Theorem m2cpminvid2 22631

Description: The transformation applied to the inverse transformation of a constant polynomial matrix over the ring 𝑅 results in the matrix itself. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 14-Dec-2019.)

Hypotheses

Ref	Expression
m2cpminvid2.s	⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
m2cpminvid2.i	⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅)
m2cpminvid2.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)

Assertion

Ref	Expression
m2cpminvid2	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝑇‘(𝐼‘𝑀)) = 𝑀)

Proof of Theorem m2cpminvid2

Dummy variables 𝑖 𝑗 𝑥 𝑦 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		m2cpminvid2.i	. . . 4 ⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅)
2		m2cpminvid2.s	. . . 4 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
3	1, 2	cpm2mval 22626	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝐼‘𝑀) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
4	3	fveq2d 6895	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝑇‘(𝐼‘𝑀)) = (𝑇‘(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))))
5		simp1 1134	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → 𝑁 ∈ Fin)
6		simp2 1135	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → 𝑅 ∈ Ring)
7		eqid 2727	. . . . 5 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
8		eqid 2727	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
9		eqid 2727	. . . . 5 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
10		eqid 2727	. . . . . . 7 ⊢ (𝑁 Mat (Poly1‘𝑅)) = (𝑁 Mat (Poly1‘𝑅))
11		eqid 2727	. . . . . . 7 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅))
12		eqid 2727	. . . . . . 7 ⊢ (Base‘(𝑁 Mat (Poly1‘𝑅))) = (Base‘(𝑁 Mat (Poly1‘𝑅)))
13		simp2 1135	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑥 ∈ 𝑁)
14		simp3 1136	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑦 ∈ 𝑁)
15		eqid 2727	. . . . . . . . 9 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅)
16	2, 15, 10, 12	cpmatpmat 22586	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → 𝑀 ∈ (Base‘(𝑁 Mat (Poly1‘𝑅))))
17	16	3ad2ant1 1131	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑀 ∈ (Base‘(𝑁 Mat (Poly1‘𝑅))))
18	10, 11, 12, 13, 14, 17	matecld 22302	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑀𝑦) ∈ (Base‘(Poly1‘𝑅)))
19		0nn0 12503	. . . . . 6 ⊢ 0 ∈ ℕ0
20		eqid 2727	. . . . . . 7 ⊢ (coe1‘(𝑥𝑀𝑦)) = (coe1‘(𝑥𝑀𝑦))
21	20, 11, 15, 8	coe1fvalcl 22105	. . . . . 6 ⊢ (((𝑥𝑀𝑦) ∈ (Base‘(Poly1‘𝑅)) ∧ 0 ∈ ℕ0) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
22	18, 19, 21	sylancl 585	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
23	7, 8, 9, 5, 6, 22	matbas2d 22299	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)) ∈ (Base‘(𝑁 Mat 𝑅)))
24		m2cpminvid2.t	. . . . 5 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
25		eqid 2727	. . . . 5 ⊢ (algSc‘(Poly1‘𝑅)) = (algSc‘(Poly1‘𝑅))
26	24, 7, 9, 15, 25	mat2pmatval 22600	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)) ∈ (Base‘(𝑁 Mat 𝑅))) → (𝑇‘(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly1‘𝑅))‘(𝑖(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))𝑗))))
27	5, 6, 23, 26	syl3anc 1369	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝑇‘(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly1‘𝑅))‘(𝑖(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))𝑗))))
28		eqidd 2728	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
29		oveq12 7423	. . . . . . . . 9 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑥𝑀𝑦) = (𝑖𝑀𝑗))
30	29	fveq2d 6895	. . . . . . . 8 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (coe1‘(𝑥𝑀𝑦)) = (coe1‘(𝑖𝑀𝑗)))
31	30	fveq1d 6893	. . . . . . 7 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → ((coe1‘(𝑥𝑀𝑦))‘0) = ((coe1‘(𝑖𝑀𝑗))‘0))
32	31	adantl 481	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ (𝑥 = 𝑖 ∧ 𝑦 = 𝑗)) → ((coe1‘(𝑥𝑀𝑦))‘0) = ((coe1‘(𝑖𝑀𝑗))‘0))
33		simp2 1135	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
34		simp3 1136	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
35		fvexd 6906	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖𝑀𝑗))‘0) ∈ V)
36	28, 32, 33, 34, 35	ovmpod 7565	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))𝑗) = ((coe1‘(𝑖𝑀𝑗))‘0))
37	36	fveq2d 6895	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((algSc‘(Poly1‘𝑅))‘(𝑖(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))𝑗)) = ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)))
38	37	mpoeq3dva 7491	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly1‘𝑅))‘(𝑖(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))𝑗))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))))
39	27, 38	eqtrd 2767	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝑇‘(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))))
40	2, 15	m2cpminvid2lem 22630	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ∀𝑛 ∈ ℕ0 ((coe1‘((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
41		simpl2 1190	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑅 ∈ Ring)
42		simprl 770	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑥 ∈ 𝑁)
43		simprr 772	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑦 ∈ 𝑁)
44	16	adantr 480	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑀 ∈ (Base‘(𝑁 Mat (Poly1‘𝑅))))
45	10, 11, 12, 42, 43, 44	matecld 22302	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥𝑀𝑦) ∈ (Base‘(Poly1‘𝑅)))
46	45, 19, 21	sylancl 585	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
47	15, 25, 8, 11	ply1sclcl 22179	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)) → ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) ∈ (Base‘(Poly1‘𝑅)))
48	41, 46, 47	syl2anc 583	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) ∈ (Base‘(Poly1‘𝑅)))
49		eqid 2727	. . . . . . . . 9 ⊢ (coe1‘((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0))) = (coe1‘((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))
50	15, 11, 49, 20	ply1coe1eq 22193	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) ∈ (Base‘(Poly1‘𝑅)) ∧ (𝑥𝑀𝑦) ∈ (Base‘(Poly1‘𝑅))) → (∀𝑛 ∈ ℕ0 ((coe1‘((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛) ↔ ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑀𝑦)))
51	50	bicomd 222	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) ∈ (Base‘(Poly1‘𝑅)) ∧ (𝑥𝑀𝑦) ∈ (Base‘(Poly1‘𝑅))) → (((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑀𝑦) ↔ ∀𝑛 ∈ ℕ0 ((coe1‘((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛)))
52	41, 48, 45, 51	syl3anc 1369	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑀𝑦) ↔ ∀𝑛 ∈ ℕ0 ((coe1‘((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛)))
53	40, 52	mpbird 257	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑀𝑦))
54	53	ralrimivva 3195	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑀𝑦))
55		eqidd 2728	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁) ∧ 𝑦 ∈ 𝑁) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))))
56		oveq12 7423	. . . . . . . . . . . 12 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑖𝑀𝑗) = (𝑥𝑀𝑦))
57	56	fveq2d 6895	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑥𝑀𝑦)))
58	57	fveq1d 6893	. . . . . . . . . 10 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → ((coe1‘(𝑖𝑀𝑗))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0))
59	58	fveq2d 6895	. . . . . . . . 9 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)) = ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))
60	59	adantl 481	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁) ∧ 𝑦 ∈ 𝑁) ∧ (𝑖 = 𝑥 ∧ 𝑗 = 𝑦)) → ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)) = ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))
61		simplr 768	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁) ∧ 𝑦 ∈ 𝑁) → 𝑥 ∈ 𝑁)
62		simpr 484	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁) ∧ 𝑦 ∈ 𝑁) → 𝑦 ∈ 𝑁)
63		fvexd 6906	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁) ∧ 𝑦 ∈ 𝑁) → ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) ∈ V)
64	55, 60, 61, 62, 63	ovmpod 7565	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁) ∧ 𝑦 ∈ 𝑁) → (𝑥(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)))𝑦) = ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))
65	64	eqeq1d 2729	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁) ∧ 𝑦 ∈ 𝑁) → ((𝑥(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)))𝑦) = (𝑥𝑀𝑦) ↔ ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑀𝑦)))
66	65	anasss 466	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ((𝑥(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)))𝑦) = (𝑥𝑀𝑦) ↔ ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑀𝑦)))
67	66	2ralbidva 3211	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)))𝑦) = (𝑥𝑀𝑦) ↔ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑀𝑦)))
68	54, 67	mpbird 257	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)))𝑦) = (𝑥𝑀𝑦))
69		fvexd 6906	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (Poly1‘𝑅) ∈ V)
70	6	3ad2ant1 1131	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
71	16	3ad2ant1 1131	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑀 ∈ (Base‘(𝑁 Mat (Poly1‘𝑅))))
72	10, 11, 12, 33, 34, 71	matecld 22302	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑀𝑗) ∈ (Base‘(Poly1‘𝑅)))
73		eqid 2727	. . . . . . . 8 ⊢ (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗))
74	73, 11, 15, 8	coe1fvalcl 22105	. . . . . . 7 ⊢ (((𝑖𝑀𝑗) ∈ (Base‘(Poly1‘𝑅)) ∧ 0 ∈ ℕ0) → ((coe1‘(𝑖𝑀𝑗))‘0) ∈ (Base‘𝑅))
75	72, 19, 74	sylancl 585	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖𝑀𝑗))‘0) ∈ (Base‘𝑅))
76	15, 25, 8, 11	ply1sclcl 22179	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘(𝑖𝑀𝑗))‘0) ∈ (Base‘𝑅)) → ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)) ∈ (Base‘(Poly1‘𝑅)))
77	70, 75, 76	syl2anc 583	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)) ∈ (Base‘(Poly1‘𝑅)))
78	10, 11, 12, 5, 69, 77	matbas2d 22299	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))) ∈ (Base‘(𝑁 Mat (Poly1‘𝑅))))
79	10, 12	eqmat 22300	. . . 4 ⊢ (((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))) ∈ (Base‘(𝑁 Mat (Poly1‘𝑅))) ∧ 𝑀 ∈ (Base‘(𝑁 Mat (Poly1‘𝑅)))) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))) = 𝑀 ↔ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)))𝑦) = (𝑥𝑀𝑦)))
80	78, 16, 79	syl2anc 583	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))) = 𝑀 ↔ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)))𝑦) = (𝑥𝑀𝑦)))
81	68, 80	mpbird 257	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))) = 𝑀)
82	4, 39, 81	3eqtrd 2771	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝑇‘(𝐼‘𝑀)) = 𝑀)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ∀wral 3056  Vcvv 3469  ‘cfv 6542  (class class class)co 7414   ∈ cmpo 7416  Fincfn 8953  0cc0 11124  ℕ₀cn0 12488  Basecbs 17165  Ringcrg 20157  algSccascl 21766  Poly₁cpl1 22070  coe₁cco1 22071   Mat cmat 22281   ConstPolyMat ccpmat 22579   matToPolyMat cmat2pmat 22580   cPolyMatToMat ccpmat2mat 22581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-ot 4633  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7677  df-ofr 7678  df-om 7863  df-1st 7985  df-2nd 7986  df-supp 8158  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8716  df-map 8836  df-pm 8837  df-ixp 8906  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-fsupp 9376  df-sup 9451  df-oi 9519  df-card 9948  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-nn 12229  df-2 12291  df-3 12292  df-4 12293  df-5 12294  df-6 12295  df-7 12296  df-8 12297  df-9 12298  df-n0 12489  df-z 12575  df-dec 12694  df-uz 12839  df-fz 13503  df-fzo 13646  df-seq 13985  df-hash 14308  df-struct 17101  df-sets 17118  df-slot 17136  df-ndx 17148  df-base 17166  df-ress 17195  df-plusg 17231  df-mulr 17232  df-sca 17234  df-vsca 17235  df-ip 17236  df-tset 17237  df-ple 17238  df-ds 17240  df-hom 17242  df-cco 17243  df-0g 17408  df-gsum 17409  df-prds 17414  df-pws 17416  df-mre 17551  df-mrc 17552  df-acs 17554  df-mgm 18585  df-sgrp 18664  df-mnd 18680  df-mhm 18725  df-submnd 18726  df-grp 18878  df-minusg 18879  df-sbg 18880  df-mulg 19008  df-subg 19062  df-ghm 19152  df-cntz 19252  df-cmn 19721  df-abl 19722  df-mgp 20059  df-rng 20077  df-ur 20106  df-srg 20111  df-ring 20159  df-subrng 20465  df-subrg 20490  df-lmod 20727  df-lss 20798  df-sra 21040  df-rgmod 21041  df-dsmm 21646  df-frlm 21661  df-ascl 21769  df-psr 21822  df-mvr 21823  df-mpl 21824  df-opsr 21826  df-psr1 22073  df-vr1 22074  df-ply1 22075  df-coe1 22076  df-mat 22282  df-cpmat 22582  df-mat2pmat 22583  df-cpmat2mat 22584

This theorem is referenced by:  m2cpmfo  22632  m2cpminv  22636  cpmadumatpoly  22759  cayhamlem4  22764