Theorem decpmatmul 22728

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 2738	. . . . 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 7375	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑖 → (𝑥(𝑈 decompPMat 𝑘)𝑡) = (𝑖(𝑈 decompPMat 𝑘)𝑡))
3		oveq2 7376	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑗 → (𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦) = (𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗))
4	2, 3	oveqan12d 7387	. . . . . . . . . 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 5194	. . . . . . . . 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 7384	. . . . . . . 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 5194	. . . . . . 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 7384	. . . . . 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 481	. . . . 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 771	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
11		simprr 773	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁)
12		ovexd 7403	. . . . 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 7520	. . . 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 22368	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑃 ∈ V))
17	16	simpld 494	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ 𝐵 → 𝑁 ∈ Fin)
18	17	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → 𝑁 ∈ Fin)
19	18	anim2i 618	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
20	19	ancomd 461	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
21	20	3adant3 1133	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
22		decpmatmul.a	. . . . . . . . . . . . . . 15 ⊢ 𝐴 = (𝑁 Mat 𝑅)
23		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
24	22, 23	matmulr 22394	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r‘𝐴))
27	26	adantr 480	. . . . . . . . . . 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 2743	. . . . . . . . . 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 7385	. . . . . . . . 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 2737	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
31		eqid 2737	. . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅)
32		simp1 1137	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ Ring)
33	32	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Ring)
34	33	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑅 ∈ Ring)
35	21	simpld 494	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑁 ∈ Fin)
36	35	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑁 ∈ Fin)
37	36	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑁 ∈ Fin)
38		simpl2l 1228	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑈 ∈ 𝐵)
39	38	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑈 ∈ 𝐵)
40		elfznn0 13548	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝐾) → 𝑘 ∈ ℕ0)
41	40	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑘 ∈ ℕ0)
42	34, 39, 41	3jca 1129	. . . . . . . . . . . 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 2737	. . . . . . . . . . . . 13 ⊢ (Base‘𝐴) = (Base‘𝐴)
45	43, 14, 15, 22, 44	decpmatcl 22723	. . . . . . . . . . . 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 22378	. . . . . . . . . . 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 1229	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑊 ∈ 𝐵)
50	49	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑊 ∈ 𝐵)
51		fznn0sub 13484	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝐾) → (𝐾 − 𝑘) ∈ ℕ0)
52	51	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝐾 − 𝑘) ∈ ℕ0)
53	34, 50, 52	3jca 1129	. . . . . . . . . . . 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 22723	. . . . . . . . . . . 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 22378	. . . . . . . . . . 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 22349	. . . . . . . . 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 2772	. . . . . . . 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 5193	. . . . . . 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 7384	. . . . . 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 2737	. . . . . . 7 ⊢ (0g‘𝐴) = (0g‘𝐴)
63		ovexd 7403	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (0...𝐾) ∈ V)
64		ringcmn 20229	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
65	32, 64	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ CMnd)
66	65	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ CMnd)
67	66	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑅 ∈ CMnd)
68	67	3ad2ant1 1134	. . . . . . . . 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 1134	. . . . . . . . 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 1134	. . . . . . . . . . . 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 480	. . . . . . . . . . 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 1194	. . . . . . . . . . . 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 484	. . . . . . . . . . . 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 1134	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 22382	. . . . . . . . . . 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 1195	. . . . . . . . . . . 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 1134	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 22382	. . . . . . . . . . 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 20197	. . . . . . . . . . 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 1374	. . . . . . . . . 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 3130	. . . . . . . . 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 19908	. . . . . . . 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 22379	. . . . . . 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 2737	. . . . . . . 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 13908	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (0...𝐾) ∈ Fin)
89		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ V) → 𝑁 ∈ Fin)
90	89, 89	jca 511	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ V) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
91	16, 90	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
92	91	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
93	92	3ad2ant2 1135	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
94	93	adantr 480	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
95	94	adantr 480	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
96		mpoexga 8031	. . . . . . . . 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 17	. . . . . . . 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 6857	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (0g‘𝐴) ∈ V)
99	87, 88, 97, 98	fsuppmptdm 9291	. . . . . . 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 22393	. . . . . 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 2772	. . . . 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 7385	. . . 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 1194	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵))
104		simpl3 1195	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝐾 ∈ ℕ0)
105	43, 14, 15	decpmatmullem 22727	. . . . . 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 1392	. . . . 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 1214	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑅 ∈ Ring)
108		simplrl 777	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑖 ∈ 𝑁)
109		simprl 771	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑡 ∈ 𝑁)
110	15	eleq2i 2829	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ 𝐵 ↔ 𝑈 ∈ (Base‘𝐶))
111	110	biimpi 216	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ 𝐵 → 𝑈 ∈ (Base‘𝐶))
112	111	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → 𝑈 ∈ (Base‘𝐶))
113	112	3ad2ant2 1135	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑈 ∈ (Base‘𝐶))
114	113	adantr 480	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑈 ∈ (Base‘𝐶))
115	114	adantr 480	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑈 ∈ (Base‘𝐶))
116		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘𝑃) = (Base‘𝑃)
117	14, 116	matecl 22381	. . . . . . . . 9 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑡 ∈ 𝑁 ∧ 𝑈 ∈ (Base‘𝐶)) → (𝑖𝑈𝑡) ∈ (Base‘𝑃))
118	108, 109, 115, 117	syl3anc 1374	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → (𝑖𝑈𝑡) ∈ (Base‘𝑃))
119	40	ad2antll 730	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑘 ∈ ℕ0)
120		eqid 2737	. . . . . . . . 9 ⊢ (coe1‘(𝑖𝑈𝑡)) = (coe1‘(𝑖𝑈𝑡))
121	120, 116, 43, 30	coe1fvalcl 22165	. . . . . . . 8 ⊢ (((𝑖𝑈𝑡) ∈ (Base‘𝑃) ∧ 𝑘 ∈ ℕ0) → ((coe1‘(𝑖𝑈𝑡))‘𝑘) ∈ (Base‘𝑅))
122	118, 119, 121	syl2anc 585	. . . . . . 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 778	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑗 ∈ 𝑁)
124	49	adantr 480	. . . . . . . . 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 22382	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → (𝑡𝑊𝑗) ∈ (Base‘𝑃))
126	51	ad2antll 730	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → (𝐾 − 𝑘) ∈ ℕ0)
127		eqid 2737	. . . . . . . . 9 ⊢ (coe1‘(𝑡𝑊𝑗)) = (coe1‘(𝑡𝑊𝑗))
128	127, 116, 43, 30	coe1fvalcl 22165	. . . . . . . 8 ⊢ (((𝑡𝑊𝑗) ∈ (Base‘𝑃) ∧ (𝐾 − 𝑘) ∈ ℕ0) → ((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘)) ∈ (Base‘𝑅))
129	125, 126, 128	syl2anc 585	. . . . . . 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 20197	. . . . . . 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 1374	. . . . . 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 19920	. . . . 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))))))))
133	10	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑖 ∈ 𝑁)
134	133	anim1i 616	. . . . . . . . . . . 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 22722	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐵 ∧ 𝑘 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑡 ∈ 𝑁)) → (𝑖(𝑈 decompPMat 𝑘)𝑡) = ((coe1‘(𝑖𝑈𝑡))‘𝑘))
136	42, 134, 135	syl2an2r 686	. . . . . . . . . . 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 778	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑗 ∈ 𝑁)
138	137	anim1ci 617	. . . . . . . . . . . 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 22722	. . . . . . . . . . . 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 686	. . . . . . . . . . 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 7386	. . . . . . . . . 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 2743	. . . . . . . . 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 5193	. . . . . . . 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 7384	. . . . . . 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)))))
145	144	mpteq2dva 5193	. . . . . 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))))))
146	145	oveq2d 7384	. . . . 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)))))))
147	106, 132, 146	3eqtrd 2776	. . . 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)))))))
148	13, 102, 147	3eqtr4rd 2783	. . 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))
149	148	ralrimivva 3181	. 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))
150	43, 14	pmatring 22648	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
151	20, 150	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝐶 ∈ Ring)
152		simprl 771	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑈 ∈ 𝐵)
153		simprr 773	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑊 ∈ 𝐵)
154		eqid 2737	. . . . . . 7 ⊢ (.r‘𝐶) = (.r‘𝐶)
155	15, 154	ringcl 20197	. . . . . 6 ⊢ ((𝐶 ∈ Ring ∧ 𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑈(.r‘𝐶)𝑊) ∈ 𝐵)
156	151, 152, 153, 155	syl3anc 1374	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (𝑈(.r‘𝐶)𝑊) ∈ 𝐵)
157	156	3adant3 1133	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑈(.r‘𝐶)𝑊) ∈ 𝐵)
158	43, 14, 15, 22, 44	decpmatcl 22723	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑈(.r‘𝐶)𝑊) ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) → ((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾) ∈ (Base‘𝐴))
159	157, 158	syld3an2 1414	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → ((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾) ∈ (Base‘𝐴))
160	22	matring 22399	. . . . . 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 20229	. . . . 5 ⊢ (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
163	161, 162	syl 17	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → 𝐴 ∈ CMnd)
164		fzfid 13908	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → (0...𝐾) ∈ Fin)
165	161	adantr 480	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝐴 ∈ Ring)
166	32	adantr 480	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑅 ∈ Ring)
167		simpl2l 1228	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑈 ∈ 𝐵)
168	40	adantl 481	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑘 ∈ ℕ0)
169	166, 167, 168	3jca 1129	. . . . . . 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 1229	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑊 ∈ 𝐵)
172	51	adantl 481	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → (𝐾 − 𝑘) ∈ ℕ0)
173	166, 171, 172	3jca 1129	. . . . . . 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 2737	. . . . . . 7 ⊢ (.r‘𝐴) = (.r‘𝐴)
176	44, 175	ringcl 20197	. . . . . 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 1374	. . . . 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 3130	. . . 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 19908	. . 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))
180	22, 44	eqmat 22380	. . 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)))
181	159, 179, 180	syl2anc 585	. 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)))
182	149, 181	mpbird 257	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))))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3442  ⟨cotp 4590   ↦ cmpt 5181   × cxp 5630  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370   ↑_m cmap 8775  Fincfn 8895  0cc0 11038   − cmin 11376  ℕ₀cn0 12413  ...cfz 13435  Basecbs 17148  ._rcmulr 17190  0_gc0g 17371   Σ_g cgsu 17372  CMndccmn 19721  Ringcrg 20180  Poly₁cpl1 22129  coe₁cco1 22130   maMul cmmul 22346   Mat cmat 22363   decompPMat cdecpmat 22718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-of 7632  df-ofr 7633  df-om 7819  df-1st 7943  df-2nd 7944  df-supp 8113  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-er 8645  df-map 8777  df-pm 8778  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9277  df-sup 9357  df-oi 9427  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-z 12501  df-dec 12620  df-uz 12764  df-fz 13436  df-fzo 13583  df-seq 13937  df-hash 14266  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-mulr 17203  df-sca 17205  df-vsca 17206  df-ip 17207  df-tset 17208  df-ple 17209  df-ds 17211  df-hom 17213  df-cco 17214  df-0g 17373  df-gsum 17374  df-prds 17379  df-pws 17381  df-mre 17517  df-mrc 17518  df-acs 17520  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-mhm 18720  df-submnd 18721  df-grp 18878  df-minusg 18879  df-sbg 18880  df-mulg 19010  df-subg 19065  df-ghm 19154  df-cntz 19258  df-cmn 19723  df-abl 19724  df-mgp 20088  df-rng 20100  df-ur 20129  df-ring 20182  df-subrng 20491  df-subrg 20515  df-lmod 20825  df-lss 20895  df-sra 21137  df-rgmod 21138  df-dsmm 21699  df-frlm 21714  df-psr 21877  df-mpl 21879  df-opsr 21881  df-psr1 22132  df-ply1 22134  df-coe1 22135  df-mamu 22347  df-mat 22364  df-decpmat 22719

This theorem is referenced by:  decpmatmulsumfsupp  22729  pm2mpmhmlem1  22774  pm2mpmhmlem2  22775