Theorem m2cpminvid2 21812

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 21807	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝐼‘𝑀) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
4	3	fveq2d 6760	. 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 2738	. . . . 5 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
8		eqid 2738	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
9		eqid 2738	. . . . 5 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
10		eqid 2738	. . . . . . 7 ⊢ (𝑁 Mat (Poly1‘𝑅)) = (𝑁 Mat (Poly1‘𝑅))
11		eqid 2738	. . . . . . 7 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅))
12		eqid 2738	. . . . . . 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 2738	. . . . . . . . 9 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅)
16	2, 15, 10, 12	cpmatpmat 21767	. . . . . . . 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 21483	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑀𝑦) ∈ (Base‘(Poly1‘𝑅)))
19		0nn0 12178	. . . . . 6 ⊢ 0 ∈ ℕ0
20		eqid 2738	. . . . . . 7 ⊢ (coe1‘(𝑥𝑀𝑦)) = (coe1‘(𝑥𝑀𝑦))
21	20, 11, 15, 8	coe1fvalcl 21293	. . . . . 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 21480	. . . 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 2738	. . . . 5 ⊢ (algSc‘(Poly1‘𝑅)) = (algSc‘(Poly1‘𝑅))
26	24, 7, 9, 15, 25	mat2pmatval 21781	. . . 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 2739	. . . . . 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 7264	. . . . . . . . 9 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑥𝑀𝑦) = (𝑖𝑀𝑗))
30	29	fveq2d 6760	. . . . . . . 8 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (coe1‘(𝑥𝑀𝑦)) = (coe1‘(𝑖𝑀𝑗)))
31	30	fveq1d 6758	. . . . . . 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 6771	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖𝑀𝑗))‘0) ∈ V)
36	28, 32, 33, 34, 35	ovmpod 7403	. . . . 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 6760	. . . 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 7330	. . 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 2778	. 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 21811	. . . . . 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 767	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑥 ∈ 𝑁)
43		simprr 769	. . . . . . . . . 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 21483	. . . . . . . . 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 21367	. . . . . . . 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 2738	. . . . . . . . 9 ⊢ (coe1‘((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0))) = (coe1‘((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))
50	15, 11, 49, 20	ply1coe1eq 21379	. . . . . . . 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 256	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑀𝑦))
54	53	ralrimivva 3114	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑀𝑦))
55		eqidd 2739	. . . . . . . 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 7264	. . . . . . . . . . . 12 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑖𝑀𝑗) = (𝑥𝑀𝑦))
57	56	fveq2d 6760	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑥𝑀𝑦)))
58	57	fveq1d 6758	. . . . . . . . . 10 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → ((coe1‘(𝑖𝑀𝑗))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0))
59	58	fveq2d 6760	. . . . . . . . 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 765	. . . . . . . 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 6771	. . . . . . . 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 7403	. . . . . . 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 2740	. . . . . 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 3121	. . . 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 6771	. . . . 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 21483	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑀𝑗) ∈ (Base‘(Poly1‘𝑅)))
73		eqid 2738	. . . . . . . 8 ⊢ (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗))
74	73, 11, 15, 8	coe1fvalcl 21293	. . . . . . 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 21367	. . . . . 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 21480	. . . 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 21481	. . . 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 256	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘(Poly1‘𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))) = 𝑀)
82	4, 39, 81	3eqtrd 2782	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝑇‘(𝐼‘𝑀)) = 𝑀)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  Fincfn 8691  0cc0 10802  ℕ₀cn0 12163  Basecbs 16840  Ringcrg 19698  algSccascl 20969  Poly₁cpl1 21258  coe₁cco1 21259   Mat cmat 21464   ConstPolyMat ccpmat 21760   matToPolyMat cmat2pmat 21761   cPolyMatToMat ccpmat2mat 21762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-ot 4567  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-ofr 7512  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-pm 8576  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-sup 9131  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-fz 13169  df-fzo 13312  df-seq 13650  df-hash 13973  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-sca 16904  df-vsca 16905  df-ip 16906  df-tset 16907  df-ple 16908  df-ds 16910  df-hom 16912  df-cco 16913  df-0g 17069  df-gsum 17070  df-prds 17075  df-pws 17077  df-mre 17212  df-mrc 17213  df-acs 17215  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-mhm 18345  df-submnd 18346  df-grp 18495  df-minusg 18496  df-sbg 18497  df-mulg 18616  df-subg 18667  df-ghm 18747  df-cntz 18838  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-srg 19657  df-ring 19700  df-subrg 19937  df-lmod 20040  df-lss 20109  df-sra 20349  df-rgmod 20350  df-dsmm 20849  df-frlm 20864  df-ascl 20972  df-psr 21022  df-mvr 21023  df-mpl 21024  df-opsr 21026  df-psr1 21261  df-vr1 21262  df-ply1 21263  df-coe1 21264  df-mat 21465  df-cpmat 21763  df-mat2pmat 21764  df-cpmat2mat 21765

This theorem is referenced by:  m2cpmfo  21813  m2cpminv  21817  cpmadumatpoly  21940  cayhamlem4  21945