Theorem mp2pm2mp 22304

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 2732	. . . 4 ⊢ (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃)
10		eqid 2732	. . . 4 ⊢ (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃))
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	mply1topmatcl 22298	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)))
12		eqid 2732	. . . 4 ⊢ ( ·𝑠 ‘𝑄) = ( ·𝑠 ‘𝑄)
13		eqid 2732	. . . 4 ⊢ (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘𝑄))
14		eqid 2732	. . . 4 ⊢ (var1‘𝐴) = (var1‘𝐴)
15		mp2pm2mp.t	. . . 4 ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)
16	4, 9, 10, 12, 13, 14, 1, 2, 15	pm2mpfval 22289	. . 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 1409	. 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 21936	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
19	18	3adant3 1132	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝐴 ∈ Ring)
20		eqid 2732	. . . . 5 ⊢ (0g‘𝑄) = (0g‘𝑄)
21	2	ply1ring 21761	. . . . . . 7 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring)
22		ringcmn 20092	. . . . . . 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 12474	. . . . . 6 ⊢ ℕ0 ∈ V
26	25	a1i 11	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → ℕ0 ∈ V)
27	19	adantr 481	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring)
28		simpl2 1192	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
29	11	adantr 481	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑛 ∈ ℕ0) → (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)))
30		simpr 485	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
31		eqid 2732	. . . . . . . . 9 ⊢ (Base‘𝐴) = (Base‘𝐴)
32	4, 9, 10, 1, 31	decpmatcl 22260	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)) ∧ 𝑛 ∈ ℕ0) → ((𝐼‘𝑂) decompPMat 𝑛) ∈ (Base‘𝐴))
33	28, 29, 30, 32	syl3anc 1371	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑛 ∈ ℕ0) → ((𝐼‘𝑂) decompPMat 𝑛) ∈ (Base‘𝐴))
34		eqid 2732	. . . . . . . 8 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
35	31, 2, 14, 12, 34, 13, 3	ply1tmcl 21785	. . . . . . 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 1371	. . . . . 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 7111	. . . . 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 6888	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → ((coe1‘𝑝)‘𝑘) = ((coe1‘𝑝)‘𝑛))
39	38	oveqd 7422	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑖((coe1‘𝑝)‘𝑘)𝑗) = (𝑖((coe1‘𝑝)‘𝑛)𝑗))
40		oveq1 7412	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑘𝐸𝑌) = (𝑛𝐸𝑌))
41	39, 40	oveq12d 7423	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)) = ((𝑖((coe1‘𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌)))
42	41	cbvmptv 5260	. . . . . . . . . . 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 7421	. . . . . . . . 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 7483	. . . . . . . 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 5252	. . . . . . 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 2760	. . . . . 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 22303	. . . . 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 19777	. . . 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 22302	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑛 ∈ ℕ0) → ((𝐼‘𝑂) decompPMat 𝑛) = ((coe1‘𝑂)‘𝑛))
53	52	oveq1d 7420	. . . . . . . . . 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 713	. . . . . . . . 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 5247	. . . . . . . 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 7421	. . . . . . 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 6892	. . . . . 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 6890	. . . . 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 512	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝐴 ∈ Ring ∧ 𝑂 ∈ 𝐿))
60	59	adantr 481	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑙 ∈ ℕ0) → (𝐴 ∈ Ring ∧ 𝑂 ∈ 𝐿))
61		eqid 2732	. . . . . . . . . 10 ⊢ (coe1‘𝑂) = (coe1‘𝑂)
62	2, 14, 3, 12, 34, 13, 61	ply1coe 21811	. . . . . . . . 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 2738	. . . . . . 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 6892	. . . . . 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 6890	. . . . 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 2772	. . . 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 3146	. . 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 2732	. . . 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 21812	. . 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 2772	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑇‘(𝐼‘𝑂)) = 𝑂)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  Vcvv 3474   ↦ cmpt 5230  ‘cfv 6540  (class class class)co 7405   ∈ cmpo 7407  Fincfn 8935  ℕ₀cn0 12468  Basecbs 17140   ·_𝑠 cvsca 17197  0_gc0g 17381   Σ_g cgsu 17382  ._gcmg 18944  CMndccmn 19642  mulGrpcmgp 19981  Ringcrg 20049  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-srg 20003  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  df-pm2mp 22286

This theorem is referenced by:  pm2mpfo  22307