Theorem mp2pm2mp 21960

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 21954	. . 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 21945	. . 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 1408	. 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 21592	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
19	18	3adant3 1131	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝐴 ∈ Ring)
20		eqid 2738	. . . . 5 ⊢ (0g‘𝑄) = (0g‘𝑄)
21	2	ply1ring 21419	. . . . . . 7 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring)
22		ringcmn 19820	. . . . . . 7 ⊢ (𝑄 ∈ Ring → 𝑄 ∈ CMnd)
23	18, 21, 22	3syl 18	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ CMnd)
24	23	3adant3 1131	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑄 ∈ CMnd)
25		nn0ex 12239	. . . . . 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 1191	. . . . . . . 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 2738	. . . . . . . . 9 ⊢ (Base‘𝐴) = (Base‘𝐴)
32	4, 9, 10, 1, 31	decpmatcl 21916	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)) ∧ 𝑛 ∈ ℕ0) → ((𝐼‘𝑂) decompPMat 𝑛) ∈ (Base‘𝐴))
33	28, 29, 30, 32	syl3anc 1370	. . . . . . 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 21443	. . . . . . 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 1370	. . . . . 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 6989	. . . . 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 6774	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → ((coe1‘𝑝)‘𝑘) = ((coe1‘𝑝)‘𝑛))
39	38	oveqd 7292	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑖((coe1‘𝑝)‘𝑘)𝑗) = (𝑖((coe1‘𝑝)‘𝑛)𝑗))
40		oveq1 7282	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑘𝐸𝑌) = (𝑛𝐸𝑌))
41	39, 40	oveq12d 7293	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)) = ((𝑖((coe1‘𝑝)‘𝑛)𝑗) · (𝑛𝐸𝑌)))
42	41	cbvmptv 5187	. . . . . . . . . . 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 7291	. . . . . . . . 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 7353	. . . . . . . 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 5179	. . . . . . 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 21959	. . . . 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 19516	. . . 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 1137	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑂 ∈ 𝐿)
51	19, 49, 50	3jca 1127	. . 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 21958	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑛 ∈ ℕ0) → ((𝐼‘𝑂) decompPMat 𝑛) = ((coe1‘𝑂)‘𝑛))
53	52	oveq1d 7290	. . . . . . . . . 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 712	. . . . . . . . 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 5174	. . . . . . . 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 7291	. . . . . . 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 6778	. . . . . 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 6776	. . . . 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 2738	. . . . . . . . . 10 ⊢ (coe1‘𝑂) = (coe1‘𝑂)
62	2, 14, 3, 12, 34, 13, 61	ply1coe 21467	. . . . . . . . 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 6778	. . . . . 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 6776	. . . . 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 3103	. . 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 21468	. . 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 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ↦ cmpt 5157  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  Fincfn 8733  ℕ₀cn0 12233  Basecbs 16912   ·_𝑠 cvsca 16966  0_gc0g 17150   Σ_g cgsu 17151  ._gcmg 18700  CMndccmn 19386  mulGrpcmgp 19720  Ringcrg 19783  var₁cv1 21347  Poly₁cpl1 21348  coe₁cco1 21349   Mat cmat 21554   decompPMat cdecpmat 21911   pMatToMatPoly cpm2mp 21941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-ot 4570  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-ofr 7534  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-sup 9201  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-fz 13240  df-fzo 13383  df-seq 13722  df-hash 14045  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-sca 16978  df-vsca 16979  df-ip 16980  df-tset 16981  df-ple 16982  df-ds 16984  df-hom 16986  df-cco 16987  df-0g 17152  df-gsum 17153  df-prds 17158  df-pws 17160  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-submnd 18431  df-grp 18580  df-minusg 18581  df-sbg 18582  df-mulg 18701  df-subg 18752  df-ghm 18832  df-cntz 18923  df-cmn 19388  df-abl 19389  df-mgp 19721  df-ur 19738  df-srg 19742  df-ring 19785  df-subrg 20022  df-lmod 20125  df-lss 20194  df-sra 20434  df-rgmod 20435  df-dsmm 20939  df-frlm 20954  df-psr 21112  df-mvr 21113  df-mpl 21114  df-opsr 21116  df-psr1 21351  df-vr1 21352  df-ply1 21353  df-coe1 21354  df-mamu 21533  df-mat 21555  df-decpmat 21912  df-pm2mp 21942

This theorem is referenced by:  pm2mpfo  21963