Theorem pm2mpmhmlem1 20842

Description: Lemma 1 for pm2mpmhm 20844. (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 6344	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (0g‘𝑄) ∈ V)
2		ovexd 6824	. 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 6800	. . . . 5 ⊢ (𝑙 = 𝑛 → (0...𝑙) = (0...𝑛))
4		oveq1 6799	. . . . . . 7 ⊢ (𝑙 = 𝑛 → (𝑙 − 𝑘) = (𝑛 − 𝑘))
5	4	oveq2d 6808	. . . . . 6 ⊢ (𝑙 = 𝑛 → (𝑦 decompPMat (𝑙 − 𝑘)) = (𝑦 decompPMat (𝑛 − 𝑘)))
6	5	oveq2d 6808	. . . . 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 4865	. . . 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 6808	. . 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 6799	. . 3 ⊢ (𝑙 = 𝑛 → (𝑙 ↑ 𝑋) = (𝑛 ↑ 𝑋))
10	8, 9	oveq12d 6810	. 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 742	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑁 ∈ Fin)
12		simplr 744	. . . . . 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 20717	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
16	15	anim1i 594	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐶 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
17		3anass 1079	. . . . . . . 8 ⊢ ((𝐶 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ (𝐶 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
18	16, 17	sylibr 224	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐶 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
19		pm2mpfo.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
20		eqid 2770	. . . . . . . 8 ⊢ (.r‘𝐶) = (.r‘𝐶)
21	19, 20	ringcl 18768	. . . . . . 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 2770	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
24	13, 14, 19, 23	pmatcoe1fsupp 20725	. . . . . 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 1475	. . . . 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 6815	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑖 → (coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏)) = (coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏)))
27	26	fveq1d 6334	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑖 → ((coe1‘(𝑎(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛))
28	27	eqeq1d 2772	. . . . . . . . . . . . . . . . 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 6800	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑗 → (𝑖(𝑥(.r‘𝐶)𝑦)𝑏) = (𝑖(𝑥(.r‘𝐶)𝑦)𝑗))
30	29	fveq2d 6336	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑗 → (coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏)) = (coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗)))
31	30	fveq1d 6334	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑗 → ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑏))‘𝑛) = ((coe1‘(𝑖(𝑥(.r‘𝐶)𝑦)𝑗))‘𝑛))
32	31	eqeq1d 2772	. . . . . . . . . . . . . . . . 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 3471	. . . . . . . . . . . . . . . 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 398	. . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . 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 1107	. . . . . . . . . . . . 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	mpt2eq3dva 6865	. . . . . . . . . . . 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 20441	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑅)))
40	39	ad3antrrr 701	. . . . . . . . . . . 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 20465	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
42		pm2mpfo.q	. . . . . . . . . . . . . . . 16 ⊢ 𝑄 = (Poly1‘𝐴)
43	42	ply1sca 19837	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
44	41, 43	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 = (Scalar‘𝑄))
45	44	ad3antrrr 701	. . . . . . . . . . . . 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 6336	. . . . . . . . . . . 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 2810	. . . . . . . . . . 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 6807	. . . . . . . . . 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 19836	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod)
50	41, 49	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
51	50	adantr 466	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑄 ∈ LMod)
52	51	adantr 466	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑄 ∈ LMod)
53	41	adantr 466	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐴 ∈ Ring)
54		pm2mpfo.x	. . . . . . . . . . . . . 14 ⊢ 𝑋 = (var1‘𝐴)
55		eqid 2770	. . . . . . . . . . . . . 14 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
56		pm2mpfo.e	. . . . . . . . . . . . . 14 ⊢ ↑ = (.g‘(mulGrp‘𝑄))
57		pm2mpfo.l	. . . . . . . . . . . . . 14 ⊢ 𝐿 = (Base‘𝑄)
58	42, 54, 55, 56, 57	ply1moncl 19855	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Ring ∧ 𝑛 ∈ ℕ0) → (𝑛 ↑ 𝑋) ∈ 𝐿)
59	53, 58	sylan 561	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑛 ↑ 𝑋) ∈ 𝐿)
60		eqid 2770	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝑄) = (Scalar‘𝑄)
61		pm2mpfo.m	. . . . . . . . . . . . 13 ⊢ ∗ = ( ·𝑠 ‘𝑄)
62		eqid 2770	. . . . . . . . . . . . 13 ⊢ (0g‘(Scalar‘𝑄)) = (0g‘(Scalar‘𝑄))
63		eqid 2770	. . . . . . . . . . . . 13 ⊢ (0g‘𝑄) = (0g‘𝑄)
64	57, 60, 61, 62, 63	lmod0vs 19105	. . . . . . . . . . . 12 ⊢ ((𝑄 ∈ LMod ∧ (𝑛 ↑ 𝑋) ∈ 𝐿) → ((0g‘(Scalar‘𝑄)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄))
65	52, 59, 64	syl2anc 565	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((0g‘(Scalar‘𝑄)) ∗ (𝑛 ↑ 𝑋)) = (0g‘𝑄))
66	65	adantr 466	. . . . . . . . . 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))
67	48, 66	eqtrd 2804	. . . . . . . . 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))
68	67	ex 397	. . . . . . . 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)))
69	68	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))))
70	69	ralimdva 3110	. . . . . 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))))
71	70	reximdv 3163	. . . . 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))))
72	25, 71	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)))
73	14, 19	decpmatval 20789	. . . . . . . . . 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)))
74	22, 73	sylan 561	. . . . . . . . 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)))
75	74	oveq1d 6807	. . . . . . . 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)))
76	75	eqeq1d 2772	. . . . . . 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)))
77	76	imbi2d 329	. . . . . 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))))
78	77	ralbidva 3133	. . . . 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))))
79	78	rexbidv 3199	. . . 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))))
80	72, 79	mpbird 247	. . 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)))
81		simpllr 752	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
82		simplr 744	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
83		simpr 471	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
84	13, 14, 19, 38	decpmatmul 20796	. . . . . . . . . 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))))))
85	81, 82, 83, 84	syl3anc 1475	. . . . . . . . 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))))))
86	85	eqcomd 2776	. . . . . . . 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))
87	86	oveq1d 6807	. . . . . . 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)))
88	87	eqeq1d 2772	. . . . . 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)))
89	88	imbi2d 329	. . . . 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))))
90	89	ralbidva 3133	. . . 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))))
91	90	rexbidv 3199	. . 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))))
92	80, 91	mpbird 247	. 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)))
93	1, 2, 10, 92	mptnn0fsuppd 13004	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 382   ∧ w3a 1070   = wceq 1630   ∈ wcel 2144  ∀wral 3060  ∃wrex 3061  Vcvv 3349   class class class wbr 4784   ↦ cmpt 4861  ‘cfv 6031  (class class class)co 6792   ↦ cmpt2 6794  Fincfn 8108   finSupp cfsupp 8430  0cc0 10137   < clt 10275   − cmin 10467  ℕ₀cn0 11493  ...cfz 12532  Basecbs 16063  ._rcmulr 16149  Scalarcsca 16151   ·_𝑠 cvsca 16152  0_gc0g 16307   Σ_g cgsu 16308  ._gcmg 17747  mulGrpcmgp 18696  Ringcrg 18754  LModclmod 19072  var₁cv1 19760  Poly₁cpl1 19761  coe₁cco1 19762   Mat cmat 20429   decompPMat cdecpmat 20786   pMatToMatPoly cpm2mp 20816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-ot 4323  df-uni 4573  df-int 4610  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-of 7043  df-ofr 7044  df-om 7212  df-1st 7314  df-2nd 7315  df-supp 7446  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-2o 7713  df-oadd 7716  df-er 7895  df-map 8010  df-pm 8011  df-ixp 8062  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-fsupp 8431  df-sup 8503  df-oi 8570  df-card 8964  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-7 11285  df-8 11286  df-9 11287  df-n0 11494  df-z 11579  df-dec 11695  df-uz 11888  df-fz 12533  df-fzo 12673  df-seq 13008  df-hash 13321  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-ress 16071  df-plusg 16161  df-mulr 16162  df-sca 16164  df-vsca 16165  df-ip 16166  df-tset 16167  df-ple 16168  df-ds 16171  df-hom 16173  df-cco 16174  df-0g 16309  df-gsum 16310  df-prds 16315  df-pws 16317  df-mre 16453  df-mrc 16454  df-acs 16456  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-mhm 17542  df-submnd 17543  df-grp 17632  df-minusg 17633  df-sbg 17634  df-mulg 17748  df-subg 17798  df-ghm 17865  df-cntz 17956  df-cmn 18401  df-abl 18402  df-mgp 18697  df-ur 18709  df-ring 18756  df-subrg 18987  df-lmod 19074  df-lss 19142  df-sra 19386  df-rgmod 19387  df-psr 19570  df-mvr 19571  df-mpl 19572  df-opsr 19574  df-psr1 19764  df-vr1 19765  df-ply1 19766  df-coe1 19767  df-dsmm 20292  df-frlm 20307  df-mamu 20406  df-mat 20430  df-decpmat 20787

This theorem is referenced by:  pm2mpmhmlem2  20843