Theorem pm2mpval 22652

Description: Value of the transformation of a polynomial matrix into a polynomial over matrices. (Contributed by AV, 5-Dec-2019.)

Hypotheses

Ref	Expression
pm2mpval.p	⊢ 𝑃 = (Poly1‘𝑅)
pm2mpval.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
pm2mpval.b	⊢ 𝐵 = (Base‘𝐶)
pm2mpval.m	⊢ ∗ = ( ·𝑠 ‘𝑄)
pm2mpval.e	⊢ ↑ = (.g‘(mulGrp‘𝑄))
pm2mpval.x	⊢ 𝑋 = (var1‘𝐴)
pm2mpval.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
pm2mpval.q	⊢ 𝑄 = (Poly1‘𝐴)
pm2mpval.t	⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)

Assertion

Ref	Expression
pm2mpval	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))))

Distinct variable groups:   𝐵,𝑚   𝑘,𝑁,𝑚   𝑅,𝑘,𝑚   𝑚,𝑉

Allowed substitution hints:   𝐴(𝑘,𝑚)   𝐵(𝑘)   𝐶(𝑘,𝑚)   𝑃(𝑘,𝑚)   𝑄(𝑘,𝑚)   𝑇(𝑘,𝑚)   ↑ (𝑘,𝑚)   ∗ (𝑘,𝑚)   𝑉(𝑘)   𝑋(𝑘,𝑚)

Proof of Theorem pm2mpval

Dummy variables 𝑛 𝑟 𝑎 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm2mpval.t	. 2 ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)
2		df-pm2mp 22650	. . . 4 ⊢ pMatToMatPoly = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat (Poly1‘𝑟))) ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌⦋(Poly1‘𝑎) / 𝑞⦌(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1‘𝑎)))))))
3	2	a1i 11	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → pMatToMatPoly = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat (Poly1‘𝑟))) ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌⦋(Poly1‘𝑎) / 𝑞⦌(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1‘𝑎))))))))
4		simpl 482	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 𝑛 = 𝑁)
5		fveq2 6885	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
6	5	adantl 481	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Poly1‘𝑟) = (Poly1‘𝑅))
7	4, 6	oveq12d 7423	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat (Poly1‘𝑟)) = (𝑁 Mat (Poly1‘𝑅)))
8	7	fveq2d 6889	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat (Poly1‘𝑟))) = (Base‘(𝑁 Mat (Poly1‘𝑅))))
9		pm2mpval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐶)
10		pm2mpval.c	. . . . . . . . 9 ⊢ 𝐶 = (𝑁 Mat 𝑃)
11		pm2mpval.p	. . . . . . . . . 10 ⊢ 𝑃 = (Poly1‘𝑅)
12	11	oveq2i 7416	. . . . . . . . 9 ⊢ (𝑁 Mat 𝑃) = (𝑁 Mat (Poly1‘𝑅))
13	10, 12	eqtri 2754	. . . . . . . 8 ⊢ 𝐶 = (𝑁 Mat (Poly1‘𝑅))
14	13	fveq2i 6888	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘(𝑁 Mat (Poly1‘𝑅)))
15	9, 14	eqtri 2754	. . . . . 6 ⊢ 𝐵 = (Base‘(𝑁 Mat (Poly1‘𝑅)))
16	8, 15	eqtr4di 2784	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat (Poly1‘𝑟))) = 𝐵)
17	16	adantl 481	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) ∧ (𝑛 = 𝑁 ∧ 𝑟 = 𝑅)) → (Base‘(𝑛 Mat (Poly1‘𝑟))) = 𝐵)
18		ovex 7438	. . . . . 6 ⊢ (𝑛 Mat 𝑟) ∈ V
19		fvexd 6900	. . . . . . 7 ⊢ (𝑎 = (𝑛 Mat 𝑟) → (Poly1‘𝑎) ∈ V)
20		simpr 484	. . . . . . . . 9 ⊢ ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1‘𝑎)) → 𝑞 = (Poly1‘𝑎))
21		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑎 = (𝑛 Mat 𝑟) → (Poly1‘𝑎) = (Poly1‘(𝑛 Mat 𝑟)))
22	21	adantr 480	. . . . . . . . 9 ⊢ ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1‘𝑎)) → (Poly1‘𝑎) = (Poly1‘(𝑛 Mat 𝑟)))
23	20, 22	eqtrd 2766	. . . . . . . 8 ⊢ ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1‘𝑎)) → 𝑞 = (Poly1‘(𝑛 Mat 𝑟)))
24	23	fveq2d 6889	. . . . . . . . . 10 ⊢ ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1‘𝑎)) → ( ·𝑠 ‘𝑞) = ( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟))))
25		eqidd 2727	. . . . . . . . . 10 ⊢ ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1‘𝑎)) → (𝑚 decompPMat 𝑘) = (𝑚 decompPMat 𝑘))
26	23	fveq2d 6889	. . . . . . . . . . . 12 ⊢ ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1‘𝑎)) → (mulGrp‘𝑞) = (mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))
27	26	fveq2d 6889	. . . . . . . . . . 11 ⊢ ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1‘𝑎)) → (.g‘(mulGrp‘𝑞)) = (.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟)))))
28		eqidd 2727	. . . . . . . . . . 11 ⊢ ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1‘𝑎)) → 𝑘 = 𝑘)
29		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑛 Mat 𝑟) → (var1‘𝑎) = (var1‘(𝑛 Mat 𝑟)))
30	29	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1‘𝑎)) → (var1‘𝑎) = (var1‘(𝑛 Mat 𝑟)))
31	27, 28, 30	oveq123d 7426	. . . . . . . . . 10 ⊢ ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1‘𝑎)) → (𝑘(.g‘(mulGrp‘𝑞))(var1‘𝑎)) = (𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟))))
32	24, 25, 31	oveq123d 7426	. . . . . . . . 9 ⊢ ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1‘𝑎)) → ((𝑚 decompPMat 𝑘)( ·𝑠 ‘𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1‘𝑎))) = ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟)))))
33	32	mpteq2dv 5243	. . . . . . . 8 ⊢ ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1‘𝑎)) → (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1‘𝑎)))) = (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟))))))
34	23, 33	oveq12d 7423	. . . . . . 7 ⊢ ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1‘𝑎)) → (𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1‘𝑎))))) = ((Poly1‘(𝑛 Mat 𝑟)) Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟)))))))
35	19, 34	csbied 3926	. . . . . 6 ⊢ (𝑎 = (𝑛 Mat 𝑟) → ⦋(Poly1‘𝑎) / 𝑞⦌(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1‘𝑎))))) = ((Poly1‘(𝑛 Mat 𝑟)) Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟)))))))
36	18, 35	csbie 3924	. . . . 5 ⊢ ⦋(𝑛 Mat 𝑟) / 𝑎⦌⦋(Poly1‘𝑎) / 𝑞⦌(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1‘𝑎))))) = ((Poly1‘(𝑛 Mat 𝑟)) Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟))))))
37		oveq12 7414	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
38	37	fveq2d 6889	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Poly1‘(𝑛 Mat 𝑟)) = (Poly1‘(𝑁 Mat 𝑅)))
39		pm2mpval.q	. . . . . . . . 9 ⊢ 𝑄 = (Poly1‘𝐴)
40		pm2mpval.a	. . . . . . . . . 10 ⊢ 𝐴 = (𝑁 Mat 𝑅)
41	40	fveq2i 6888	. . . . . . . . 9 ⊢ (Poly1‘𝐴) = (Poly1‘(𝑁 Mat 𝑅))
42	39, 41	eqtri 2754	. . . . . . . 8 ⊢ 𝑄 = (Poly1‘(𝑁 Mat 𝑅))
43	38, 42	eqtr4di 2784	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Poly1‘(𝑛 Mat 𝑟)) = 𝑄)
44	38	fveq2d 6889	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟))) = ( ·𝑠 ‘(Poly1‘(𝑁 Mat 𝑅))))
45		pm2mpval.m	. . . . . . . . . . 11 ⊢ ∗ = ( ·𝑠 ‘𝑄)
46	42	fveq2i 6888	. . . . . . . . . . 11 ⊢ ( ·𝑠 ‘𝑄) = ( ·𝑠 ‘(Poly1‘(𝑁 Mat 𝑅)))
47	45, 46	eqtri 2754	. . . . . . . . . 10 ⊢ ∗ = ( ·𝑠 ‘(Poly1‘(𝑁 Mat 𝑅)))
48	44, 47	eqtr4di 2784	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟))) = ∗ )
49		eqidd 2727	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑚 decompPMat 𝑘) = (𝑚 decompPMat 𝑘))
50	38	fveq2d 6889	. . . . . . . . . . . 12 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (mulGrp‘(Poly1‘(𝑛 Mat 𝑟))) = (mulGrp‘(Poly1‘(𝑁 Mat 𝑅))))
51	50	fveq2d 6889	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟)))) = (.g‘(mulGrp‘(Poly1‘(𝑁 Mat 𝑅)))))
52		pm2mpval.e	. . . . . . . . . . . 12 ⊢ ↑ = (.g‘(mulGrp‘𝑄))
53	42	fveq2i 6888	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑄) = (mulGrp‘(Poly1‘(𝑁 Mat 𝑅)))
54	53	fveq2i 6888	. . . . . . . . . . . 12 ⊢ (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘(Poly1‘(𝑁 Mat 𝑅))))
55	52, 54	eqtri 2754	. . . . . . . . . . 11 ⊢ ↑ = (.g‘(mulGrp‘(Poly1‘(𝑁 Mat 𝑅))))
56	51, 55	eqtr4di 2784	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟)))) = ↑ )
57		eqidd 2727	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 𝑘 = 𝑘)
58	37	fveq2d 6889	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (var1‘(𝑛 Mat 𝑟)) = (var1‘(𝑁 Mat 𝑅)))
59		pm2mpval.x	. . . . . . . . . . . 12 ⊢ 𝑋 = (var1‘𝐴)
60	40	fveq2i 6888	. . . . . . . . . . . 12 ⊢ (var1‘𝐴) = (var1‘(𝑁 Mat 𝑅))
61	59, 60	eqtri 2754	. . . . . . . . . . 11 ⊢ 𝑋 = (var1‘(𝑁 Mat 𝑅))
62	58, 61	eqtr4di 2784	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (var1‘(𝑛 Mat 𝑟)) = 𝑋)
63	56, 57, 62	oveq123d 7426	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟))) = (𝑘 ↑ 𝑋))
64	48, 49, 63	oveq123d 7426	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟)))) = ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))
65	64	mpteq2dv 5243	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟))))) = (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))
66	43, 65	oveq12d 7423	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((Poly1‘(𝑛 Mat 𝑟)) Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟)))))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))
67	66	adantl 481	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) ∧ (𝑛 = 𝑁 ∧ 𝑟 = 𝑅)) → ((Poly1‘(𝑛 Mat 𝑟)) Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟)))))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))
68	36, 67	eqtrid 2778	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) ∧ (𝑛 = 𝑁 ∧ 𝑟 = 𝑅)) → ⦋(𝑛 Mat 𝑟) / 𝑎⦌⦋(Poly1‘𝑎) / 𝑞⦌(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1‘𝑎))))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))
69	17, 68	mpteq12dv 5232	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) ∧ (𝑛 = 𝑁 ∧ 𝑟 = 𝑅)) → (𝑚 ∈ (Base‘(𝑛 Mat (Poly1‘𝑟))) ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌⦋(Poly1‘𝑎) / 𝑞⦌(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1‘𝑎)))))) = (𝑚 ∈ 𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))))
70		simpl 482	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ Fin)
71		elex 3487	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
72	71	adantl 481	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ V)
73	9	fvexi 6899	. . . . 5 ⊢ 𝐵 ∈ V
74	73	mptex 7220	. . . 4 ⊢ (𝑚 ∈ 𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))) ∈ V
75	74	a1i 11	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑚 ∈ 𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))) ∈ V)
76	3, 69, 70, 72, 75	ovmpod 7556	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑁 pMatToMatPoly 𝑅) = (𝑚 ∈ 𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))))
77	1, 76	eqtrid 2778	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468  ⦋csb 3888   ↦ cmpt 5224  ‘cfv 6537  (class class class)co 7405   ∈ cmpo 7407  Fincfn 8941  ℕ₀cn0 12476  Basecbs 17153   ·_𝑠 cvsca 17210   Σ_g cgsu 17395  ._gcmg 18995  mulGrpcmgp 20039  var₁cv1 22050  Poly₁cpl1 22051   Mat cmat 22262   decompPMat cdecpmat 22619   pMatToMatPoly cpm2mp 22649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-pm2mp 22650

This theorem is referenced by:  pm2mpfval  22653  pm2mpf  22655