pmatcollpwlem - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > pmatcollpwlem		Structured version Visualization version GIF version

Theorem pmatcollpwlem 22695

Description: Lemma for pmatcollpw 22696. (Contributed by AV, 26-Oct-2019.) (Revised by AV, 4-Dec-2019.)

**Hypotheses**
Ref	Expression
pmatcollpw.p	⊢ 𝑃 = (Poly₁‘𝑅)
pmatcollpw.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
pmatcollpw.b	⊢ 𝐵 = (Base‘𝐶)
pmatcollpw.m	⊢ ∗ = ( ·_𝑠 ‘𝐶)
pmatcollpw.e	⊢ ↑ = (._g‘(mulGrp‘𝑃))
pmatcollpw.x	⊢ 𝑋 = (var₁‘𝑅)
pmatcollpw.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)

**Assertion**
Ref	Expression
pmatcollpwlem	⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·_𝑠 ‘𝑃)(𝑛 ↑ 𝑋)) = (𝑎((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))𝑏))

Proof of Theorem pmatcollpwlem Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pmatcollpw.p	. . . . . . . 8 ⊢ 𝑃 = (Poly₁‘𝑅)
2	1	ply1assa 22122	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
3	2	3ad2ant2 1131	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ AssAlg)
4	3	adantr 479	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) → 𝑃 ∈ AssAlg)
5	4	3ad2ant1 1130	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑃 ∈ AssAlg)
6		eqid 2725	. . . . . 6 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
7		eqid 2725	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		eqid 2725	. . . . . 6 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
9		simp2 1134	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑎 ∈ 𝑁)
10		simp3 1135	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑏 ∈ 𝑁)
11		simp2 1134	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ CRing)
12	11	adantr 479	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) → 𝑅 ∈ CRing)
13		simp3 1135	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵)
14	13	adantr 479	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) → 𝑀 ∈ 𝐵)
15		simpr 483	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℕ₀)
16		pmatcollpw.c	. . . . . . . . 9 ⊢ 𝐶 = (𝑁 Mat 𝑃)
17		pmatcollpw.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐶)
18	1, 16, 17, 6, 8	decpmatcl 22682	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑛 ∈ ℕ₀) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
19	12, 14, 15, 18	syl3anc 1368	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
20	19	3ad2ant1 1130	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
21	6, 7, 8, 9, 10, 20	matecld 22341	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘𝑅))
22		crngring 20184	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
23	22	3ad2ant2 1131	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
24	1	ply1sca 22175	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
25	23, 24	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 = (Scalar‘𝑃))
26	25	eqcomd 2731	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑃) = 𝑅)
27	26	fveq2d 6894	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
28	27	eleq2d 2811	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘(Scalar‘𝑃)) ↔ (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘𝑅)))
29	28	adantr 479	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) → ((𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘(Scalar‘𝑃)) ↔ (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘𝑅)))
30	29	3ad2ant1 1130	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘(Scalar‘𝑃)) ↔ (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘𝑅)))
31	21, 30	mpbird 256	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘(Scalar‘𝑃)))
32		pmatcollpw.x	. . . . . . 7 ⊢ 𝑋 = (var₁‘𝑅)
33		eqid 2725	. . . . . . 7 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
34		pmatcollpw.e	. . . . . . 7 ⊢ ↑ = (._g‘(mulGrp‘𝑃))
35		eqid 2725	. . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑃)
36	1, 32, 33, 34, 35	ply1moncl 22194	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑛 ∈ ℕ₀) → (𝑛 ↑ 𝑋) ∈ (Base‘𝑃))
37	23, 36	sylan 578	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) → (𝑛 ↑ 𝑋) ∈ (Base‘𝑃))
38	37	3ad2ant1 1130	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑛 ↑ 𝑋) ∈ (Base‘𝑃))
39		eqid 2725	. . . . 5 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
40		eqid 2725	. . . . 5 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
41		eqid 2725	. . . . 5 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
42		eqid 2725	. . . . 5 ⊢ (._r‘𝑃) = (._r‘𝑃)
43		eqid 2725	. . . . 5 ⊢ ( ·_𝑠 ‘𝑃) = ( ·_𝑠 ‘𝑃)
44	39, 40, 41, 35, 42, 43	asclmul2 21819	. . . 4 ⊢ ((𝑃 ∈ AssAlg ∧ (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑛 ↑ 𝑋) ∈ (Base‘𝑃)) → ((𝑛 ↑ 𝑋)(._r‘𝑃)((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏))) = ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·_𝑠 ‘𝑃)(𝑛 ↑ 𝑋)))
45	5, 31, 38, 44	syl3anc 1368	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑛 ↑ 𝑋)(._r‘𝑃)((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏))) = ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·_𝑠 ‘𝑃)(𝑛 ↑ 𝑋)))
46		eqidd 2726	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))))
47		oveq12 7422	. . . . . . . 8 ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → (𝑖(𝑀 decompPMat 𝑛)𝑗) = (𝑎(𝑀 decompPMat 𝑛)𝑏))
48	47	fveq2d 6894	. . . . . . 7 ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)) = ((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏)))
49	48	adantl 480	. . . . . 6 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ (𝑖 = 𝑎 ∧ 𝑗 = 𝑏)) → ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)) = ((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏)))
50		fvexd 6905	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏)) ∈ V)
51	46, 49, 9, 10, 50	ovmpod 7567	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)))𝑏) = ((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏)))
52	51	eqcomd 2731	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏)) = (𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)))𝑏))
53	52	oveq2d 7429	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑛 ↑ 𝑋)(._r‘𝑃)((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏))) = ((𝑛 ↑ 𝑋)(._r‘𝑃)(𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)))𝑏)))
54	45, 53	eqtr3d 2767	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·_𝑠 ‘𝑃)(𝑛 ↑ 𝑋)) = ((𝑛 ↑ 𝑋)(._r‘𝑃)(𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)))𝑏)))
55	1	ply1ring 22170	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
56	22, 55	syl 17	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
57	56	3ad2ant2 1131	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ Ring)
58	57	adantr 479	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) → 𝑃 ∈ Ring)
59	58	3ad2ant1 1130	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑃 ∈ Ring)
60		simpl1 1188	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) → 𝑁 ∈ Fin)
61	12, 22	syl 17	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) → 𝑅 ∈ Ring)
62	61	3ad2ant1 1130	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
63		simp2 1134	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
64		simp3 1135	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
65	19	3ad2ant1 1130	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
66	6, 7, 8, 63, 64, 65	matecld 22341	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖(𝑀 decompPMat 𝑛)𝑗) ∈ (Base‘𝑅))
67	1, 39, 7, 35	ply1sclcl 22209	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑖(𝑀 decompPMat 𝑛)𝑗) ∈ (Base‘𝑅)) → ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)) ∈ (Base‘𝑃))
68	62, 66, 67	syl2anc 582	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)) ∈ (Base‘𝑃))
69	16, 35, 17, 60, 58, 68	matbas2d 22338	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))) ∈ 𝐵)
70	37, 69	jca 510	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) → ((𝑛 ↑ 𝑋) ∈ (Base‘𝑃) ∧ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))) ∈ 𝐵))
71	70	3ad2ant1 1130	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑛 ↑ 𝑋) ∈ (Base‘𝑃) ∧ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))) ∈ 𝐵))
72	9, 10	jca 510	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁))
73		pmatcollpw.m	. . . 4 ⊢ ∗ = ( ·_𝑠 ‘𝐶)
74	16, 17, 35, 73, 42	matvscacell 22351	. . 3 ⊢ ((𝑃 ∈ Ring ∧ ((𝑛 ↑ 𝑋) ∈ (Base‘𝑃) ∧ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))) ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎((𝑛 ↑ 𝑋) ∗ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))))𝑏) = ((𝑛 ↑ 𝑋)(._r‘𝑃)(𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)))𝑏)))
75	59, 71, 72, 74	syl3anc 1368	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑎((𝑛 ↑ 𝑋) ∗ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))))𝑏) = ((𝑛 ↑ 𝑋)(._r‘𝑃)(𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)))𝑏)))
76	23	adantr 479	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) → 𝑅 ∈ Ring)
77		pmatcollpw.t	. . . . . . . 8 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
78	77, 6, 8, 1, 39	mat2pmatval 22639	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅))) → (𝑇‘(𝑀 decompPMat 𝑛)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))))
79	60, 76, 19, 78	syl3anc 1368	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) → (𝑇‘(𝑀 decompPMat 𝑛)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))))
80	79	eqcomd 2731	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))) = (𝑇‘(𝑀 decompPMat 𝑛)))
81	80	oveq2d 7429	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) → ((𝑛 ↑ 𝑋) ∗ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)))) = ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))
82	81	oveqd 7430	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) → (𝑎((𝑛 ↑ 𝑋) ∗ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))))𝑏) = (𝑎((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))𝑏))
83	82	3ad2ant1 1130	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑎((𝑛 ↑ 𝑋) ∗ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))))𝑏) = (𝑎((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))𝑏))
84	54, 75, 83	3eqtr2d 2771	1 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·_𝑠 ‘𝑃)(𝑛 ↑ 𝑋)) = (𝑎((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))𝑏))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ‘cfv 6543 (class class class)co 7413 ∈ cmpo 7415 Fincfn 8957 ℕ₀cn0 12497 Basecbs 17174 ._rcmulr 17228 Scalarcsca 17230 ·_𝑠 cvsca 17231 ._gcmg 19022 mulGrpcmgp 20073 Ringcrg 20172 CRingccrg 20173 AssAlgcasa 21783 algSccascl 21785 var₁cv1 22098 Poly₁cpl1 22099 Mat cmat 22320 matToPolyMat cmat2pmat 22619 decompPMat cdecpmat 22677

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-ot 4634 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-ofr 7680 df-om 7866 df-1st 7987 df-2nd 7988 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9381 df-sup 9460 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-fz 13512 df-fzo 13655 df-seq 13994 df-hash 14317 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-hom 17251 df-cco 17252 df-0g 17417 df-gsum 17418 df-prds 17423 df-pws 17425 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-mhm 18734 df-submnd 18735 df-grp 18892 df-minusg 18893 df-sbg 18894 df-mulg 19023 df-subg 19077 df-ghm 19167 df-cntz 19267 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-cring 20175 df-subrng 20482 df-subrg 20507 df-lmod 20744 df-lss 20815 df-sra 21057 df-rgmod 21058 df-dsmm 21665 df-frlm 21680 df-assa 21786 df-ascl 21788 df-psr 21841 df-mvr 21842 df-mpl 21843 df-opsr 21845 df-psr1 22102 df-vr1 22103 df-ply1 22104 df-coe1 22105 df-mat 22321 df-mat2pmat 22622 df-decpmat 22678

This theorem is referenced by: pmatcollpw 22696

W3C validator