Theorem m2cpminvid2 22782

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 22777	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝐼‘𝑀) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
4	3	fveq2d 6924	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝑇‘(𝐼‘𝑀)) = (𝑇‘(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))))
5		simp1 1136	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → 𝑁 ∈ Fin)
6		simp2 1137	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → 𝑅 ∈ Ring)
7		eqid 2740	. . . . 5 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
8		eqid 2740	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
9		eqid 2740	. . . . 5 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
10		eqid 2740	. . . . . . 7 ⊢ (𝑁 Mat (Poly1‘𝑅)) = (𝑁 Mat (Poly1‘𝑅))
11		eqid 2740	. . . . . . 7 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅))
12		eqid 2740	. . . . . . 7 ⊢ (Base‘(𝑁 Mat (Poly1‘𝑅))) = (Base‘(𝑁 Mat (Poly1‘𝑅)))
13		simp2 1137	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑥 ∈ 𝑁)
14		simp3 1138	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑦 ∈ 𝑁)
15		eqid 2740	. . . . . . . . 9 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅)
16	2, 15, 10, 12	cpmatpmat 22737	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → 𝑀 ∈ (Base‘(𝑁 Mat (Poly1‘𝑅))))
17	16	3ad2ant1 1133	. . . . . . 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 22453	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑀𝑦) ∈ (Base‘(Poly1‘𝑅)))
19		0nn0 12568	. . . . . 6 ⊢ 0 ∈ ℕ0
20		eqid 2740	. . . . . . 7 ⊢ (coe1‘(𝑥𝑀𝑦)) = (coe1‘(𝑥𝑀𝑦))
21	20, 11, 15, 8	coe1fvalcl 22235	. . . . . 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 22450	. . . 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 2740	. . . . 5 ⊢ (algSc‘(Poly1‘𝑅)) = (algSc‘(Poly1‘𝑅))
26	24, 7, 9, 15, 25	mat2pmatval 22751	. . . 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 1371	. . 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 2741	. . . . . 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 7457	. . . . . . . . 9 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑥𝑀𝑦) = (𝑖𝑀𝑗))
30	29	fveq2d 6924	. . . . . . . 8 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (coe1‘(𝑥𝑀𝑦)) = (coe1‘(𝑖𝑀𝑗)))
31	30	fveq1d 6922	. . . . . . 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 1137	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
34		simp3 1138	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
35		fvexd 6935	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖𝑀𝑗))‘0) ∈ V)
36	28, 32, 33, 34, 35	ovmpod 7602	. . . . 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 6924	. . . 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 7527	. . 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 2780	. 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 22781	. . . . . 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 1192	. . . . . . 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 22453	. . . . . . . . 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 22310	. . . . . . . 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 2740	. . . . . . . . 9 ⊢ (coe1‘((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0))) = (coe1‘((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))
50	15, 11, 49, 20	ply1coe1eq 22325	. . . . . . . 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 1371	. . . . . 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 3208	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑀𝑦))
55		eqidd 2741	. . . . . . . 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 7457	. . . . . . . . . . . 12 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑖𝑀𝑗) = (𝑥𝑀𝑦))
57	56	fveq2d 6924	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑥𝑀𝑦)))
58	57	fveq1d 6922	. . . . . . . . . 10 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → ((coe1‘(𝑖𝑀𝑗))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0))
59	58	fveq2d 6924	. . . . . . . . 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 6935	. . . . . . . 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 7602	. . . . . . 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 2742	. . . . . 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 3225	. . . 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 6935	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (Poly1‘𝑅) ∈ V)
70	6	3ad2ant1 1133	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
71	16	3ad2ant1 1133	. . . . . . . 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 22453	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑀𝑗) ∈ (Base‘(Poly1‘𝑅)))
73		eqid 2740	. . . . . . . 8 ⊢ (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗))
74	73, 11, 15, 8	coe1fvalcl 22235	. . . . . . 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 22310	. . . . . 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 22450	. . . 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 22451	. . . 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 2784	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝑇‘(𝐼‘𝑀)) = 𝑀)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  Fincfn 9003  0cc0 11184  ℕ₀cn0 12553  Basecbs 17258  Ringcrg 20260  algSccascl 21895  Poly₁cpl1 22199  coe₁cco1 22200   Mat cmat 22432   ConstPolyMat ccpmat 22730   matToPolyMat cmat2pmat 22731   cPolyMatToMat ccpmat2mat 22732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-ofr 7715  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-pm 8887  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-sup 9511  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-fzo 13712  df-seq 14053  df-hash 14380  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-hom 17335  df-cco 17336  df-0g 17501  df-gsum 17502  df-prds 17507  df-pws 17509  df-mre 17644  df-mrc 17645  df-acs 17647  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-mhm 18818  df-submnd 18819  df-grp 18976  df-minusg 18977  df-sbg 18978  df-mulg 19108  df-subg 19163  df-ghm 19253  df-cntz 19357  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-srg 20214  df-ring 20262  df-subrng 20572  df-subrg 20597  df-lmod 20882  df-lss 20953  df-sra 21195  df-rgmod 21196  df-dsmm 21775  df-frlm 21790  df-ascl 21898  df-psr 21952  df-mvr 21953  df-mpl 21954  df-opsr 21956  df-psr1 22202  df-vr1 22203  df-ply1 22204  df-coe1 22205  df-mat 22433  df-cpmat 22733  df-mat2pmat 22734  df-cpmat2mat 22735

This theorem is referenced by:  m2cpmfo  22783  m2cpminv  22787  cpmadumatpoly  22910  cayhamlem4  22915