Theorem decpmatmulsumfsupp 22895

Description: Lemma 0 for pm2mpmhm 22942. (Contributed by AV, 21-Oct-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)
decpmatmulsumfsupp.m	⊢ · = (.r‘𝐴)
decpmatmulsumfsupp.0	⊢ 0 = (0g‘𝐴)

Assertion

Ref	Expression
decpmatmulsumfsupp	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑙 ∈ ℕ0 ↦ (𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘) · (𝑦 decompPMat (𝑙 − 𝑘)))))) finSupp 0 )

Distinct variable groups:   𝐵,𝑘,𝑙   𝑘,𝑁   𝑃,𝑘   𝑅,𝑘,𝑙   𝐴,𝑘   𝑥,𝐵,𝑦,𝑘   𝑥,𝑁,𝑦   𝑥,𝑅,𝑦   𝐴,𝑙   𝐵,𝑙   𝑁,𝑙,𝑥,𝑦   · ,𝑙

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐶(𝑥,𝑦,𝑘,𝑙)   𝑃(𝑥,𝑦,𝑙)   · (𝑥,𝑦,𝑘)   0 (𝑥,𝑦,𝑘,𝑙)

Proof of Theorem decpmatmulsumfsupp

Dummy variables 𝑖 𝑗 𝑛 𝑠 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		decpmatmulsumfsupp.0	. . . 4 ⊢ 0 = (0g‘𝐴)
2	1	fvexi 6893	. . 3 ⊢ 0 ∈ V
3	2	a1i 11	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 0 ∈ V)
4		ovexd 7443	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑙 ∈ ℕ0) → (𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘) · (𝑦 decompPMat (𝑙 − 𝑘))))) ∈ V)
5		oveq2 7416	. . . 4 ⊢ (𝑙 = 𝑛 → (0...𝑙) = (0...𝑛))
6		oveq1 7415	. . . . . 6 ⊢ (𝑙 = 𝑛 → (𝑙 − 𝑘) = (𝑛 − 𝑘))
7	6	oveq2d 7424	. . . . 5 ⊢ (𝑙 = 𝑛 → (𝑦 decompPMat (𝑙 − 𝑘)) = (𝑦 decompPMat (𝑛 − 𝑘)))
8	7	oveq2d 7424	. . . 4 ⊢ (𝑙 = 𝑛 → ((𝑥 decompPMat 𝑘) · (𝑦 decompPMat (𝑙 − 𝑘))) = ((𝑥 decompPMat 𝑘) · (𝑦 decompPMat (𝑛 − 𝑘))))
9	5, 8	mpteq12dv 5199	. . 3 ⊢ (𝑙 = 𝑛 → (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘) · (𝑦 decompPMat (𝑙 − 𝑘)))) = (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘) · (𝑦 decompPMat (𝑛 − 𝑘)))))
10	9	oveq2d 7424	. 2 ⊢ (𝑙 = 𝑛 → (𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘) · (𝑦 decompPMat (𝑙 − 𝑘))))) = (𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘) · (𝑦 decompPMat (𝑛 − 𝑘))))))
11		simpll 778	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑁 ∈ Fin)
12		simplr 780	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑅 ∈ Ring)
13		decpmatmul.p	. . . . . . . . 9 ⊢ 𝑃 = (Poly1‘𝑅)
14		decpmatmul.c	. . . . . . . . 9 ⊢ 𝐶 = (𝑁 Mat 𝑃)
15	13, 14	pmatring 22814	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
16	15	anim1i 626	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐶 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
17		3anass 1109	. . . . . . 7 ⊢ ((𝐶 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ (𝐶 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
18	16, 17	sylibr 237	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐶 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
19		decpmatmul.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐶)
20		eqid 2769	. . . . . . 7 ⊢ (.r‘𝐶) = (.r‘𝐶)
21	19, 20	ringcl 20328	. . . . . 6 ⊢ ((𝐶 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(.r‘𝐶)𝑦) ∈ 𝐵)
22	18, 21	syl 18	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐶)𝑦) ∈ 𝐵)
23		eqid 2769	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
24	13, 14, 19, 23	pmatcoe1fsupp 22823	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑥(.r‘𝐶)𝑦) ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)))
25	11, 12, 22, 24	syl3anc 1396	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)))
26		fvoveq1 7431	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑖 → (coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏)) = (coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏)))
27	26	fveq1d 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑖 → ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛))
28	27	eqeq1d 2771	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑖 → (((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅) ↔ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)))
29		oveq2 7416	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑗 → (𝑖(𝑥(.r‘𝐶)𝑦)𝑏) = (𝑖(𝑥(.r‘𝐶)𝑦)𝑗))
30	29	fveq2d 6883	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑗 → (coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏)) = (coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗)))
31	30	fveq1d 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑗 → ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛))
32	31	eqeq1d 2771	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑗 → (((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅) ↔ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛) = (0g‘𝑅)))
33	28, 32	rspc2va 3602	. . . . . . . . . . . . 13 ⊢ (((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) → ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛) = (0g‘𝑅))
34	33	expcom 418	. . . . . . . . . . . 12 ⊢ (∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅) → ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛) = (0g‘𝑅)))
35	34	adantl 486	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) → ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛) = (0g‘𝑅)))
36	35	3impib 1132	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛) = (0g‘𝑅))
37	36	mpoeq3dva 7485	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑅)))
38		decpmatmul.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (𝑁 Mat 𝑅)
39	38, 23	mat0op 22541	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑅)))
40	1, 39	eqtrid 2816	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑅)))
41	40	ad3antrrr 742	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) → 0 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑅)))
42	37, 41	eqtr4d 2807	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) = 0 )
43	42	ex 417	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → (∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) = 0 ))
44	43	imim2d 58	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑠 < 𝑛 → ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) → (𝑠 < 𝑛 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) = 0 )))
45	44	ralimdva 3183	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) → ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) = 0 )))
46	45	reximdv 3186	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) = 0 )))
47	25, 46	mpd 16	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) = 0 ))
48		decpmatmulsumfsupp.m	. . . . . . . . . . . 12 ⊢ · = (.r‘𝐴)
49	48	oveqi 7421	. . . . . . . . . . 11 ⊢ ((𝑥 decompPMat 𝑘) · (𝑦 decompPMat (𝑛 − 𝑘))) = ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘)))
50	49	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑥 decompPMat 𝑘) · (𝑦 decompPMat (𝑛 − 𝑘))) = ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘))))
51	50	mpteq2dv 5206	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘) · (𝑦 decompPMat (𝑛 − 𝑘)))) = (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘)))))
52	51	oveq2d 7424	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘) · (𝑦 decompPMat (𝑛 − 𝑘))))) = (𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘))))))
53	13, 14, 19, 38	decpmatmul 22894	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → ((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) = (𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘))))))
54	53	ad4ant234 1192	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) = (𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘))))))
55	14, 19	decpmatval 22887	. . . . . . . . 9 ⊢ (((𝑥(.r‘𝐶)𝑦) ∈ 𝐵 ∧ 𝑛 ∈ ℕ0) → ((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)))
56	22, 55	sylan 591	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)))
57	52, 54, 56	3eqtr2d 2810	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘) · (𝑦 decompPMat (𝑛 − 𝑘))))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)))
58	57	eqeq1d 2771	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘) · (𝑦 decompPMat (𝑛 − 𝑘))))) = 0 ↔ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) = 0 ))
59	58	imbi2d 343	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑠 < 𝑛 → (𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘) · (𝑦 decompPMat (𝑛 − 𝑘))))) = 0 ) ↔ (𝑠 < 𝑛 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) = 0 )))
60	59	ralbidva 3192	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘) · (𝑦 decompPMat (𝑛 − 𝑘))))) = 0 ) ↔ ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) = 0 )))
61	60	rexbidv 3195	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘) · (𝑦 decompPMat (𝑛 − 𝑘))))) = 0 ) ↔ ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) = 0 )))
62	47, 61	mpbird 260	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘) · (𝑦 decompPMat (𝑛 − 𝑘))))) = 0 ))
63	3, 4, 10, 62	mptnn0fsuppd 14030	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑙 ∈ ℕ0 ↦ (𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘) · (𝑦 decompPMat (𝑙 − 𝑘)))))) finSupp 0 )

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  Vcvv 3463   class class class wbr 5110   ↦ cmpt 5193  ‘cfv 6533  (class class class)co 7408   ∈ cmpo 7410  Fincfn 8939   finSupp cfsupp 9317  0cc0 11096   < clt 11239   − cmin 11437  ℕ₀cn0 12500  ...cfz 13531  Basecbs 17265  ._rcmulr 17307  0_gc0g 17488   Σ_g cgsu 17489  Ringcrg 20311  Poly₁cpl1 22302  coe₁cco1 22303   Mat cmat 22529   decompPMat cdecpmat 22884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-ot 4600  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7672  df-ofr 7673  df-om 7859  df-1st 7982  df-2nd 7983  df-supp 8153  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-er 8690  df-map 8822  df-pm 8823  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9318  df-sup 9398  df-oi 9468  df-card 9921  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-nn 12230  df-2 12299  df-3 12300  df-4 12301  df-5 12302  df-6 12303  df-7 12304  df-8 12305  df-9 12306  df-n0 12501  df-z 12588  df-dec 12708  df-uz 12859  df-fz 13532  df-fzo 13679  df-seq 14034  df-hash 14363  df-struct 17203  df-sets 17220  df-slot 17238  df-ndx 17250  df-base 17266  df-ress 17287  df-plusg 17319  df-mulr 17320  df-sca 17322  df-vsca 17323  df-ip 17324  df-tset 17325  df-ple 17326  df-ds 17328  df-hom 17330  df-cco 17331  df-0g 17490  df-gsum 17491  df-prds 17496  df-pws 17498  df-mre 17634  df-mrc 17635  df-acs 17637  df-mgm 18694  df-sgrp 18773  df-mnd 18789  df-mhm 18837  df-submnd 18838  df-grp 18999  df-minusg 19000  df-sbg 19001  df-mulg 19130  df-subg 19185  df-ghm 19280  df-cntz 19383  df-cmn 19848  df-abl 19849  df-mgp 20213  df-rng 20227  df-ur 20260  df-ring 20313  df-subrng 20627  df-subrg 20651  df-lmod 20957  df-lss 21027  df-sra 21268  df-rgmod 21269  df-dsmm 21847  df-frlm 21862  df-psr 22024  df-mpl 22026  df-opsr 22028  df-psr1 22305  df-ply1 22307  df-coe1 22308  df-mamu 22513  df-mat 22530  df-decpmat 22885

This theorem is referenced by:  pm2mpmhmlem2  22941