Theorem mp2pm2mp 22844

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 2756	. . . 4 ⊢ (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃)
10		eqid 2756	. . . 4 ⊢ (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃))
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	mply1topmatcl 22838	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)))
12		eqid 2756	. . . 4 ⊢ ( ·𝑠 ‘𝑄) = ( ·𝑠 ‘𝑄)
13		eqid 2756	. . . 4 ⊢ (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘𝑄))
14		eqid 2756	. . . 4 ⊢ (var1‘𝐴) = (var1‘𝐴)
15		mp2pm2mp.t	. . . 4 ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)
16	4, 9, 10, 12, 13, 14, 1, 2, 15	pm2mpfval 22829	. . 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 1424	. 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 22476	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
19	18	3adant3 1141	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝐴 ∈ Ring)
20		eqid 2756	. . . . 5 ⊢ (0g‘𝑄) = (0g‘𝑄)
21	2	ply1ring 22282	. . . . . . 7 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring)
22		ringcmn 20304	. . . . . . 7 ⊢ (𝑄 ∈ Ring → 𝑄 ∈ CMnd)
23	18, 21, 22	3syl 18	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ CMnd)
24	23	3adant3 1141	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑄 ∈ CMnd)
25		nn0ex 12477	. . . . . 6 ⊢ ℕ0 ∈ V
26	25	a1i 11	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → ℕ0 ∈ V)
27	19	adantr 483	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring)
28		simpl2 1202	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
29	11	adantr 483	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑛 ∈ ℕ0) → (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)))
30		simpr 487	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
31		eqid 2756	. . . . . . . . 9 ⊢ (Base‘𝐴) = (Base‘𝐴)
32	4, 9, 10, 1, 31	decpmatcl 22800	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)) ∧ 𝑛 ∈ ℕ0) → ((𝐼‘𝑂) decompPMat 𝑛) ∈ (Base‘𝐴))
33	28, 29, 30, 32	syl3anc 1386	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑛 ∈ ℕ0) → ((𝐼‘𝑂) decompPMat 𝑛) ∈ (Base‘𝐴))
34		eqid 2756	. . . . . . . 8 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
35	31, 2, 14, 12, 34, 13, 3	ply1tmcl 22308	. . . . . . 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 1386	. . . . . 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 7085	. . . . 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 6856	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → ((coe1‘𝑝)‘𝑘) = ((coe1‘𝑝)‘𝑛))
39	38	oveqd 7402	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑖((coe1‘𝑝)‘𝑘)𝑗) = (𝑖((coe1‘𝑝)‘𝑛)𝑗))
40		oveq1 7392	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑘𝐸𝑌) = (𝑛𝐸𝑌))
41	39, 40	oveq12d 7403	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)) = ((𝑖((coe1‘𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌)))
42	41	cbvmptv 5198	. . . . . . . . . . 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 7401	. . . . . . . . 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 7463	. . . . . . . 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 5190	. . . . . . 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 2779	. . . . . 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 22843	. . . . 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 19931	. . . 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 1147	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑂 ∈ 𝐿)
51	19, 49, 50	3jca 1137	. . 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 22842	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑛 ∈ ℕ0) → ((𝐼‘𝑂) decompPMat 𝑛) = ((coe1‘𝑂)‘𝑛))
53	52	oveq1d 7400	. . . . . . . . . 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 723	. . . . . . . . 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 5187	. . . . . . . 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 7401	. . . . . . 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 6860	. . . . . 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 6858	. . . . 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 518	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝐴 ∈ Ring ∧ 𝑂 ∈ 𝐿))
60	59	adantr 483	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑙 ∈ ℕ0) → (𝐴 ∈ Ring ∧ 𝑂 ∈ 𝐿))
61		eqid 2756	. . . . . . . . . 10 ⊢ (coe1‘𝑂) = (coe1‘𝑂)
62	2, 14, 3, 12, 34, 13, 61	ply1coe 22334	. . . . . . . . 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 2762	. . . . . . 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 6860	. . . . . 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 6858	. . . . 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 2791	. . . 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 3148	. . 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 2756	. . . 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 22335	. . 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 2791	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑇‘(𝐼‘𝑂)) = 𝑂)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1095   = wceq 1554   ∈ wcel 2136  ∀wral 3070  Vcvv 3448   ↦ cmpt 5175  ‘cfv 6510  (class class class)co 7385   ∈ cmpo 7387  Fincfn 8916  ℕ₀cn0 12471  Basecbs 17221   ·_𝑠 cvsca 17266  0_gc0g 17444   Σ_g cgsu 17445  ._gcmg 19085  CMndccmn 19796  mulGrpcmgp 20162  Ringcrg 20255  var₁cv1 22211  Poly₁cpl1 22212  coe₁cco1 22213   Mat cmat 22440   decompPMat cdecpmat 22795   pMatToMatPoly cpm2mp 22825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707  ax-cnex 11119  ax-resscn 11120  ax-1cn 11121  ax-icn 11122  ax-addcl 11123  ax-addrcl 11124  ax-mulcl 11125  ax-mulrcl 11126  ax-mulcom 11127  ax-addass 11128  ax-mulass 11129  ax-distr 11130  ax-i2m1 11131  ax-1ne0 11132  ax-1rid 11133  ax-rnegex 11134  ax-rrecex 11135  ax-cnre 11136  ax-pre-lttri 11137  ax-pre-lttrn 11138  ax-pre-ltadd 11139  ax-pre-mulgt0 11140

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-nel 3056  df-ral 3071  df-rex 3081  df-rmo 3361  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-ot 4585  df-uni 4860  df-int 4900  df-iun 4945  df-iin 4946  df-br 5095  df-opab 5157  df-mpt 5176  df-tr 5202  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-se 5594  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-isom 6519  df-riota 7342  df-ov 7388  df-oprab 7389  df-mpo 7390  df-of 7649  df-ofr 7650  df-om 7836  df-1st 7959  df-2nd 7960  df-supp 8129  df-frecs 8250  df-wrecs 8281  df-recs 8330  df-rdg 8369  df-1o 8425  df-2o 8426  df-er 8666  df-map 8798  df-pm 8799  df-ixp 8869  df-en 8917  df-dom 8918  df-sdom 8919  df-fin 8920  df-fsupp 9298  df-sup 9378  df-oi 9448  df-card 9887  df-pnf 11208  df-mnf 11209  df-xr 11210  df-ltxr 11211  df-le 11212  df-sub 11406  df-neg 11407  df-nn 12201  df-2 12270  df-3 12271  df-4 12272  df-5 12273  df-6 12274  df-7 12275  df-8 12276  df-9 12277  df-n0 12472  df-z 12559  df-dec 12679  df-uz 12830  df-fz 13503  df-fzo 13650  df-seq 14005  df-hash 14334  df-struct 17159  df-sets 17176  df-slot 17194  df-ndx 17206  df-base 17222  df-ress 17243  df-plusg 17275  df-mulr 17276  df-sca 17278  df-vsca 17279  df-ip 17280  df-tset 17281  df-ple 17282  df-ds 17284  df-hom 17286  df-cco 17287  df-0g 17446  df-gsum 17447  df-prds 17452  df-pws 17454  df-mre 17590  df-mrc 17591  df-acs 17593  df-mgm 18650  df-sgrp 18729  df-mnd 18745  df-mhm 18793  df-submnd 18794  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19230  df-cntz 19333  df-cmn 19798  df-abl 19799  df-mgp 20163  df-rng 20175  df-ur 20204  df-srg 20209  df-ring 20257  df-subrng 20568  df-subrg 20592  df-lmod 20902  df-lss 20972  df-sra 21213  df-rgmod 21214  df-dsmm 21757  df-frlm 21772  df-psr 21934  df-mvr 21935  df-mpl 21936  df-opsr 21938  df-psr1 22215  df-vr1 22216  df-ply1 22217  df-coe1 22218  df-mamu 22424  df-mat 22441  df-decpmat 22796  df-pm2mp 22826

This theorem is referenced by:  pm2mpfo  22847