Theorem pmatcollpw3lem 23128

Description: Lemma for pmatcollpw3 23129 and pmatcollpw3fi 23130: Write a polynomial matrix (over a commutative ring) as a sum of products of variable powers and constant matrices with scalar entries. (Contributed by AV, 8-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)
pmatcollpw3.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
pmatcollpw3.d	⊢ 𝐷 = (Base‘𝐴)

Assertion

Ref	Expression
pmatcollpw3lem	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → (𝑀 = (𝐶 Σg (𝑛 ∈ 𝐼 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))) → ∃𝑓 ∈ (𝐷 ↑m 𝐼)𝑀 = (𝐶 Σg (𝑛 ∈ 𝐼 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))))))

Distinct variable groups:   𝐵,𝑛   𝑛,𝑀   𝑛,𝑁   𝑃,𝑛   𝑅,𝑛   𝑛,𝑋   ↑ ,𝑛   𝐶,𝑛   𝐵,𝑓   𝐶,𝑓,𝑛   𝐷,𝑓   𝑓,𝐼,𝑛   𝑓,𝑀   𝑓,𝑁   𝑅,𝑓   𝑇,𝑓   𝑓,𝑋   ↑ ,𝑓   ∗ ,𝑓

Allowed substitution hints:   𝐴(𝑓, 𝑛)   𝐷(𝑛)   𝑃(𝑓)   𝑇(𝑛)   ∗ (𝑛)

Proof of Theorem pmatcollpw3lem

Dummy variables 𝑖 𝑗 𝑘 𝑙 𝑥 𝑦 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmeq 5891	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → dom 𝑥 = dom 𝑦)
2	1	dmeqd 5893	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → dom dom 𝑥 = dom dom 𝑦)
3		oveq 7446	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑖𝑥𝑗) = (𝑖𝑦𝑗))
4	3	fveq2d 6909	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (coe1‘(𝑖𝑥𝑗)) = (coe1‘(𝑖𝑦𝑗)))
5	4	fveq1d 6907	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((coe1‘(𝑖𝑥𝑗))‘𝑘) = ((coe1‘(𝑖𝑦𝑗))‘𝑘))
6	2, 2, 5	mpoeq123dv 7515	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑘)))
7		fveq2 6905	. . . . . . . 8 ⊢ (𝑘 = 𝑙 → ((coe1‘(𝑖𝑦𝑗))‘𝑘) = ((coe1‘(𝑖𝑦𝑗))‘𝑙))
8	7	mpoeq3dv 7519	. . . . . . 7 ⊢ (𝑘 = 𝑙 → (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙)))
9	6, 8	cbvmpov 7535	. . . . . 6 ⊢ (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))) = (𝑦 ∈ 𝐵, 𝑙 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙)))
10		dmexg 7934	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 → dom 𝑦 ∈ V)
11	10	dmexd 7936	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 → dom dom 𝑦 ∈ V)
12	11, 11	jca 521	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → (dom dom 𝑦 ∈ V ∧ dom dom 𝑦 ∈ V))
13	12	ad2antrl 741	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑙 ∈ 𝐼)) → (dom dom 𝑦 ∈ V ∧ dom dom 𝑦 ∈ V))
14		mpoexga 8112	. . . . . . . 8 ⊢ ((dom dom 𝑦 ∈ V ∧ dom dom 𝑦 ∈ V) → (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙)) ∈ V)
15	13, 14	syl 18	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑙 ∈ 𝐼)) → (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙)) ∈ V)
16	15	ralrimivva 3209	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → ∀𝑦 ∈ 𝐵 ∀𝑙 ∈ 𝐼 (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙)) ∈ V)
17		simprr 785	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → 𝐼 ≠ ∅)
18		nn0ex 12636	. . . . . . . 8 ⊢ ℕ0 ∈ V
19	18	ssex 5286	. . . . . . 7 ⊢ (𝐼 ⊆ ℕ0 → 𝐼 ∈ V)
20	19	ad2antrl 741	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → 𝐼 ∈ V)
21		simp3 1156	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵)
22	21	adantr 486	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → 𝑀 ∈ 𝐵)
23	9, 16, 17, 20, 22	mpocurryvald 8312	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → (curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀) = (𝑙 ∈ 𝐼 ↦ ⦋𝑀 / 𝑦⦌(𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙))))
24		fveq2 6905	. . . . . . . . 9 ⊢ (𝑙 = 𝑘 → ((coe1‘(𝑖𝑦𝑗))‘𝑙) = ((coe1‘(𝑖𝑦𝑗))‘𝑘))
25	24	mpoeq3dv 7519	. . . . . . . 8 ⊢ (𝑙 = 𝑘 → (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙)) = (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑘)))
26	25	csbeq2dv 3858	. . . . . . 7 ⊢ (𝑙 = 𝑘 → ⦋𝑀 / 𝑦⦌(𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙)) = ⦋𝑀 / 𝑦⦌(𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑘)))
27		eqcom 2770	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
28		eqcom 2770	. . . . . . . . 9 ⊢ ((𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑘)) ↔ (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))
29	6, 27, 28	3imtr3i 294	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))
30	29	cbvcsbv 3863	. . . . . . 7 ⊢ ⦋𝑀 / 𝑦⦌(𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑘)) = ⦋𝑀 / 𝑥⦌(𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))
31	26, 30	eqtrdi 2814	. . . . . 6 ⊢ (𝑙 = 𝑘 → ⦋𝑀 / 𝑦⦌(𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙)) = ⦋𝑀 / 𝑥⦌(𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))
32	31	cbvmptv 5212	. . . . 5 ⊢ (𝑙 ∈ 𝐼 ↦ ⦋𝑀 / 𝑦⦌(𝑖 ∈ dom dom 𝑦, 𝑗 ∈ dom dom 𝑦 ↦ ((coe1‘(𝑖𝑦𝑗))‘𝑙))) = (𝑘 ∈ 𝐼 ↦ ⦋𝑀 / 𝑥⦌(𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))
33	23, 32	eqtrdi 2814	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → (curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀) = (𝑘 ∈ 𝐼 ↦ ⦋𝑀 / 𝑥⦌(𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))))
34		dmeq 5891	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑀 → dom 𝑥 = dom 𝑀)
35	34	dmeqd 5893	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → dom dom 𝑥 = dom dom 𝑀)
36		oveq 7446	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑀 → (𝑖𝑥𝑗) = (𝑖𝑀𝑗))
37	36	fveq2d 6909	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑀 → (coe1‘(𝑖𝑥𝑗)) = (coe1‘(𝑖𝑀𝑗)))
38	37	fveq1d 6907	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → ((coe1‘(𝑖𝑥𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝑘))
39	35, 35, 38	mpoeq123dv 7515	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)))
40	39	adantl 487	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 = 𝑀) → (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)))
41	21, 40	csbied 3887	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ⦋𝑀 / 𝑥⦌(𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)))
42		pmatcollpw.c	. . . . . . . . . . . . 13 ⊢ 𝐶 = (𝑁 Mat 𝑃)
43		eqid 2763	. . . . . . . . . . . . 13 ⊢ (Base‘𝑃) = (Base‘𝑃)
44		pmatcollpw.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝐶)
45	42, 43, 44	matbas2i 22764	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ ((Base‘𝑃) ↑m (𝑁 × 𝑁)))
46		elmapi 8894	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ((Base‘𝑃) ↑m (𝑁 × 𝑁)) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑃))
47		fdm 6738	. . . . . . . . . . . . . 14 ⊢ (𝑀:(𝑁 × 𝑁)⟶(Base‘𝑃) → dom 𝑀 = (𝑁 × 𝑁))
48	47	dmeqd 5893	. . . . . . . . . . . . 13 ⊢ (𝑀:(𝑁 × 𝑁)⟶(Base‘𝑃) → dom dom 𝑀 = dom (𝑁 × 𝑁))
49		dmxpid 5918	. . . . . . . . . . . . 13 ⊢ dom (𝑁 × 𝑁) = 𝑁
50	48, 49	eqtr2di 2815	. . . . . . . . . . . 12 ⊢ (𝑀:(𝑁 × 𝑁)⟶(Base‘𝑃) → 𝑁 = dom dom 𝑀)
51	45, 46, 50	3syl 19	. . . . . . . . . . 11 ⊢ (𝑀 ∈ 𝐵 → 𝑁 = dom dom 𝑀)
52	51	3ad2ant3 1153	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑁 = dom dom 𝑀)
53	52	adantr 486	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → 𝑁 = dom dom 𝑀)
54		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
55	54	oveqd 7457	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
56	55	fveq2d 6909	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
57	56	fveq1d 6907	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → ((coe1‘(𝑖𝑚𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝑘))
58	53, 53, 57	mpoeq123dv 7515	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)))
59	21, 58	csbied 3887	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ⦋𝑀 / 𝑚⦌(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)))
60	41, 59	eqtr4d 2801	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ⦋𝑀 / 𝑥⦌(𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) = ⦋𝑀 / 𝑚⦌(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)))
61	60	adantr 486	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → ⦋𝑀 / 𝑥⦌(𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) = ⦋𝑀 / 𝑚⦌(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)))
62	61	mpteq2dv 5202	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → (𝑘 ∈ 𝐼 ↦ ⦋𝑀 / 𝑥⦌(𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))) = (𝑘 ∈ 𝐼 ↦ ⦋𝑀 / 𝑚⦌(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))))
63	33, 62	eqtrd 2798	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → (curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀) = (𝑘 ∈ 𝐼 ↦ ⦋𝑀 / 𝑚⦌(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))))
64		oveq 7446	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
65	64	adantl 487	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
66	65	fveq2d 6909	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
67	66	fveq1d 6907	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → ((coe1‘(𝑖𝑚𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝑘))
68	67	mpoeq3dv 7519	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)))
69	21, 68	csbied 3887	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ⦋𝑀 / 𝑚⦌(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)))
70	69	ad2antrr 739	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑘 ∈ 𝐼) → ⦋𝑀 / 𝑚⦌(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)))
71		pmatcollpw3.a	. . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅)
72		eqid 2763	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
73		pmatcollpw3.d	. . . . . . 7 ⊢ 𝐷 = (Base‘𝐴)
74		simpll1 1231	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑘 ∈ 𝐼) → 𝑁 ∈ Fin)
75		simpll2 1232	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑘 ∈ 𝐼) → 𝑅 ∈ CRing)
76		simp2 1155	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
77		simp3 1156	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
78	22	adantr 486	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑘 ∈ 𝐼) → 𝑀 ∈ 𝐵)
79	78	3ad2ant1 1151	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑀 ∈ 𝐵)
80	42, 43, 44, 76, 77, 79	matecld 22768	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑀𝑗) ∈ (Base‘𝑃))
81		ssel 3907	. . . . . . . . . . 11 ⊢ (𝐼 ⊆ ℕ0 → (𝑘 ∈ 𝐼 → 𝑘 ∈ ℕ0))
82	81	ad2antrl 741	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → (𝑘 ∈ 𝐼 → 𝑘 ∈ ℕ0))
83	82	imp 412	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑘 ∈ 𝐼) → 𝑘 ∈ ℕ0)
84	83	3ad2ant1 1151	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑘 ∈ ℕ0)
85		eqid 2763	. . . . . . . . 9 ⊢ (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗))
86		pmatcollpw.p	. . . . . . . . 9 ⊢ 𝑃 = (Poly1‘𝑅)
87	85, 43, 86, 72	coe1fvalcl 22557	. . . . . . . 8 ⊢ (((𝑖𝑀𝑗) ∈ (Base‘𝑃) ∧ 𝑘 ∈ ℕ0) → ((coe1‘(𝑖𝑀𝑗))‘𝑘) ∈ (Base‘𝑅))
88	80, 84, 87	syl2anc 596	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖𝑀𝑗))‘𝑘) ∈ (Base‘𝑅))
89	71, 72, 73, 74, 75, 88	matbas2d 22765	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑘 ∈ 𝐼) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)) ∈ 𝐷)
90	70, 89	eqeltrd 2864	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑘 ∈ 𝐼) → ⦋𝑀 / 𝑚⦌(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) ∈ 𝐷)
91	90	fmpttd 7136	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → (𝑘 ∈ 𝐼 ↦ ⦋𝑀 / 𝑚⦌(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))):𝐼⟶𝐷)
92	73	fvexi 6919	. . . . . 6 ⊢ 𝐷 ∈ V
93	92	a1i 11	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐷 ∈ V)
94	19	adantr 486	. . . . 5 ⊢ ((𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅) → 𝐼 ∈ V)
95		elmapg 8884	. . . . 5 ⊢ ((𝐷 ∈ V ∧ 𝐼 ∈ V) → ((𝑘 ∈ 𝐼 ↦ ⦋𝑀 / 𝑚⦌(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))) ∈ (𝐷 ↑m 𝐼) ↔ (𝑘 ∈ 𝐼 ↦ ⦋𝑀 / 𝑚⦌(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))):𝐼⟶𝐷))
96	93, 94, 95	syl2an 608	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → ((𝑘 ∈ 𝐼 ↦ ⦋𝑀 / 𝑚⦌(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))) ∈ (𝐷 ↑m 𝐼) ↔ (𝑘 ∈ 𝐼 ↦ ⦋𝑀 / 𝑚⦌(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))):𝐼⟶𝐷))
97	91, 96	mpbird 260	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → (𝑘 ∈ 𝐼 ↦ ⦋𝑀 / 𝑚⦌(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))) ∈ (𝐷 ↑m 𝐼))
98	63, 97	eqeltrd 2864	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → (curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀) ∈ (𝐷 ↑m 𝐼))
99		fveq1 6904	. . . . . . . . . . 11 ⊢ (𝑓 = (curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀) → (𝑓‘𝑛) = ((curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)‘𝑛))
100	99	adantl 487	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) → (𝑓‘𝑛) = ((curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)‘𝑛))
101	100	adantr 486	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) ∧ 𝑛 ∈ 𝐼) → (𝑓‘𝑛) = ((curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)‘𝑛))
102		eqid 2763	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))) = (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))
103		dmexg 7934	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝐵 → dom 𝑥 ∈ V)
104	103	dmexd 7936	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐵 → dom dom 𝑥 ∈ V)
105	104, 104	jca 521	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐵 → (dom dom 𝑥 ∈ V ∧ dom dom 𝑥 ∈ V))
106	105	ad2antrl 741	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑛 ∈ 𝐼) ∧ (𝑥 ∈ 𝐵 ∧ 𝑘 ∈ 𝐼)) → (dom dom 𝑥 ∈ V ∧ dom dom 𝑥 ∈ V))
107		mpoexga 8112	. . . . . . . . . . . . . 14 ⊢ ((dom dom 𝑥 ∈ V ∧ dom dom 𝑥 ∈ V) → (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) ∈ V)
108	106, 107	syl 18	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑛 ∈ 𝐼) ∧ (𝑥 ∈ 𝐵 ∧ 𝑘 ∈ 𝐼)) → (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) ∈ V)
109	108	ralrimivva 3209	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑛 ∈ 𝐼) → ∀𝑥 ∈ 𝐵 ∀𝑘 ∈ 𝐼 (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)) ∈ V)
110	20	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑛 ∈ 𝐼) → 𝐼 ∈ V)
111	22	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑛 ∈ 𝐼) → 𝑀 ∈ 𝐵)
112		simpr 490	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑛 ∈ 𝐼) → 𝑛 ∈ 𝐼)
113	102, 109, 110, 111, 112	fvmpocurryd 8313	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑛 ∈ 𝐼) → ((curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)‘𝑛) = (𝑀(𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))𝑛))
114		df-decpmat 23108	. . . . . . . . . . . . . 14 ⊢ decompPMat = (𝑥 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))
115	114	reseq1i 5976	. . . . . . . . . . . . 13 ⊢ ( decompPMat ↾ (𝐵 × 𝐼)) = ((𝑥 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))) ↾ (𝐵 × 𝐼))
116		ssv 3937	. . . . . . . . . . . . . . . . 17 ⊢ 𝐵 ⊆ V
117	116	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐵 ⊆ V)
118		simpl 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅) → 𝐼 ⊆ ℕ0)
119	117, 118	anim12i 625	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → (𝐵 ⊆ V ∧ 𝐼 ⊆ ℕ0))
120	119	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑛 ∈ 𝐼) → (𝐵 ⊆ V ∧ 𝐼 ⊆ ℕ0))
121		resmpo 7560	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ⊆ V ∧ 𝐼 ⊆ ℕ0) → ((𝑥 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))) ↾ (𝐵 × 𝐼)) = (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))))
122	120, 121	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑛 ∈ 𝐼) → ((𝑥 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))) ↾ (𝐵 × 𝐼)) = (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))))
123	115, 122	eqtr2id 2811	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑛 ∈ 𝐼) → (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))) = ( decompPMat ↾ (𝐵 × 𝐼)))
124	123	oveqd 7457	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑛 ∈ 𝐼) → (𝑀(𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))𝑛) = (𝑀( decompPMat ↾ (𝐵 × 𝐼))𝑛))
125	113, 124	eqtrd 2798	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑛 ∈ 𝐼) → ((curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)‘𝑛) = (𝑀( decompPMat ↾ (𝐵 × 𝐼))𝑛))
126	125	adantlr 728	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) ∧ 𝑛 ∈ 𝐼) → ((curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)‘𝑛) = (𝑀( decompPMat ↾ (𝐵 × 𝐼))𝑛))
127	101, 126	eqtrd 2798	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) ∧ 𝑛 ∈ 𝐼) → (𝑓‘𝑛) = (𝑀( decompPMat ↾ (𝐵 × 𝐼))𝑛))
128	127	fveq2d 6909	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) ∧ 𝑛 ∈ 𝐼) → (𝑇‘(𝑓‘𝑛)) = (𝑇‘(𝑀( decompPMat ↾ (𝐵 × 𝐼))𝑛)))
129	21	ad2antrr 739	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) → 𝑀 ∈ 𝐵)
130		ovres 7607	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑛 ∈ 𝐼) → (𝑀( decompPMat ↾ (𝐵 × 𝐼))𝑛) = (𝑀 decompPMat 𝑛))
131	129, 130	sylan 592	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) ∧ 𝑛 ∈ 𝐼) → (𝑀( decompPMat ↾ (𝐵 × 𝐼))𝑛) = (𝑀 decompPMat 𝑛))
132	131	fveq2d 6909	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) ∧ 𝑛 ∈ 𝐼) → (𝑇‘(𝑀( decompPMat ↾ (𝐵 × 𝐼))𝑛)) = (𝑇‘(𝑀 decompPMat 𝑛)))
133	128, 132	eqtrd 2798	. . . . . 6 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) ∧ 𝑛 ∈ 𝐼) → (𝑇‘(𝑓‘𝑛)) = (𝑇‘(𝑀 decompPMat 𝑛)))
134	133	oveq2d 7456	. . . . 5 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) ∧ 𝑛 ∈ 𝐼) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))) = ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))
135	134	mpteq2dva 5201	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) → (𝑛 ∈ 𝐼 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))) = (𝑛 ∈ 𝐼 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))))
136	135	oveq2d 7456	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) → (𝐶 Σg (𝑛 ∈ 𝐼 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))) = (𝐶 Σg (𝑛 ∈ 𝐼 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))))
137	136	eqeq2d 2774	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑓 = (curry (𝑥 ∈ 𝐵, 𝑘 ∈ 𝐼 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))‘𝑀)) → (𝑀 = (𝐶 Σg (𝑛 ∈ 𝐼 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))) ↔ 𝑀 = (𝐶 Σg (𝑛 ∈ 𝐼 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))))))
138	98, 137	rspcedv 3573	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → (𝑀 = (𝐶 Σg (𝑛 ∈ 𝐼 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))) → ∃𝑓 ∈ (𝐷 ↑m 𝐼)𝑀 = (𝐶 Σg (𝑛 ∈ 𝐼 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∃wrex 3090  Vcvv 3454  ⦋csb 3851   ⊆ wss 3900  ∅c0 4281   ↦ cmpt 5189   × cxp 5653  dom cdm 5655   ↾ cres 5657  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440   ∈ cmpo 7442  curry ccur 8307   ↑_m cmap 8872  Fincfn 8998  ℕ₀cn0 12630  Basecbs 17413   ·_𝑠 cvsca 17458   Σ_g cgsu 17637  ._gcmg 19303  mulGrpcmgp 20387  CRingccrg 20487  var₁cv1 22521  Poly₁cpl1 22522  coe₁cco1 22523   Mat cmat 22749   matToPolyMat cmat2pmat 23049   decompPMat cdecpmat 23107

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-of 7714  df-om 7899  df-1st 8022  df-2nd 8023  df-supp 8202  df-cur 8309  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-er 8742  df-map 8874  df-ixp 8951  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-fsupp 9379  df-sup 9458  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-3 12430  df-4 12431  df-5 12432  df-6 12433  df-7 12434  df-8 12435  df-9 12436  df-n0 12631  df-z 12718  df-dec 12839  df-uz 12990  df-fz 13664  df-struct 17351  df-sets 17368  df-slot 17386  df-ndx 17398  df-base 17414  df-ress 17435  df-plusg 17467  df-mulr 17468  df-sca 17470  df-vsca 17471  df-ip 17472  df-tset 17473  df-ple 17474  df-ds 17476  df-hom 17478  df-cco 17479  df-0g 17638  df-prds 17644  df-pws 17646  df-sra 21475  df-rgmod 21476  df-dsmm 22065  df-frlm 22080  df-psr 22244  df-opsr 22248  df-psr1 22525  df-ply1 22527  df-coe1 22528  df-mat 22750  df-decpmat 23108

This theorem is used by:  pmatcollpw3  23129  pmatcollpw3fi  23130