Theorem pmatcollpw1 21925

Description: Write a polynomial matrix as a matrix of sums of scaled monomials. (Contributed by AV, 29-Sep-2019.) (Revised by AV, 3-Dec-2019.)

Hypotheses

Ref	Expression
pmatcollpw1.p	⊢ 𝑃 = (Poly1‘𝑅)
pmatcollpw1.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
pmatcollpw1.b	⊢ 𝐵 = (Base‘𝐶)
pmatcollpw1.m	⊢ × = ( ·𝑠 ‘𝑃)
pmatcollpw1.e	⊢ ↑ = (.g‘(mulGrp‘𝑃))
pmatcollpw1.x	⊢ 𝑋 = (var1‘𝑅)

Assertion

Ref	Expression
pmatcollpw1	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))))

Distinct variable groups:   𝐵,𝑛   𝑛,𝑀   𝑛,𝑁   𝑅,𝑛   𝑛,𝑋   × ,𝑛   ↑ ,𝑛   𝑃,𝑛   𝐵,𝑖,𝑗   𝑖,𝑀,𝑗   𝑖,𝑁,𝑗   𝑃,𝑖,𝑗,𝑛   𝑅,𝑖,𝑗   𝑖,𝑋,𝑗   × ,𝑖,𝑗   ↑ ,𝑖,𝑗

Allowed substitution hints:   𝐶(𝑖,𝑗,𝑛)

Proof of Theorem pmatcollpw1

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pmatcollpw1.p	. . . . 5 ⊢ 𝑃 = (Poly1‘𝑅)
2		pmatcollpw1.c	. . . . 5 ⊢ 𝐶 = (𝑁 Mat 𝑃)
3		pmatcollpw1.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
4		pmatcollpw1.m	. . . . 5 ⊢ × = ( ·𝑠 ‘𝑃)
5		pmatcollpw1.e	. . . . 5 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
6		pmatcollpw1.x	. . . . 5 ⊢ 𝑋 = (var1‘𝑅)
7	1, 2, 3, 4, 5, 6	pmatcollpw1lem2 21924	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎𝑀𝑏) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑎(𝑀 decompPMat 𝑛)𝑏) × (𝑛 ↑ 𝑋)))))
8		eqidd 2739	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))))
9		oveq12 7284	. . . . . . . . 9 ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → (𝑖(𝑀 decompPMat 𝑛)𝑗) = (𝑎(𝑀 decompPMat 𝑛)𝑏))
10	9	oveq1d 7290	. . . . . . . 8 ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)) = ((𝑎(𝑀 decompPMat 𝑛)𝑏) × (𝑛 ↑ 𝑋)))
11	10	mpteq2dv 5176	. . . . . . 7 ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → (𝑛 ∈ ℕ0 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))) = (𝑛 ∈ ℕ0 ↦ ((𝑎(𝑀 decompPMat 𝑛)𝑏) × (𝑛 ↑ 𝑋))))
12	11	oveq2d 7291	. . . . . 6 ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑎(𝑀 decompPMat 𝑛)𝑏) × (𝑛 ↑ 𝑋)))))
13	12	adantl 482	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ (𝑖 = 𝑎 ∧ 𝑗 = 𝑏)) → (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑎(𝑀 decompPMat 𝑛)𝑏) × (𝑛 ↑ 𝑋)))))
14		simprl 768	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑎 ∈ 𝑁)
15		simprr 770	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑏 ∈ 𝑁)
16		eqid 2738	. . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃)
17		eqid 2738	. . . . . 6 ⊢ (0g‘𝑃) = (0g‘𝑃)
18	1	ply1ring 21419	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
19		ringcmn 19820	. . . . . . . . 9 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
20	18, 19	syl 17	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ CMnd)
21	20	3ad2ant2 1133	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ CMnd)
22	21	adantr 481	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑃 ∈ CMnd)
23		nn0ex 12239	. . . . . . 7 ⊢ ℕ0 ∈ V
24	23	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → ℕ0 ∈ V)
25		simpl2 1191	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑅 ∈ Ring)
26	25	adantr 481	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
27		eqid 2738	. . . . . . . . 9 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
28		eqid 2738	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
29		eqid 2738	. . . . . . . . 9 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
30		simplrl 774	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ0) → 𝑎 ∈ 𝑁)
31	15	adantr 481	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ0) → 𝑏 ∈ 𝑁)
32		simpl3 1192	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑀 ∈ 𝐵)
33	32	adantr 481	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ0) → 𝑀 ∈ 𝐵)
34		simpr 485	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
35	1, 2, 3, 27, 29	decpmatcl 21916	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑛 ∈ ℕ0) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
36	26, 33, 34, 35	syl3anc 1370	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ0) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
37	27, 28, 29, 30, 31, 36	matecld 21575	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ0) → (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘𝑅))
38		eqid 2738	. . . . . . . . 9 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
39	28, 1, 6, 4, 38, 5, 16	ply1tmcl 21443	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘𝑅) ∧ 𝑛 ∈ ℕ0) → ((𝑎(𝑀 decompPMat 𝑛)𝑏) × (𝑛 ↑ 𝑋)) ∈ (Base‘𝑃))
40	26, 37, 34, 39	syl3anc 1370	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ0) → ((𝑎(𝑀 decompPMat 𝑛)𝑏) × (𝑛 ↑ 𝑋)) ∈ (Base‘𝑃))
41	40	fmpttd 6989	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑛 ∈ ℕ0 ↦ ((𝑎(𝑀 decompPMat 𝑛)𝑏) × (𝑛 ↑ 𝑋))):ℕ0⟶(Base‘𝑃))
42	1, 2, 3, 4, 5, 6	pmatcollpw1lem1 21923	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑛 ∈ ℕ0 ↦ ((𝑎(𝑀 decompPMat 𝑛)𝑏) × (𝑛 ↑ 𝑋))) finSupp (0g‘𝑃))
43	42	3expb 1119	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑛 ∈ ℕ0 ↦ ((𝑎(𝑀 decompPMat 𝑛)𝑏) × (𝑛 ↑ 𝑋))) finSupp (0g‘𝑃))
44	16, 17, 22, 24, 41, 43	gsumcl 19516	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑎(𝑀 decompPMat 𝑛)𝑏) × (𝑛 ↑ 𝑋)))) ∈ (Base‘𝑃))
45	8, 13, 14, 15, 44	ovmpod 7425	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))))𝑏) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑎(𝑀 decompPMat 𝑛)𝑏) × (𝑛 ↑ 𝑋)))))
46	7, 45	eqtr4d 2781	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎𝑀𝑏) = (𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))))𝑏))
47	46	ralrimivva 3123	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 (𝑎𝑀𝑏) = (𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))))𝑏))
48		simp3 1137	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵)
49		simp1 1135	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin)
50	18	3ad2ant2 1133	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ Ring)
51	21	3ad2ant1 1132	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑃 ∈ CMnd)
52	23	a1i 11	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ℕ0 ∈ V)
53		simpl12 1248	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
54		simpl2 1191	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑛 ∈ ℕ0) → 𝑖 ∈ 𝑁)
55		simpl3 1192	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑛 ∈ ℕ0) → 𝑗 ∈ 𝑁)
56	48	3ad2ant1 1132	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑀 ∈ 𝐵)
57	56	adantr 481	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑛 ∈ ℕ0) → 𝑀 ∈ 𝐵)
58		simpr 485	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
59	53, 57, 58, 35	syl3anc 1370	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑛 ∈ ℕ0) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
60	27, 28, 29, 54, 55, 59	matecld 21575	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑛 ∈ ℕ0) → (𝑖(𝑀 decompPMat 𝑛)𝑗) ∈ (Base‘𝑅))
61	28, 1, 6, 4, 38, 5, 16	ply1tmcl 21443	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑖(𝑀 decompPMat 𝑛)𝑗) ∈ (Base‘𝑅) ∧ 𝑛 ∈ ℕ0) → ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)) ∈ (Base‘𝑃))
62	53, 60, 58, 61	syl3anc 1370	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑛 ∈ ℕ0) → ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)) ∈ (Base‘𝑃))
63	62	fmpttd 6989	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑛 ∈ ℕ0 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))):ℕ0⟶(Base‘𝑃))
64	1, 2, 3, 4, 5, 6	pmatcollpw1lem1 21923	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑛 ∈ ℕ0 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))) finSupp (0g‘𝑃))
65	16, 17, 51, 52, 63, 64	gsumcl 19516	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) ∈ (Base‘𝑃))
66	2, 16, 3, 49, 50, 65	matbas2d 21572	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))) ∈ 𝐵)
67	2, 3	eqmat 21573	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))) ∈ 𝐵) → (𝑀 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))) ↔ ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 (𝑎𝑀𝑏) = (𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))))𝑏)))
68	48, 66, 67	syl2anc 584	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑀 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))) ↔ ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 (𝑎𝑀𝑏) = (𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))))𝑏)))
69	47, 68	mpbird 256	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   class class class wbr 5074   ↦ cmpt 5157  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  Fincfn 8733   finSupp cfsupp 9128  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   Mat cmat 21554   decompPMat cdecpmat 21911

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-mat 21555  df-decpmat 21912

This theorem is referenced by:  pmatcollpw2  21927