Theorem pm2mpmhmlem1 22733

Description: Lemma 1 for pm2mpmhm 22735. (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 6905	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (0g‘𝑄) ∈ V)
2		ovexd 7448	. 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 7421	. . . . 5 ⊢ (𝑙 = 𝑛 → (0...𝑙) = (0...𝑛))
4		oveq1 7420	. . . . . . 7 ⊢ (𝑙 = 𝑛 → (𝑙 − 𝑘) = (𝑛 − 𝑘))
5	4	oveq2d 7429	. . . . . 6 ⊢ (𝑙 = 𝑛 → (𝑦 decompPMat (𝑙 − 𝑘)) = (𝑦 decompPMat (𝑛 − 𝑘)))
6	5	oveq2d 7429	. . . . 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 5235	. . . 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 7429	. . 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 7420	. . 3 ⊢ (𝑙 = 𝑛 → (𝑙 ↑ 𝑋) = (𝑛 ↑ 𝑋))
10	8, 9	oveq12d 7431	. 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 22607	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
16	15	anim1i 613	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐶 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
17		3anass 1092	. . . . . . . 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 2725	. . . . . . . 8 ⊢ (.r‘𝐶) = (.r‘𝐶)
21	19, 20	ringcl 20189	. . . . . . 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 2725	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
24	13, 14, 19, 23	pmatcoe1fsupp 22616	. . . . . 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 1368	. . . . 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 7436	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑖 → (coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏)) = (coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏)))
27	26	fveq1d 6892	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑖 → ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛))
28	27	eqeq1d 2727	. . . . . . . . . . . . . . . . 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 7421	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑗 → (𝑖(𝑥(.r‘𝐶)𝑦)𝑏) = (𝑖(𝑥(.r‘𝐶)𝑦)𝑗))
30	29	fveq2d 6894	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑗 → (coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏)) = (coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗)))
31	30	fveq1d 6892	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑗 → ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛))
32	31	eqeq1d 2727	. . . . . . . . . . . . . . . . 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 3615	. . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 1113	. . . . . . . . . . . . 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 7491	. . . . . . . . . . . 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 22334	. . . . . . . . . . . . 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 22358	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
42		pm2mpfo.q	. . . . . . . . . . . . . . . 16 ⊢ 𝑄 = (Poly1‘𝐴)
43	42	ply1sca 22175	. . . . . . . . . . . . . . 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 6894	. . . . . . . . . . . 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 2771	. . . . . . . . . . 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 7428	. . . . . . . . . 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 22174	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod)
50	41, 49	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
51	50	adantr 479	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑄 ∈ LMod)
52	41	adantr 479	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐴 ∈ Ring)
53		pm2mpfo.x	. . . . . . . . . . . . . 14 ⊢ 𝑋 = (var1‘𝐴)
54		eqid 2725	. . . . . . . . . . . . . 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 22194	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Ring ∧ 𝑛 ∈ ℕ0) → (𝑛 ↑ 𝑋) ∈ 𝐿)
58	52, 57	sylan 578	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑛 ↑ 𝑋) ∈ 𝐿)
59		eqid 2725	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝑄) = (Scalar‘𝑄)
60		pm2mpfo.m	. . . . . . . . . . . . 13 ⊢ ∗ = ( ·𝑠 ‘𝑄)
61		eqid 2725	. . . . . . . . . . . . 13 ⊢ (0g‘(Scalar‘𝑄)) = (0g‘(Scalar‘𝑄))
62		eqid 2725	. . . . . . . . . . . . 13 ⊢ (0g‘𝑄) = (0g‘𝑄)
63	56, 59, 60, 61, 62	lmod0vs 20777	. . . . . . . . . . . 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 479	. . . . . . . . . 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 2765	. . . . . . . . 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 411	. . . . . . . 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 3157	. . . . . 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 3160	. . . . 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 22680	. . . . . . . . . 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 578	. . . . . . . . 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 7428	. . . . . . . 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 2727	. . . . . . 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 339	. . . . . 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 3166	. . . . 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 3169	. . . 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 22687	. . . . . . . . . 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 1172	. . . . . . . . 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 2731	. . . . . . . 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 7428	. . . . . . 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 2727	. . . . . 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 339	. . . . 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 3166	. . . 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 3169	. . 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 13990	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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3051  ∃wrex 3060  Vcvv 3463   class class class wbr 5144   ↦ cmpt 5227  ‘cfv 6543  (class class class)co 7413   ∈ cmpo 7415  Fincfn 8957   finSupp cfsupp 9380  0cc0 11133   < clt 11273   − cmin 11469  ℕ₀cn0 12497  ...cfz 13511  Basecbs 17174  ._rcmulr 17228  Scalarcsca 17230   ·_𝑠 cvsca 17231  0_gc0g 17415   Σ_g cgsu 17416  ._gcmg 19022  mulGrpcmgp 20073  Ringcrg 20172  LModclmod 20742  var₁cv1 22098  Poly₁cpl1 22099  coe₁cco1 22100   Mat cmat 22320   decompPMat cdecpmat 22677   pMatToMatPoly cpm2mp 22707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4905  df-int 4946  df-iun 4994  df-iin 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-se 5629  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7679  df-ofr 7680  df-om 7866  df-1st 7987  df-2nd 7988  df-supp 8159  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8840  df-pm 8841  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fsupp 9381  df-sup 9460  df-oi 9528  df-card 9957  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-2 12300  df-3 12301  df-4 12302  df-5 12303  df-6 12304  df-7 12305  df-8 12306  df-9 12307  df-n0 12498  df-z 12584  df-dec 12703  df-uz 12848  df-fz 13512  df-fzo 13655  df-seq 13994  df-hash 14317  df-struct 17110  df-sets 17127  df-slot 17145  df-ndx 17157  df-base 17175  df-ress 17204  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-hom 17251  df-cco 17252  df-0g 17417  df-gsum 17418  df-prds 17423  df-pws 17425  df-mre 17560  df-mrc 17561  df-acs 17563  df-mgm 18594  df-sgrp 18673  df-mnd 18689  df-mhm 18734  df-submnd 18735  df-grp 18892  df-minusg 18893  df-sbg 18894  df-mulg 19023  df-subg 19077  df-ghm 19167  df-cntz 19267  df-cmn 19736  df-abl 19737  df-mgp 20074  df-rng 20092  df-ur 20121  df-ring 20174  df-subrng 20482  df-subrg 20507  df-lmod 20744  df-lss 20815  df-sra 21057  df-rgmod 21058  df-dsmm 21665  df-frlm 21680  df-psr 21841  df-mvr 21842  df-mpl 21843  df-opsr 21845  df-psr1 22102  df-vr1 22103  df-ply1 22104  df-coe1 22105  df-mamu 22304  df-mat 22321  df-decpmat 22678

This theorem is referenced by:  pm2mpmhmlem2  22734