Theorem m2cpminvid2 22771

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 22766	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝐼‘𝑀) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
4	3	fveq2d 6909	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝑇‘(𝐼‘𝑀)) = (𝑇‘(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))))
5		simp1 1133	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → 𝑁 ∈ Fin)
6		simp2 1134	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → 𝑅 ∈ Ring)
7		eqid 2729	. . . . 5 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
8		eqid 2729	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
9		eqid 2729	. . . . 5 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
10		eqid 2729	. . . . . . 7 ⊢ (𝑁 Mat (Poly1‘𝑅)) = (𝑁 Mat (Poly1‘𝑅))
11		eqid 2729	. . . . . . 7 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅))
12		eqid 2729	. . . . . . 7 ⊢ (Base‘(𝑁 Mat (Poly1‘𝑅))) = (Base‘(𝑁 Mat (Poly1‘𝑅)))
13		simp2 1134	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑥 ∈ 𝑁)
14		simp3 1135	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑦 ∈ 𝑁)
15		eqid 2729	. . . . . . . . 9 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅)
16	2, 15, 10, 12	cpmatpmat 22726	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → 𝑀 ∈ (Base‘(𝑁 Mat (Poly1‘𝑅))))
17	16	3ad2ant1 1130	. . . . . . 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 22442	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑀𝑦) ∈ (Base‘(Poly1‘𝑅)))
19		0nn0 12549	. . . . . 6 ⊢ 0 ∈ ℕ0
20		eqid 2729	. . . . . . 7 ⊢ (coe1‘(𝑥𝑀𝑦)) = (coe1‘(𝑥𝑀𝑦))
21	20, 11, 15, 8	coe1fvalcl 22224	. . . . . 6 ⊢ (((𝑥𝑀𝑦) ∈ (Base‘(Poly1‘𝑅)) ∧ 0 ∈ ℕ0) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
22	18, 19, 21	sylancl 584	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
23	7, 8, 9, 5, 6, 22	matbas2d 22439	. . . 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 2729	. . . . 5 ⊢ (algSc‘(Poly1‘𝑅)) = (algSc‘(Poly1‘𝑅))
26	24, 7, 9, 15, 25	mat2pmatval 22740	. . . 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 1368	. . 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 2730	. . . . . 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 7439	. . . . . . . . 9 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑥𝑀𝑦) = (𝑖𝑀𝑗))
30	29	fveq2d 6909	. . . . . . . 8 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (coe1‘(𝑥𝑀𝑦)) = (coe1‘(𝑖𝑀𝑗)))
31	30	fveq1d 6907	. . . . . . 7 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → ((coe1‘(𝑥𝑀𝑦))‘0) = ((coe1‘(𝑖𝑀𝑗))‘0))
32	31	adantl 480	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ (𝑥 = 𝑖 ∧ 𝑦 = 𝑗)) → ((coe1‘(𝑥𝑀𝑦))‘0) = ((coe1‘(𝑖𝑀𝑗))‘0))
33		simp2 1134	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
34		simp3 1135	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
35		fvexd 6920	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖𝑀𝑗))‘0) ∈ V)
36	28, 32, 33, 34, 35	ovmpod 7583	. . . . 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 6909	. . . 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 7508	. . 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 2769	. 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 22770	. . . . . 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 1189	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑅 ∈ Ring)
42		simprl 769	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑥 ∈ 𝑁)
43		simprr 771	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑦 ∈ 𝑁)
44	16	adantr 479	. . . . . . . . . 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 22442	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥𝑀𝑦) ∈ (Base‘(Poly1‘𝑅)))
46	45, 19, 21	sylancl 584	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
47	15, 25, 8, 11	ply1sclcl 22299	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)) → ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) ∈ (Base‘(Poly1‘𝑅)))
48	41, 46, 47	syl2anc 582	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) ∈ (Base‘(Poly1‘𝑅)))
49		eqid 2729	. . . . . . . . 9 ⊢ (coe1‘((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0))) = (coe1‘((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))
50	15, 11, 49, 20	ply1coe1eq 22314	. . . . . . . 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 1368	. . . . . 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 256	. . . . 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 2730	. . . . . . . 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 7439	. . . . . . . . . . . 12 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑖𝑀𝑗) = (𝑥𝑀𝑦))
57	56	fveq2d 6909	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑥𝑀𝑦)))
58	57	fveq1d 6907	. . . . . . . . . 10 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → ((coe1‘(𝑖𝑀𝑗))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0))
59	58	fveq2d 6909	. . . . . . . . 9 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)) = ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))
60	59	adantl 480	. . . . . . . 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 767	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁) ∧ 𝑦 ∈ 𝑁) → 𝑥 ∈ 𝑁)
62		simpr 483	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁) ∧ 𝑦 ∈ 𝑁) → 𝑦 ∈ 𝑁)
63		fvexd 6920	. . . . . . . 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 7583	. . . . . . 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 2731	. . . . . 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 465	. . . . 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 256	. . 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 6920	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (Poly1‘𝑅) ∈ V)
70	6	3ad2ant1 1130	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
71	16	3ad2ant1 1130	. . . . . . . 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 22442	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑀𝑗) ∈ (Base‘(Poly1‘𝑅)))
73		eqid 2729	. . . . . . . 8 ⊢ (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗))
74	73, 11, 15, 8	coe1fvalcl 22224	. . . . . . 7 ⊢ (((𝑖𝑀𝑗) ∈ (Base‘(Poly1‘𝑅)) ∧ 0 ∈ ℕ0) → ((coe1‘(𝑖𝑀𝑗))‘0) ∈ (Base‘𝑅))
75	72, 19, 74	sylancl 584	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖𝑀𝑗))‘0) ∈ (Base‘𝑅))
76	15, 25, 8, 11	ply1sclcl 22299	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘(𝑖𝑀𝑗))‘0) ∈ (Base‘𝑅)) → ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)) ∈ (Base‘(Poly1‘𝑅)))
77	70, 75, 76	syl2anc 582	. . . . 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 22439	. . . 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 22440	. . . 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 582	. . 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 256	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))) = 𝑀)
82	4, 39, 81	3eqtrd 2773	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝑇‘(𝐼‘𝑀)) = 𝑀)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2100  ∀wral 3054  Vcvv 3472  ‘cfv 6558  (class class class)co 7430   ∈ cmpo 7432  Fincfn 8982  0cc0 11165  ℕ₀cn0 12534  Basecbs 17234  Ringcrg 20238  algSccascl 21872  Poly₁cpl1 22188  coe₁cco1 22189   Mat cmat 22421   ConstPolyMat ccpmat 22719   matToPolyMat cmat2pmat 22720   cPolyMatToMat ccpmat2mat 22721

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 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-rep 5293  ax-sep 5307  ax-nul 5314  ax-pow 5373  ax-pr 5437  ax-un 7751  ax-cnex 11221  ax-resscn 11222  ax-1cn 11223  ax-icn 11224  ax-addcl 11225  ax-addrcl 11226  ax-mulcl 11227  ax-mulrcl 11228  ax-mulcom 11229  ax-addass 11230  ax-mulass 11231  ax-distr 11232  ax-i2m1 11233  ax-1ne0 11234  ax-1rid 11235  ax-rnegex 11236  ax-rrecex 11237  ax-cnre 11238  ax-pre-lttri 11239  ax-pre-lttrn 11240  ax-pre-ltadd 11241  ax-pre-mulgt0 11242

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ne 2934  df-nel 3040  df-ral 3055  df-rex 3064  df-rmo 3373  df-reu 3374  df-rab 3429  df-v 3474  df-sbc 3789  df-csb 3905  df-dif 3962  df-un 3964  df-in 3966  df-ss 3976  df-pss 3979  df-nul 4336  df-if 4537  df-pw 4612  df-sn 4637  df-pr 4639  df-tp 4641  df-op 4643  df-ot 4645  df-uni 4919  df-int 4960  df-iun 5008  df-iin 5009  df-br 5157  df-opab 5219  df-mpt 5240  df-tr 5274  df-id 5584  df-eprel 5590  df-po 5598  df-so 5599  df-fr 5641  df-se 5642  df-we 5643  df-xp 5692  df-rel 5693  df-cnv 5694  df-co 5695  df-dm 5696  df-rn 5697  df-res 5698  df-ima 5699  df-pred 6317  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7386  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7695  df-ofr 7696  df-om 7885  df-1st 8011  df-2nd 8012  df-supp 8183  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8742  df-map 8865  df-pm 8866  df-ixp 8935  df-en 8983  df-dom 8984  df-sdom 8985  df-fin 8986  df-fsupp 9413  df-sup 9492  df-oi 9560  df-card 9989  df-pnf 11307  df-mnf 11308  df-xr 11309  df-ltxr 11310  df-le 11311  df-sub 11503  df-neg 11504  df-nn 12275  df-2 12337  df-3 12338  df-4 12339  df-5 12340  df-6 12341  df-7 12342  df-8 12343  df-9 12344  df-n0 12535  df-z 12621  df-dec 12740  df-uz 12885  df-fz 13549  df-fzo 13692  df-seq 14033  df-hash 14360  df-struct 17170  df-sets 17187  df-slot 17205  df-ndx 17217  df-base 17235  df-ress 17264  df-plusg 17300  df-mulr 17301  df-sca 17303  df-vsca 17304  df-ip 17305  df-tset 17306  df-ple 17307  df-ds 17309  df-hom 17311  df-cco 17312  df-0g 17477  df-gsum 17478  df-prds 17483  df-pws 17485  df-mre 17620  df-mrc 17621  df-acs 17623  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mhm 18794  df-submnd 18795  df-grp 18952  df-minusg 18953  df-sbg 18954  df-mulg 19084  df-subg 19139  df-ghm 19229  df-cntz 19333  df-cmn 19802  df-abl 19803  df-mgp 20140  df-rng 20158  df-ur 20187  df-srg 20192  df-ring 20240  df-subrng 20550  df-subrg 20575  df-lmod 20860  df-lss 20931  df-sra 21173  df-rgmod 21174  df-dsmm 21752  df-frlm 21767  df-ascl 21875  df-psr 21928  df-mvr 21929  df-mpl 21930  df-opsr 21932  df-psr1 22191  df-vr1 22192  df-ply1 22193  df-coe1 22194  df-mat 22422  df-cpmat 22722  df-mat2pmat 22723  df-cpmat2mat 22724

This theorem is referenced by:  m2cpmfo  22772  m2cpminv  22776  cpmadumatpoly  22899  cayhamlem4  22904