Theorem m2cpminvid2 22761

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 22756	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝐼‘𝑀) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
4	3	fveq2d 6910	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝑇‘(𝐼‘𝑀)) = (𝑇‘(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))))
5		simp1 1137	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → 𝑁 ∈ Fin)
6		simp2 1138	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → 𝑅 ∈ Ring)
7		eqid 2737	. . . . 5 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
8		eqid 2737	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
9		eqid 2737	. . . . 5 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
10		eqid 2737	. . . . . . 7 ⊢ (𝑁 Mat (Poly1‘𝑅)) = (𝑁 Mat (Poly1‘𝑅))
11		eqid 2737	. . . . . . 7 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅))
12		eqid 2737	. . . . . . 7 ⊢ (Base‘(𝑁 Mat (Poly1‘𝑅))) = (Base‘(𝑁 Mat (Poly1‘𝑅)))
13		simp2 1138	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑥 ∈ 𝑁)
14		simp3 1139	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑦 ∈ 𝑁)
15		eqid 2737	. . . . . . . . 9 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅)
16	2, 15, 10, 12	cpmatpmat 22716	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → 𝑀 ∈ (Base‘(𝑁 Mat (Poly1‘𝑅))))
17	16	3ad2ant1 1134	. . . . . . 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 22432	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑀𝑦) ∈ (Base‘(Poly1‘𝑅)))
19		0nn0 12541	. . . . . 6 ⊢ 0 ∈ ℕ0
20		eqid 2737	. . . . . . 7 ⊢ (coe1‘(𝑥𝑀𝑦)) = (coe1‘(𝑥𝑀𝑦))
21	20, 11, 15, 8	coe1fvalcl 22214	. . . . . 6 ⊢ (((𝑥𝑀𝑦) ∈ (Base‘(Poly1‘𝑅)) ∧ 0 ∈ ℕ0) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
22	18, 19, 21	sylancl 586	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
23	7, 8, 9, 5, 6, 22	matbas2d 22429	. . . 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 2737	. . . . 5 ⊢ (algSc‘(Poly1‘𝑅)) = (algSc‘(Poly1‘𝑅))
26	24, 7, 9, 15, 25	mat2pmatval 22730	. . . 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 1373	. . 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 2738	. . . . . 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 7440	. . . . . . . . 9 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑥𝑀𝑦) = (𝑖𝑀𝑗))
30	29	fveq2d 6910	. . . . . . . 8 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (coe1‘(𝑥𝑀𝑦)) = (coe1‘(𝑖𝑀𝑗)))
31	30	fveq1d 6908	. . . . . . 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 1138	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
34		simp3 1139	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
35		fvexd 6921	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖𝑀𝑗))‘0) ∈ V)
36	28, 32, 33, 34, 35	ovmpod 7585	. . . . 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 6910	. . . 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 7510	. . 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 2777	. 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 22760	. . . . . 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 1193	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑅 ∈ Ring)
42		simprl 771	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑥 ∈ 𝑁)
43		simprr 773	. . . . . . . . . 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 22432	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥𝑀𝑦) ∈ (Base‘(Poly1‘𝑅)))
46	45, 19, 21	sylancl 586	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
47	15, 25, 8, 11	ply1sclcl 22289	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)) → ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) ∈ (Base‘(Poly1‘𝑅)))
48	41, 46, 47	syl2anc 584	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) ∈ (Base‘(Poly1‘𝑅)))
49		eqid 2737	. . . . . . . . 9 ⊢ (coe1‘((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0))) = (coe1‘((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))
50	15, 11, 49, 20	ply1coe1eq 22304	. . . . . . . 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 223	. . . . . . 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 1373	. . . . . 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 3202	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑀𝑦))
55		eqidd 2738	. . . . . . . 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 7440	. . . . . . . . . . . 12 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑖𝑀𝑗) = (𝑥𝑀𝑦))
57	56	fveq2d 6910	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑥𝑀𝑦)))
58	57	fveq1d 6908	. . . . . . . . . 10 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → ((coe1‘(𝑖𝑀𝑗))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0))
59	58	fveq2d 6910	. . . . . . . . 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 769	. . . . . . . 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 6921	. . . . . . . 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 7585	. . . . . . 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 2739	. . . . . 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 3219	. . . 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 6921	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (Poly1‘𝑅) ∈ V)
70	6	3ad2ant1 1134	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
71	16	3ad2ant1 1134	. . . . . . . 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 22432	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑀𝑗) ∈ (Base‘(Poly1‘𝑅)))
73		eqid 2737	. . . . . . . 8 ⊢ (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗))
74	73, 11, 15, 8	coe1fvalcl 22214	. . . . . . 7 ⊢ (((𝑖𝑀𝑗) ∈ (Base‘(Poly1‘𝑅)) ∧ 0 ∈ ℕ0) → ((coe1‘(𝑖𝑀𝑗))‘0) ∈ (Base‘𝑅))
75	72, 19, 74	sylancl 586	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖𝑀𝑗))‘0) ∈ (Base‘𝑅))
76	15, 25, 8, 11	ply1sclcl 22289	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘(𝑖𝑀𝑗))‘0) ∈ (Base‘𝑅)) → ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)) ∈ (Base‘(Poly1‘𝑅)))
77	70, 75, 76	syl2anc 584	. . . . 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 22429	. . . 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 22430	. . . 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 584	. . 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 2781	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝑇‘(𝐼‘𝑀)) = 𝑀)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  Vcvv 3480  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  Fincfn 8985  0cc0 11155  ℕ₀cn0 12526  Basecbs 17247  Ringcrg 20230  algSccascl 21872  Poly₁cpl1 22178  coe₁cco1 22179   Mat cmat 22411   ConstPolyMat ccpmat 22709   matToPolyMat cmat2pmat 22710   cPolyMatToMat ccpmat2mat 22711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-srg 20184  df-ring 20232  df-subrng 20546  df-subrg 20570  df-lmod 20860  df-lss 20930  df-sra 21172  df-rgmod 21173  df-dsmm 21752  df-frlm 21767  df-ascl 21875  df-psr 21929  df-mvr 21930  df-mpl 21931  df-opsr 21933  df-psr1 22181  df-vr1 22182  df-ply1 22183  df-coe1 22184  df-mat 22412  df-cpmat 22712  df-mat2pmat 22713  df-cpmat2mat 22714

This theorem is referenced by:  m2cpmfo  22762  m2cpminv  22766  cpmadumatpoly  22889  cayhamlem4  22894