Theorem mp2pm2mp 21868

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 2738	. . . 4 ⊢ (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃)
10		eqid 2738	. . . 4 ⊢ (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃))
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	mply1topmatcl 21862	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)))
12		eqid 2738	. . . 4 ⊢ ( ·𝑠 ‘𝑄) = ( ·𝑠 ‘𝑄)
13		eqid 2738	. . . 4 ⊢ (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘𝑄))
14		eqid 2738	. . . 4 ⊢ (var1‘𝐴) = (var1‘𝐴)
15		mp2pm2mp.t	. . . 4 ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)
16	4, 9, 10, 12, 13, 14, 1, 2, 15	pm2mpfval 21853	. . 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 1407	. 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 21500	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
19	18	3adant3 1130	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝐴 ∈ Ring)
20		eqid 2738	. . . . 5 ⊢ (0g‘𝑄) = (0g‘𝑄)
21	2	ply1ring 21329	. . . . . . 7 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring)
22		ringcmn 19735	. . . . . . 7 ⊢ (𝑄 ∈ Ring → 𝑄 ∈ CMnd)
23	18, 21, 22	3syl 18	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ CMnd)
24	23	3adant3 1130	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑄 ∈ CMnd)
25		nn0ex 12169	. . . . . 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 1190	. . . . . . . 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 2738	. . . . . . . . 9 ⊢ (Base‘𝐴) = (Base‘𝐴)
32	4, 9, 10, 1, 31	decpmatcl 21824	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)) ∧ 𝑛 ∈ ℕ0) → ((𝐼‘𝑂) decompPMat 𝑛) ∈ (Base‘𝐴))
33	28, 29, 30, 32	syl3anc 1369	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑛 ∈ ℕ0) → ((𝐼‘𝑂) decompPMat 𝑛) ∈ (Base‘𝐴))
34		eqid 2738	. . . . . . . 8 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
35	31, 2, 14, 12, 34, 13, 3	ply1tmcl 21353	. . . . . . 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 1369	. . . . . 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 6971	. . . . 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 6756	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → ((coe1‘𝑝)‘𝑘) = ((coe1‘𝑝)‘𝑛))
39	38	oveqd 7272	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑖((coe1‘𝑝)‘𝑘)𝑗) = (𝑖((coe1‘𝑝)‘𝑛)𝑗))
40		oveq1 7262	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑘𝐸𝑌) = (𝑛𝐸𝑌))
41	39, 40	oveq12d 7273	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)) = ((𝑖((coe1‘𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌)))
42	41	cbvmptv 5183	. . . . . . . . . . 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 7271	. . . . . . . . 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 7331	. . . . . . . 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 5175	. . . . . . 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 2766	. . . . . 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 21867	. . . . 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 19431	. . . 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 1136	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑂 ∈ 𝐿)
51	19, 49, 50	3jca 1126	. . 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 21866	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑛 ∈ ℕ0) → ((𝐼‘𝑂) decompPMat 𝑛) = ((coe1‘𝑂)‘𝑛))
53	52	oveq1d 7270	. . . . . . . . . 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 711	. . . . . . . . 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 5170	. . . . . . . 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 7271	. . . . . . 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 6760	. . . . . 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 6758	. . . . 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 2738	. . . . . . . . . 10 ⊢ (coe1‘𝑂) = (coe1‘𝑂)
62	2, 14, 3, 12, 34, 13, 61	ply1coe 21377	. . . . . . . . 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 2744	. . . . . . 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 6760	. . . . . 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 6758	. . . . 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 2778	. . . 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 3107	. . 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 2738	. . . 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 21378	. . 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 2778	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑇‘(𝐼‘𝑂)) = 𝑂)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ↦ cmpt 5153  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  Fincfn 8691  ℕ₀cn0 12163  Basecbs 16840   ·_𝑠 cvsca 16892  0_gc0g 17067   Σ_g cgsu 17068  ._gcmg 18615  CMndccmn 19301  mulGrpcmgp 19635  Ringcrg 19698  var₁cv1 21257  Poly₁cpl1 21258  coe₁cco1 21259   Mat cmat 21464   decompPMat cdecpmat 21819   pMatToMatPoly cpm2mp 21849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-ot 4567  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-ofr 7512  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-pm 8576  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-sup 9131  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-fz 13169  df-fzo 13312  df-seq 13650  df-hash 13973  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-sca 16904  df-vsca 16905  df-ip 16906  df-tset 16907  df-ple 16908  df-ds 16910  df-hom 16912  df-cco 16913  df-0g 17069  df-gsum 17070  df-prds 17075  df-pws 17077  df-mre 17212  df-mrc 17213  df-acs 17215  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-mhm 18345  df-submnd 18346  df-grp 18495  df-minusg 18496  df-sbg 18497  df-mulg 18616  df-subg 18667  df-ghm 18747  df-cntz 18838  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-srg 19657  df-ring 19700  df-subrg 19937  df-lmod 20040  df-lss 20109  df-sra 20349  df-rgmod 20350  df-dsmm 20849  df-frlm 20864  df-psr 21022  df-mvr 21023  df-mpl 21024  df-opsr 21026  df-psr1 21261  df-vr1 21262  df-ply1 21263  df-coe1 21264  df-mamu 21443  df-mat 21465  df-decpmat 21820  df-pm2mp 21850

This theorem is referenced by:  pm2mpfo  21871