Theorem mp2pm2mp 22719

Description: A polynomial over matrices transformed into a polynomial matrix transformed back into the polynomial over matrices. (Contributed by AV, 12-Oct-2019.)

Hypotheses

Ref	Expression
mp2pm2mp.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
mp2pm2mp.q	⊢ 𝑄 = (Poly1‘𝐴)
mp2pm2mp.l	⊢ 𝐿 = (Base‘𝑄)
mp2pm2mp.m	⊢ · = ( ·𝑠 ‘𝑃)
mp2pm2mp.e	⊢ 𝐸 = (.g‘(mulGrp‘𝑃))
mp2pm2mp.y	⊢ 𝑌 = (var1‘𝑅)
mp2pm2mp.i	⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))
mp2pm2mplem2.p	⊢ 𝑃 = (Poly1‘𝑅)
mp2pm2mp.t	⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)

Assertion

Ref	Expression
mp2pm2mp	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑇‘(𝐼‘𝑂)) = 𝑂)

Distinct variable groups:   𝐸,𝑝   𝐿,𝑝   𝑖,𝑁,𝑗,𝑝   𝑖,𝑂,𝑗,𝑝,𝑘   𝑃,𝑝   𝑅,𝑝   𝑌,𝑝   · ,𝑝   𝑘,𝐿   𝑃,𝑖,𝑗,𝑘   𝑅,𝑘   · ,𝑘   𝑖,𝐸,𝑗   𝑖,𝐿,𝑗   𝑘,𝑁   𝑅,𝑖,𝑗   𝑖,𝑌,𝑗   · ,𝑖,𝑗   𝐴,𝑖,𝑗,𝑘   𝑘,𝐸   𝑘,𝑌

Allowed substitution hints:   𝐴(𝑝)   𝑄(𝑖,𝑗,𝑘,𝑝)   𝑇(𝑖,𝑗,𝑘,𝑝)   𝐼(𝑖,𝑗,𝑘,𝑝)

Proof of Theorem mp2pm2mp

Dummy variables 𝑛 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mp2pm2mp.a	. . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		mp2pm2mp.q	. . . 4 ⊢ 𝑄 = (Poly1‘𝐴)
3		mp2pm2mp.l	. . . 4 ⊢ 𝐿 = (Base‘𝑄)
4		mp2pm2mplem2.p	. . . 4 ⊢ 𝑃 = (Poly1‘𝑅)
5		mp2pm2mp.m	. . . 4 ⊢ · = ( ·𝑠 ‘𝑃)
6		mp2pm2mp.e	. . . 4 ⊢ 𝐸 = (.g‘(mulGrp‘𝑃))
7		mp2pm2mp.y	. . . 4 ⊢ 𝑌 = (var1‘𝑅)
8		mp2pm2mp.i	. . . 4 ⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))
9		eqid 2730	. . . 4 ⊢ (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃)
10		eqid 2730	. . . 4 ⊢ (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃))
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	mply1topmatcl 22713	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)))
12		eqid 2730	. . . 4 ⊢ ( ·𝑠 ‘𝑄) = ( ·𝑠 ‘𝑄)
13		eqid 2730	. . . 4 ⊢ (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘𝑄))
14		eqid 2730	. . . 4 ⊢ (var1‘𝐴) = (var1‘𝐴)
15		mp2pm2mp.t	. . . 4 ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)
16	4, 9, 10, 12, 13, 14, 1, 2, 15	pm2mpfval 22704	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃))) → (𝑇‘(𝐼‘𝑂)) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼‘𝑂) decompPMat 𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))))))
17	11, 16	syld3an3 1411	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑇‘(𝐼‘𝑂)) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼‘𝑂) decompPMat 𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))))))
18	1	matring 22351	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
19	18	3adant3 1132	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝐴 ∈ Ring)
20		eqid 2730	. . . . 5 ⊢ (0g‘𝑄) = (0g‘𝑄)
21	2	ply1ring 22153	. . . . . . 7 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring)
22		ringcmn 20193	. . . . . . 7 ⊢ (𝑄 ∈ Ring → 𝑄 ∈ CMnd)
23	18, 21, 22	3syl 18	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ CMnd)
24	23	3adant3 1132	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑄 ∈ CMnd)
25		nn0ex 12379	. . . . . 6 ⊢ ℕ0 ∈ V
26	25	a1i 11	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → ℕ0 ∈ V)
27	19	adantr 480	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring)
28		simpl2 1193	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
29	11	adantr 480	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑛 ∈ ℕ0) → (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)))
30		simpr 484	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
31		eqid 2730	. . . . . . . . 9 ⊢ (Base‘𝐴) = (Base‘𝐴)
32	4, 9, 10, 1, 31	decpmatcl 22675	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)) ∧ 𝑛 ∈ ℕ0) → ((𝐼‘𝑂) decompPMat 𝑛) ∈ (Base‘𝐴))
33	28, 29, 30, 32	syl3anc 1373	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑛 ∈ ℕ0) → ((𝐼‘𝑂) decompPMat 𝑛) ∈ (Base‘𝐴))
34		eqid 2730	. . . . . . . 8 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
35	31, 2, 14, 12, 34, 13, 3	ply1tmcl 22179	. . . . . . 7 ⊢ ((𝐴 ∈ Ring ∧ ((𝐼‘𝑂) decompPMat 𝑛) ∈ (Base‘𝐴) ∧ 𝑛 ∈ ℕ0) → (((𝐼‘𝑂) decompPMat 𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))) ∈ 𝐿)
36	27, 33, 30, 35	syl3anc 1373	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑛 ∈ ℕ0) → (((𝐼‘𝑂) decompPMat 𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))) ∈ 𝐿)
37	36	fmpttd 7043	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑛 ∈ ℕ0 ↦ (((𝐼‘𝑂) decompPMat 𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴)))):ℕ0⟶𝐿)
38		fveq2 6817	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → ((coe1‘𝑝)‘𝑘) = ((coe1‘𝑝)‘𝑛))
39	38	oveqd 7358	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑖((coe1‘𝑝)‘𝑘)𝑗) = (𝑖((coe1‘𝑝)‘𝑛)𝑗))
40		oveq1 7348	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑘𝐸𝑌) = (𝑛𝐸𝑌))
41	39, 40	oveq12d 7359	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)) = ((𝑖((coe1‘𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌)))
42	41	cbvmptv 5193	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))) = (𝑛 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌)))
43	42	a1i 11	. . . . . . . . . 10 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))) = (𝑛 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌))))
44	43	oveq2d 7357	. . . . . . . . 9 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌)))))
45	44	mpoeq3ia 7419	. . . . . . . 8 ⊢ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌)))))
46	45	mpteq2i 5185	. . . . . . 7 ⊢ (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))) = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌))))))
47	8, 46	eqtri 2753	. . . . . 6 ⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌))))))
48	1, 2, 3, 5, 6, 7, 47, 4, 12, 13, 14	mp2pm2mplem5 22718	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑛 ∈ ℕ0 ↦ (((𝐼‘𝑂) decompPMat 𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴)))) finSupp (0g‘𝑄))
49	3, 20, 24, 26, 37, 48	gsumcl 19820	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼‘𝑂) decompPMat 𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))))) ∈ 𝐿)
50		simp3 1138	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑂 ∈ 𝐿)
51	19, 49, 50	3jca 1128	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝐴 ∈ Ring ∧ (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼‘𝑂) decompPMat 𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))))) ∈ 𝐿 ∧ 𝑂 ∈ 𝐿))
52	1, 2, 3, 5, 6, 7, 8, 4	mp2pm2mplem4 22717	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑛 ∈ ℕ0) → ((𝐼‘𝑂) decompPMat 𝑛) = ((coe1‘𝑂)‘𝑛))
53	52	oveq1d 7356	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑛 ∈ ℕ0) → (((𝐼‘𝑂) decompPMat 𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))) = (((coe1‘𝑂)‘𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))))
54	53	adantlr 715	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑙 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (((𝐼‘𝑂) decompPMat 𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))) = (((coe1‘𝑂)‘𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))))
55	54	mpteq2dva 5182	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑙 ∈ ℕ0) → (𝑛 ∈ ℕ0 ↦ (((𝐼‘𝑂) decompPMat 𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴)))) = (𝑛 ∈ ℕ0 ↦ (((coe1‘𝑂)‘𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴)))))
56	55	oveq2d 7357	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑙 ∈ ℕ0) → (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼‘𝑂) decompPMat 𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘𝑂)‘𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))))))
57	56	fveq2d 6821	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑙 ∈ ℕ0) → (coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼‘𝑂) decompPMat 𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴)))))) = (coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘𝑂)‘𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴)))))))
58	57	fveq1d 6819	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑙 ∈ ℕ0) → ((coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼‘𝑂) decompPMat 𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))))))‘𝑙) = ((coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘𝑂)‘𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))))))‘𝑙))
59	19, 50	jca 511	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝐴 ∈ Ring ∧ 𝑂 ∈ 𝐿))
60	59	adantr 480	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑙 ∈ ℕ0) → (𝐴 ∈ Ring ∧ 𝑂 ∈ 𝐿))
61		eqid 2730	. . . . . . . . . 10 ⊢ (coe1‘𝑂) = (coe1‘𝑂)
62	2, 14, 3, 12, 34, 13, 61	ply1coe 22206	. . . . . . . . 9 ⊢ ((𝐴 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑂 = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘𝑂)‘𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))))))
63	60, 62	syl 17	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑙 ∈ ℕ0) → 𝑂 = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘𝑂)‘𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))))))
64	63	eqcomd 2736	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑙 ∈ ℕ0) → (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘𝑂)‘𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))))) = 𝑂)
65	64	fveq2d 6821	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑙 ∈ ℕ0) → (coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘𝑂)‘𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴)))))) = (coe1‘𝑂))
66	65	fveq1d 6819	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑙 ∈ ℕ0) → ((coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘𝑂)‘𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))))))‘𝑙) = ((coe1‘𝑂)‘𝑙))
67	58, 66	eqtrd 2765	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑙 ∈ ℕ0) → ((coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼‘𝑂) decompPMat 𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))))))‘𝑙) = ((coe1‘𝑂)‘𝑙))
68	67	ralrimiva 3122	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → ∀𝑙 ∈ ℕ0 ((coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼‘𝑂) decompPMat 𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))))))‘𝑙) = ((coe1‘𝑂)‘𝑙))
69		eqid 2730	. . . 4 ⊢ (coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼‘𝑂) decompPMat 𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴)))))) = (coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼‘𝑂) decompPMat 𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))))))
70	2, 3, 69, 61	eqcoe1ply1eq 22207	. . 3 ⊢ ((𝐴 ∈ Ring ∧ (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼‘𝑂) decompPMat 𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))))) ∈ 𝐿 ∧ 𝑂 ∈ 𝐿) → (∀𝑙 ∈ ℕ0 ((coe1‘(𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼‘𝑂) decompPMat 𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))))))‘𝑙) = ((coe1‘𝑂)‘𝑙) → (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼‘𝑂) decompPMat 𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))))) = 𝑂))
71	51, 68, 70	sylc 65	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝐼‘𝑂) decompPMat 𝑛)( ·𝑠 ‘𝑄)(𝑛(.g‘(mulGrp‘𝑄))(var1‘𝐴))))) = 𝑂)
72	17, 71	eqtrd 2765	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑇‘(𝐼‘𝑂)) = 𝑂)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2110  ∀wral 3045  Vcvv 3434   ↦ cmpt 5170  ‘cfv 6477  (class class class)co 7341   ∈ cmpo 7343  Fincfn 8864  ℕ₀cn0 12373  Basecbs 17112   ·_𝑠 cvsca 17157  0_gc0g 17335   Σ_g cgsu 17336  ._gcmg 18972  CMndccmn 19685  mulGrpcmgp 20051  Ringcrg 20144  var₁cv1 22081  Poly₁cpl1 22082  coe₁cco1 22083   Mat cmat 22315   decompPMat cdecpmat 22670   pMatToMatPoly cpm2mp 22700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-cnex 11054  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074  ax-pre-mulgt0 11075

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-tp 4579  df-op 4581  df-ot 4583  df-uni 4858  df-int 4896  df-iun 4941  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-isom 6486  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7605  df-ofr 7606  df-om 7792  df-1st 7916  df-2nd 7917  df-supp 8086  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-er 8617  df-map 8747  df-pm 8748  df-ixp 8817  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-fsupp 9241  df-sup 9321  df-oi 9391  df-card 9824  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-nn 12118  df-2 12180  df-3 12181  df-4 12182  df-5 12183  df-6 12184  df-7 12185  df-8 12186  df-9 12187  df-n0 12374  df-z 12461  df-dec 12581  df-uz 12725  df-fz 13400  df-fzo 13547  df-seq 13901  df-hash 14230  df-struct 17050  df-sets 17067  df-slot 17085  df-ndx 17097  df-base 17113  df-ress 17134  df-plusg 17166  df-mulr 17167  df-sca 17169  df-vsca 17170  df-ip 17171  df-tset 17172  df-ple 17173  df-ds 17175  df-hom 17177  df-cco 17178  df-0g 17337  df-gsum 17338  df-prds 17343  df-pws 17345  df-mre 17480  df-mrc 17481  df-acs 17483  df-mgm 18540  df-sgrp 18619  df-mnd 18635  df-mhm 18683  df-submnd 18684  df-grp 18841  df-minusg 18842  df-sbg 18843  df-mulg 18973  df-subg 19028  df-ghm 19118  df-cntz 19222  df-cmn 19687  df-abl 19688  df-mgp 20052  df-rng 20064  df-ur 20093  df-srg 20098  df-ring 20146  df-subrng 20454  df-subrg 20478  df-lmod 20788  df-lss 20858  df-sra 21100  df-rgmod 21101  df-dsmm 21662  df-frlm 21677  df-psr 21839  df-mvr 21840  df-mpl 21841  df-opsr 21843  df-psr1 22085  df-vr1 22086  df-ply1 22087  df-coe1 22088  df-mamu 22299  df-mat 22316  df-decpmat 22671  df-pm2mp 22701

This theorem is referenced by:  pm2mpfo  22722