Theorem pm2mpmhmlem1 22311

Description: Lemma 1 for pm2mpmhm 22313. (Contributed by AV, 21-Oct-2019.) (Revised by AV, 6-Dec-2019.)

Hypotheses

Ref	Expression
pm2mpfo.p	⊢ 𝑃 = (Poly1‘𝑅)
pm2mpfo.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
pm2mpfo.b	⊢ 𝐵 = (Base‘𝐶)
pm2mpfo.m	⊢ ∗ = ( ·𝑠 ‘𝑄)
pm2mpfo.e	⊢ ↑ = (.g‘(mulGrp‘𝑄))
pm2mpfo.x	⊢ 𝑋 = (var1‘𝐴)
pm2mpfo.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
pm2mpfo.q	⊢ 𝑄 = (Poly1‘𝐴)
pm2mpfo.l	⊢ 𝐿 = (Base‘𝑄)
pm2mpfo.t	⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)

Assertion

Ref	Expression
pm2mpmhmlem1	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑙 ∈ ℕ0 ↦ ((𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑙 − 𝑘))))) ∗ (𝑙 ↑ 𝑋))) finSupp (0g‘𝑄))

Distinct variable groups:   𝐴,𝑘   𝐵,𝑘,𝑙   𝐶,𝑘,𝑙   𝑘,𝐿   𝑘,𝑁,𝑙   𝑄,𝑘   𝑅,𝑘   ∗ ,𝑘   𝐴,𝑙   𝑃,𝑘   𝑅,𝑙   𝑋,𝑙   ∗ ,𝑙   ↑ ,𝑙   𝑥,𝑦,𝑘,𝑙

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝑃(𝑥,𝑦,𝑙)   𝑄(𝑥,𝑦,𝑙)   𝑅(𝑥,𝑦)   𝑇(𝑥,𝑦,𝑘,𝑙)   ↑ (𝑥,𝑦,𝑘)   ∗ (𝑥,𝑦)   𝐿(𝑥,𝑦,𝑙)   𝑁(𝑥,𝑦)   𝑋(𝑥,𝑦,𝑘)

Proof of Theorem pm2mpmhmlem1

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

Step	Hyp	Ref	Expression
1		fvexd 6903	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (0g‘𝑄) ∈ V)
2		ovexd 7440	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑙 ∈ ℕ0) → ((𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑙 − 𝑘))))) ∗ (𝑙 ↑ 𝑋)) ∈ V)
3		oveq2 7413	. . . . 5 ⊢ (𝑙 = 𝑛 → (0...𝑙) = (0...𝑛))
4		oveq1 7412	. . . . . . 7 ⊢ (𝑙 = 𝑛 → (𝑙 − 𝑘) = (𝑛 − 𝑘))
5	4	oveq2d 7421	. . . . . 6 ⊢ (𝑙 = 𝑛 → (𝑦 decompPMat (𝑙 − 𝑘)) = (𝑦 decompPMat (𝑛 − 𝑘)))
6	5	oveq2d 7421	. . . . 5 ⊢ (𝑙 = 𝑛 → ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑙 − 𝑘))) = ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘))))
7	3, 6	mpteq12dv 5238	. . . 4 ⊢ (𝑙 = 𝑛 → (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑙 − 𝑘)))) = (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘)))))
8	7	oveq2d 7421	. . 3 ⊢ (𝑙 = 𝑛 → (𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑙 − 𝑘))))) = (𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘))))))
9		oveq1 7412	. . 3 ⊢ (𝑙 = 𝑛 → (𝑙 ↑ 𝑋) = (𝑛 ↑ 𝑋))
10	8, 9	oveq12d 7423	. 2 ⊢ (𝑙 = 𝑛 → ((𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑙 − 𝑘))))) ∗ (𝑙 ↑ 𝑋)) = ((𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘))))) ∗ (𝑛 ↑ 𝑋)))
11		simpll 765	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑁 ∈ Fin)
12		simplr 767	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑅 ∈ Ring)
13		pm2mpfo.p	. . . . . . . . . 10 ⊢ 𝑃 = (Poly1‘𝑅)
14		pm2mpfo.c	. . . . . . . . . 10 ⊢ 𝐶 = (𝑁 Mat 𝑃)
15	13, 14	pmatring 22185	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
16	15	anim1i 615	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐶 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
17		3anass 1095	. . . . . . . 8 ⊢ ((𝐶 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ (𝐶 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
18	16, 17	sylibr 233	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐶 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
19		pm2mpfo.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
20		eqid 2732	. . . . . . . 8 ⊢ (.r‘𝐶) = (.r‘𝐶)
21	19, 20	ringcl 20066	. . . . . . 7 ⊢ ((𝐶 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(.r‘𝐶)𝑦) ∈ 𝐵)
22	18, 21	syl 17	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐶)𝑦) ∈ 𝐵)
23		eqid 2732	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
24	13, 14, 19, 23	pmatcoe1fsupp 22194	. . . . . 6 ⊢ ((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 1371	. . . . 5 ⊢ (((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 7428	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑖 → (coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏)) = (coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏)))
27	26	fveq1d 6890	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑖 → ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛))
28	27	eqeq1d 2734	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑖 → (((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅) ↔ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)))
29		oveq2 7413	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑗 → (𝑖(𝑥(.r‘𝐶)𝑦)𝑏) = (𝑖(𝑥(.r‘𝐶)𝑦)𝑗))
30	29	fveq2d 6892	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑗 → (coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏)) = (coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗)))
31	30	fveq1d 6890	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑗 → ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛))
32	31	eqeq1d 2734	. . . . . . . . . . . . . . . . 17 ⊢ (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 3622	. . . . . . . . . . . . . . . 16 ⊢ (((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 414	. . . . . . . . . . . . . . 15 ⊢ (∀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 482	. . . . . . . . . . . . . 14 ⊢ (((((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 1116	. . . . . . . . . . . . 13 ⊢ ((((((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 7482	. . . . . . . . . . . 12 ⊢ (((((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		pm2mpfo.a	. . . . . . . . . . . . . 14 ⊢ 𝐴 = (𝑁 Mat 𝑅)
39	38, 23	mat0op 21912	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑅)))
40	39	ad3antrrr 728	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) → (0g‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑅)))
41	38	matring 21936	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
42		pm2mpfo.q	. . . . . . . . . . . . . . . 16 ⊢ 𝑄 = (Poly1‘𝐴)
43	42	ply1sca 21766	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
44	41, 43	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 = (Scalar‘𝑄))
45	44	ad3antrrr 728	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) → 𝐴 = (Scalar‘𝑄))
46	45	fveq2d 6892	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = (0g‘𝑅)) → (0g‘𝐴) = (0g‘(Scalar‘𝑄)))
47	37, 40, 46	3eqtr2d 2778	. . . . . . . . . . 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‘(Scalar‘𝑄)))
48	47	oveq1d 7420	. . . . . . . . . 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)) ∗ (𝑛 ↑ 𝑋)) = ((0g‘(Scalar‘𝑄)) ∗ (𝑛 ↑ 𝑋)))
49	42	ply1lmod 21765	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod)
50	41, 49	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
51	50	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑄 ∈ LMod)
52	41	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐴 ∈ Ring)
53		pm2mpfo.x	. . . . . . . . . . . . . 14 ⊢ 𝑋 = (var1‘𝐴)
54		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
55		pm2mpfo.e	. . . . . . . . . . . . . 14 ⊢ ↑ = (.g‘(mulGrp‘𝑄))
56		pm2mpfo.l	. . . . . . . . . . . . . 14 ⊢ 𝐿 = (Base‘𝑄)
57	42, 53, 54, 55, 56	ply1moncl 21784	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Ring ∧ 𝑛 ∈ ℕ0) → (𝑛 ↑ 𝑋) ∈ 𝐿)
58	52, 57	sylan 580	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑛 ↑ 𝑋) ∈ 𝐿)
59		eqid 2732	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝑄) = (Scalar‘𝑄)
60		pm2mpfo.m	. . . . . . . . . . . . 13 ⊢ ∗ = ( ·𝑠 ‘𝑄)
61		eqid 2732	. . . . . . . . . . . . 13 ⊢ (0g‘(Scalar‘𝑄)) = (0g‘(Scalar‘𝑄))
62		eqid 2732	. . . . . . . . . . . . 13 ⊢ (0g‘𝑄) = (0g‘𝑄)
63	56, 59, 60, 61, 62	lmod0vs 20497	. . . . . . . . . . . 12 ⊢ ((𝑄 ∈ LMod ∧ (𝑛 ↑ 𝑋) ∈ 𝐿) → ((0g‘(Scalar‘𝑄)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄))
64	51, 58, 63	syl2an2r 683	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((0g‘(Scalar‘𝑄)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄))
65	64	adantr 481	. . . . . . . . . 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)) → ((0g‘(Scalar‘𝑄)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄))
66	48, 65	eqtrd 2772	. . . . . . . . 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)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄))
67	66	ex 413	. . . . . . . 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)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)))
68	67	imim2d 57	. . . . . . 7 ⊢ ((((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)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄))))
69	68	ralimdva 3167	. . . . . 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)) → ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄))))
70	69	reximdv 3170	. . . . 5 ⊢ (((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)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄))))
71	25, 70	mpd 15	. . . 4 ⊢ (((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)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)))
72	14, 19	decpmatval 22258	. . . . . . . . . 10 ⊢ (((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)))
73	22, 72	sylan 580	. . . . . . . . 9 ⊢ ((((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)))
74	73	oveq1d 7420	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → (((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) ∗ (𝑛 ↑ 𝑋)) = ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) ∗ (𝑛 ↑ 𝑋)))
75	74	eqeq1d 2734	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄) ↔ ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)))
76	75	imbi2d 340	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑠 < 𝑛 → (((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)) ↔ (𝑠 < 𝑛 → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄))))
77	76	ralbidva 3175	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)) ↔ ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄))))
78	77	rexbidv 3178	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)) ↔ ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄))))
79	71, 78	mpbird 256	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)))
80	13, 14, 19, 38	decpmatmul 22265	. . . . . . . . . 10 ⊢ ((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))))))
81	80	ad4ant234 1175	. . . . . . . . 9 ⊢ ((((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))))))
82	81	eqcomd 2738	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘))))) = ((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛))
83	82	oveq1d 7420	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘))))) ∗ (𝑛 ↑ 𝑋)) = (((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) ∗ (𝑛 ↑ 𝑋)))
84	83	eqeq1d 2734	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → (((𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘))))) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄) ↔ (((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)))
85	84	imbi2d 340	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑠 < 𝑛 → ((𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘))))) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)) ↔ (𝑠 < 𝑛 → (((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄))))
86	85	ralbidva 3175	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘))))) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)) ↔ ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄))))
87	86	rexbidv 3178	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘))))) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)) ↔ ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (((𝑥(.r‘𝐶)𝑦) decompPMat 𝑛) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄))))
88	79, 87	mpbird 256	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑘))))) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄)))
89	1, 2, 10, 88	mptnn0fsuppd 13959	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑙 ∈ ℕ0 ↦ ((𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑙 − 𝑘))))) ∗ (𝑙 ↑ 𝑋))) finSupp (0g‘𝑄))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070  Vcvv 3474   class class class wbr 5147   ↦ cmpt 5230  ‘cfv 6540  (class class class)co 7405   ∈ cmpo 7407  Fincfn 8935   finSupp cfsupp 9357  0cc0 11106   < clt 11244   − cmin 11440  ℕ₀cn0 12468  ...cfz 13480  Basecbs 17140  ._rcmulr 17194  Scalarcsca 17196   ·_𝑠 cvsca 17197  0_gc0g 17381   Σ_g cgsu 17382  ._gcmg 18944  mulGrpcmgp 19981  Ringcrg 20049  LModclmod 20463  var₁cv1 21691  Poly₁cpl1 21692  coe₁cco1 21693   Mat cmat 21898   decompPMat cdecpmat 22255   pMatToMatPoly cpm2mp 22285

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-ot 4636  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-ofr 7667  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-pm 8819  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-sup 9433  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-fz 13481  df-fzo 13624  df-seq 13963  df-hash 14287  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-sca 17209  df-vsca 17210  df-ip 17211  df-tset 17212  df-ple 17213  df-ds 17215  df-hom 17217  df-cco 17218  df-0g 17383  df-gsum 17384  df-prds 17389  df-pws 17391  df-mre 17526  df-mrc 17527  df-acs 17529  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-mhm 18667  df-submnd 18668  df-grp 18818  df-minusg 18819  df-sbg 18820  df-mulg 18945  df-subg 18997  df-ghm 19084  df-cntz 19175  df-cmn 19644  df-abl 19645  df-mgp 19982  df-ur 19999  df-ring 20051  df-subrg 20353  df-lmod 20465  df-lss 20535  df-sra 20777  df-rgmod 20778  df-dsmm 21278  df-frlm 21293  df-psr 21453  df-mvr 21454  df-mpl 21455  df-opsr 21457  df-psr1 21695  df-vr1 21696  df-ply1 21697  df-coe1 21698  df-mamu 21877  df-mat 21899  df-decpmat 22256

This theorem is referenced by:  pm2mpmhmlem2  22312