝑖𝑈𝑡) ∈ (Base‘𝑃))
117	108, 109, 114, 116	syl3anc 1399	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → (𝑖𝑈𝑡) ∈ (Base‘𝑃))
118	40	ad2antll 742	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑘 ∈ ℕ0)
119		eqid 2763	. . . . . . . . 9 ⊢ (coe1‘(𝑖𝑈𝑡)) = (coe1‘(𝑖𝑈𝑡))
120	119, 115, 43, 30	coe1fvalcl 22560	. . . . . . . 8 ⊢ (((𝑖𝑈𝑡) ∈ (Base‘𝑃) ∧ 𝑘 ∈ ℕ0) → ((coe1‘(𝑖𝑈𝑡))‘𝑘) ∈ (Base‘𝑅))
121	117, 118, 120	syl2anc 596	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → ((coe1‘(𝑖𝑈𝑡))‘𝑘) ∈ (Base‘𝑅))
122		simplrr 790	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑗 ∈ 𝑁)
123	49	adantr 486	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑊 ∈ 𝐵)
124	14, 115, 15, 109, 122, 123	matecld 22771	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → (𝑡𝑊𝑗) ∈ (Base‘𝑃))
125	51	ad2antll 742	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → (𝐾 − 𝑘) ∈ ℕ0)
126		eqid 2763	. . . . . . . . 9 ⊢ (coe1‘(𝑡𝑊𝑗)) = (coe1‘(𝑡𝑊𝑗))
127	126, 115, 43, 30	coe1fvalcl 22560	. . . . . . . 8 ⊢ (((𝑡𝑊𝑗) ∈ (Base‘𝑃) ∧ (𝐾 − 𝑘) ∈ ℕ0) → ((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘)) ∈ (Base‘𝑅))
128	124, 125, 127	syl2anc 596	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → ((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘)) ∈ (Base‘𝑅))
129	30, 31	ringcl 20507	. . . . . . 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))
130	107, 121, 128, 129	syl3anc 1399	. . . . . 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))
131	30, 66, 36, 88, 130	gsumcom3fi 20223	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑅 Σg (𝑡 ∈ 𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r‘𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))))))) = (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r‘𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))))))))
132	10	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑖 ∈ 𝑁)
133	132	anim1i 627	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡 ∈ 𝑁) → (𝑖 ∈ 𝑁 ∧ 𝑡 ∈ 𝑁))
134	43, 14, 15	decpmate 23114	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐵 ∧ 𝑘 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑡 ∈ 𝑁)) → (𝑖(𝑈 decompPMat 𝑘)𝑡) = ((coe1‘(𝑖𝑈𝑡))‘𝑘))
135	42, 133, 134	syl2an2r 698	. . . . . . . . . . 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))
136		simplrr 790	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑗 ∈ 𝑁)
137	136	anim1ci 628	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡 ∈ 𝑁) → (𝑡 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
138	43, 14, 15	decpmate 23114	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑊 ∈ 𝐵 ∧ (𝐾 − 𝑘) ∈ ℕ0) ∧ (𝑡 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗) = ((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘)))
139	53, 137, 138	syl2an2r 698	. . . . . . . . . . 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)))
140	135, 139	oveq12d 7461	. . . . . . . . . 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))))
141	140	eqcomd 2769	. . . . . . . . 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)))
142	141	mpteq2dva 5201	. . . . . . . 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))))
143	142	oveq2d 7459	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 Σg (𝑡 ∈ 𝑁 ↦ (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r‘𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))))) = (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)))))
144	143	mpteq2dva 5201	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r‘𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘)))))) = (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗))))))
145	144	oveq2d 7459	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r‘𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))))))) = (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)))))))
146	106, 131, 145	3eqtrd 2802	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)))))))
147	13, 102, 146	3eqtr4rd 2809	. . 3 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑖(𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘)))))𝑗))
148	147	ralrimivva 3209	. 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑖(𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘)))))𝑗))
149	43, 14	pmatring 23040	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
150	20, 149	syl 18	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝐶 ∈ Ring)
151		simprl 783	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑈 ∈ 𝐵)
152		simprr 785	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑊 ∈ 𝐵)
153		eqid 2763	. . . . . . 7 ⊢ (.r‘𝐶) = (.r‘𝐶)
154	15, 153	ringcl 20507	. . . . . 6 ⊢ ((𝐶 ∈ Ring ∧ 𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑈(.r‘𝐶)𝑊) ∈ 𝐵)
155	150, 151, 152, 154	syl3anc 1399	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (𝑈(.r‘𝐶)𝑊) ∈ 𝐵)
156	155	3adant3 1150	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑈(.r‘𝐶)𝑊) ∈ 𝐵)
157	43, 14, 15, 22, 44	decpmatcl 23115	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑈(.r‘𝐶)𝑊) ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) → ((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾) ∈ (Base‘𝐴))
158	156, 157	syld3an2 1439	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → ((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾) ∈ (Base‘𝐴))
159	22	matring 22788	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
160	21, 159	syl 18	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → 𝐴 ∈ Ring)
161		ringcmn 20541	. . . . 5 ⊢ (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
162	160, 161	syl 18	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → 𝐴 ∈ CMnd)
163		fzfid 14143	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → (0...𝐾) ∈ Fin)
164	160	adantr 486	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝐴 ∈ Ring)
165	32	adantr 486	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑅 ∈ Ring)
166		simpl2l 1245	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑈 ∈ 𝐵)
167	40	adantl 487	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑘 ∈ ℕ0)
168	165, 166, 167	3jca 1146	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐵 ∧ 𝑘 ∈ ℕ0))
169	168, 45	syl 18	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴))
170		simpl2r 1246	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑊 ∈ 𝐵)
171	51	adantl 487	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → (𝐾 − 𝑘) ∈ ℕ0)
172	165, 170, 171	3jca 1146	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 ∈ Ring ∧ 𝑊 ∈ 𝐵 ∧ (𝐾 − 𝑘) ∈ ℕ0))
173	172, 54	syl 18	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → (𝑊 decompPMat (𝐾 − 𝑘)) ∈ (Base‘𝐴))
174		eqid 2763	. . . . . . 7 ⊢ (.r‘𝐴) = (.r‘𝐴)
175	44, 174	ringcl 20507	. . . . . 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))
176	164, 169, 173, 175	syl3anc 1399	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))) ∈ (Base‘𝐴))
177	176	ralrimiva 3158	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → ∀𝑘 ∈ (0...𝐾)((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))) ∈ (Base‘𝐴))
178	44, 162, 163, 177	gsummptcl 20211	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))))) ∈ (Base‘𝐴))
179	22, 44	eqmat 22769	. . 3 ⊢ ((((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾) ∈ (Base‘𝐴) ∧ (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))))) ∈ (Base‘𝐴)) → (((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾) = (𝐴 Σg (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 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘)))))𝑗)))
180	158, 178, 179	syl2anc 596	. 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → (((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾) = (𝐴 Σg (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 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘)))))𝑗)))
181	148, 180	mpbird 260	1 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → ((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾) = (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454  ⟨cotp 4861   ↦ cmpt 5189   × cxp 5653  ‘cfv 6558  (class class class)co 7443   ∈ cmpo 7445   ↑_m cmap 8875  Fincfn 9001  0cc0 11227   − cmin 11568  ℕ₀cn0 12633  ...cfz 13666  Basecbs 17416  ._rcmulr 17458  0_gc0g 17639   Σ_g cgsu 17640  CMndccmn 20024  Ringcrg 20489  Poly₁cpl1 22525  coe₁cco1 22526   maMul cmmul 22735   Mat cmat 22752   decompPMat cdecpmat 23110

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-of 7717  df-ofr 7718  df-om 7902  df-1st 8025  df-2nd 8026  df-supp 8205  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-er 8745  df-map 8877  df-pm 8878  df-ixp 8954  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-fsupp 9382  df-sup 9461  df-oi 9531  df-card 10047  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-5 12435  df-6 12436  df-7 12437  df-8 12438  df-9 12439  df-n0 12634  df-z 12721  df-dec 12842  df-uz 12993  df-fz 13667  df-fzo 13816  df-seq 14172  df-hash 14502  df-struct 17354  df-sets 17371  df-slot 17389  df-ndx 17401  df-base 17417  df-ress 17438  df-plusg 17470  df-mulr 17471  df-sca 17473  df-vsca 17474  df-ip 17475  df-tset 17476  df-ple 17477  df-ds 17479  df-hom 17481  df-cco 17482  df-0g 17641  df-gsum 17642  df-prds 17647  df-pws 17649  df-mre 17785  df-mrc 17786  df-acs 17788  df-mgm 18845  df-sgrp 18937  df-mnd 18953  df-mhm 19007  df-submnd 19008  df-grp 19176  df-minusg 19177  df-sbg 19178  df-mulg 19307  df-subg 19362  df-ghm 19457  df-cntz 19561  df-cmn 20026  df-abl 20027  df-mgp 20391  df-rng 20405  df-ur 20438  df-ring 20491  df-subrng 20828  df-subrg 20852  df-lmod 21167  df-lss 21237  df-sra 21478  df-rgmod 21479  df-dsmm 22068  df-frlm 22083  df-psr 22247  df-mpl 22249  df-opsr 22251  df-psr1 22528  df-ply1 22530  df-coe1 22531  df-mamu 22736  df-mat 22753  df-decpmat 23111

This theorem is used by:  decpmatmulsumfsupp  23121  pm2mpmhmlem1  23166  pm2mpmhmlem2  23167