Theorem pmatcollpw3lem 22744

Description: Lemma for pmatcollpw3 22745 and pmatcollpw3fi 22746: 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 5862	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → dom 𝑥 = dom 𝑦)
2	1	dmeqd 5864	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → dom dom 𝑥 = dom dom 𝑦)
3		oveq 7376	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑖𝑥𝑗) = (𝑖𝑦𝑗))
4	3	fveq2d 6848	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (coe1‘(𝑖𝑥𝑗)) = (coe1‘(𝑖𝑦𝑗)))
5	4	fveq1d 6846	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((coe1‘(𝑖𝑥𝑗))‘𝑘) = ((coe1‘(𝑖𝑦𝑗))‘𝑘))
6	2, 2, 5	mpoeq123dv 7445	. . . . . . 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 6844	. . . . . . . 8 ⊢ (𝑘 = 𝑙 → ((coe1‘(𝑖𝑦𝑗))‘𝑘) = ((coe1‘(𝑖𝑦𝑗))‘𝑙))
8	7	mpoeq3dv 7449	. . . . . . 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 7465	. . . . . 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 7855	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 → dom 𝑦 ∈ V)
11	10	dmexd 7857	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 → dom dom 𝑦 ∈ V)
12	11, 11	jca 511	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → (dom dom 𝑦 ∈ V ∧ dom dom 𝑦 ∈ V))
13	12	ad2antrl 729	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑙 ∈ 𝐼)) → (dom dom 𝑦 ∈ V ∧ dom dom 𝑦 ∈ V))
14		mpoexga 8033	. . . . . . . 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 17	. . . . . . 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 3181	. . . . . 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 773	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → 𝐼 ≠ ∅)
18		nn0ex 12421	. . . . . . . 8 ⊢ ℕ0 ∈ V
19	18	ssex 5270	. . . . . . 7 ⊢ (𝐼 ⊆ ℕ0 → 𝐼 ∈ V)
20	19	ad2antrl 729	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → 𝐼 ∈ V)
21		simp3 1139	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵)
22	21	adantr 480	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → 𝑀 ∈ 𝐵)
23	9, 16, 17, 20, 22	mpocurryvald 8224	. . . . 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 6844	. . . . . . . . 9 ⊢ (𝑙 = 𝑘 → ((coe1‘(𝑖𝑦𝑗))‘𝑙) = ((coe1‘(𝑖𝑦𝑗))‘𝑘))
25	24	mpoeq3dv 7449	. . . . . . . 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 2744	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
28		eqcom 2744	. . . . . . . . 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 291	. . . . . . . 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 2788	. . . . . 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 5204	. . . . 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 2788	. . . 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 5862	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑀 → dom 𝑥 = dom 𝑀)
35	34	dmeqd 5864	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → dom dom 𝑥 = dom dom 𝑀)
36		oveq 7376	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑀 → (𝑖𝑥𝑗) = (𝑖𝑀𝑗))
37	36	fveq2d 6848	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑀 → (coe1‘(𝑖𝑥𝑗)) = (coe1‘(𝑖𝑀𝑗)))
38	37	fveq1d 6846	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → ((coe1‘(𝑖𝑥𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝑘))
39	35, 35, 38	mpoeq123dv 7445	. . . . . . . . 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 481	. . . . . . . 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 2737	. . . . . . . . . . . . 13 ⊢ (Base‘𝑃) = (Base‘𝑃)
44		pmatcollpw.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝐶)
45	42, 43, 44	matbas2i 22383	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ ((Base‘𝑃) ↑m (𝑁 × 𝑁)))
46		elmapi 8800	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ((Base‘𝑃) ↑m (𝑁 × 𝑁)) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑃))
47		fdm 6681	. . . . . . . . . . . . . 14 ⊢ (𝑀:(𝑁 × 𝑁)⟶(Base‘𝑃) → dom 𝑀 = (𝑁 × 𝑁))
48	47	dmeqd 5864	. . . . . . . . . . . . 13 ⊢ (𝑀:(𝑁 × 𝑁)⟶(Base‘𝑃) → dom dom 𝑀 = dom (𝑁 × 𝑁))
49		dmxpid 5889	. . . . . . . . . . . . 13 ⊢ dom (𝑁 × 𝑁) = 𝑁
50	48, 49	eqtr2di 2789	. . . . . . . . . . . 12 ⊢ (𝑀:(𝑁 × 𝑁)⟶(Base‘𝑃) → 𝑁 = dom dom 𝑀)
51	45, 46, 50	3syl 18	. . . . . . . . . . 11 ⊢ (𝑀 ∈ 𝐵 → 𝑁 = dom dom 𝑀)
52	51	3ad2ant3 1136	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑁 = dom dom 𝑀)
53	52	adantr 480	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → 𝑁 = dom dom 𝑀)
54		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
55	54	oveqd 7387	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
56	55	fveq2d 6848	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
57	56	fveq1d 6846	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → ((coe1‘(𝑖𝑚𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝑘))
58	53, 53, 57	mpoeq123dv 7445	. . . . . . . 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 2775	. . . . . 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 480	. . . . 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 5194	. . . 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 2772	. . 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 7376	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
65	64	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
66	65	fveq2d 6848	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
67	66	fveq1d 6846	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → ((coe1‘(𝑖𝑚𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝑘))
68	67	mpoeq3dv 7449	. . . . . . . 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 727	. . . . . 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 2737	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
73		pmatcollpw3.d	. . . . . . 7 ⊢ 𝐷 = (Base‘𝐴)
74		simpll1 1214	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑘 ∈ 𝐼) → 𝑁 ∈ Fin)
75		simpll2 1215	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑘 ∈ 𝐼) → 𝑅 ∈ CRing)
76		simp2 1138	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
77		simp3 1139	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
78	22	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑘 ∈ 𝐼) → 𝑀 ∈ 𝐵)
79	78	3ad2ant1 1134	. . . . . . . . 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 22387	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑀𝑗) ∈ (Base‘𝑃))
81		ssel 3929	. . . . . . . . . . 11 ⊢ (𝐼 ⊆ ℕ0 → (𝑘 ∈ 𝐼 → 𝑘 ∈ ℕ0))
82	81	ad2antrl 729	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → (𝑘 ∈ 𝐼 → 𝑘 ∈ ℕ0))
83	82	imp 406	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑘 ∈ 𝐼) → 𝑘 ∈ ℕ0)
84	83	3ad2ant1 1134	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑘 ∈ ℕ0)
85		eqid 2737	. . . . . . . . 9 ⊢ (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗))
86		pmatcollpw.p	. . . . . . . . 9 ⊢ 𝑃 = (Poly1‘𝑅)
87	85, 43, 86, 72	coe1fvalcl 22170	. . . . . . . 8 ⊢ (((𝑖𝑀𝑗) ∈ (Base‘𝑃) ∧ 𝑘 ∈ ℕ0) → ((coe1‘(𝑖𝑀𝑗))‘𝑘) ∈ (Base‘𝑅))
88	80, 84, 87	syl2anc 585	. . . . . . 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 22384	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑘 ∈ 𝐼) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝑘)) ∈ 𝐷)
90	70, 89	eqeltrd 2837	. . . . 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 7071	. . . 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 6858	. . . . . 6 ⊢ 𝐷 ∈ V
93	92	a1i 11	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐷 ∈ V)
94	19	adantr 480	. . . . 5 ⊢ ((𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅) → 𝐼 ∈ V)
95		elmapg 8790	. . . . 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 597	. . . 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 257	. . 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 2837	. 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 6843	. . . . . . . . . . 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 481	. . . . . . . . . 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 480	. . . . . . . . 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 2737	. . . . . . . . . . . 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 7855	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝐵 → dom 𝑥 ∈ V)
104	103	dmexd 7857	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐵 → dom dom 𝑥 ∈ V)
105	104, 104	jca 511	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐵 → (dom dom 𝑥 ∈ V ∧ dom dom 𝑥 ∈ V))
106	105	ad2antrl 729	. . . . . . . . . . . . . 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 8033	. . . . . . . . . . . . . 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 17	. . . . . . . . . . . . 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 3181	. . . . . . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑛 ∈ 𝐼) → 𝐼 ∈ V)
111	22	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑛 ∈ 𝐼) → 𝑀 ∈ 𝐵)
112		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑛 ∈ 𝐼) → 𝑛 ∈ 𝐼)
113	102, 109, 110, 111, 112	fvmpocurryd 8225	. . . . . . . . . . 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 22724	. . . . . . . . . . . . . 14 ⊢ decompPMat = (𝑥 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘)))
115	114	reseq1i 5944	. . . . . . . . . . . . 13 ⊢ ( decompPMat ↾ (𝐵 × 𝐼)) = ((𝑥 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑥, 𝑗 ∈ dom dom 𝑥 ↦ ((coe1‘(𝑖𝑥𝑗))‘𝑘))) ↾ (𝐵 × 𝐼))
116		ssv 3960	. . . . . . . . . . . . . . . . 17 ⊢ 𝐵 ⊆ V
117	116	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐵 ⊆ V)
118		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅) → 𝐼 ⊆ ℕ0)
119	117, 118	anim12i 614	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → (𝐵 ⊆ V ∧ 𝐼 ⊆ ℕ0))
120	119	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) ∧ 𝑛 ∈ 𝐼) → (𝐵 ⊆ V ∧ 𝐼 ⊆ ℕ0))
121		resmpo 7490	. . . . . . . . . . . . . 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 17	. . . . . . . . . . . . 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 2785	. . . . . . . . . . . 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 7387	. . . . . . . . . . 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 2772	. . . . . . . . . 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 716	. . . . . . . . 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 2772	. . . . . . . 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 6848	. . . . . . 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 727	. . . . . . . . 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 7536	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑛 ∈ 𝐼) → (𝑀( decompPMat ↾ (𝐵 × 𝐼))𝑛) = (𝑀 decompPMat 𝑛))
131	129, 130	sylan 581	. . . . . . . 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 6848	. . . . . . 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 2772	. . . . . 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 7386	. . . . 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 5193	. . . 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 7386	. . 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 2748	. 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 3571	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)))))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062  Vcvv 3442  ⦋csb 3851   ⊆ wss 3903  ∅c0 4287   ↦ cmpt 5181   × cxp 5632  dom cdm 5634   ↾ cres 5636  ⟶wf 6498  ‘cfv 6502  (class class class)co 7370   ∈ cmpo 7372  curry ccur 8219   ↑_m cmap 8777  Fincfn 8897  ℕ₀cn0 12415  Basecbs 17150   ·_𝑠 cvsca 17195   Σ_g cgsu 17374  ._gcmg 19014  mulGrpcmgp 20092  CRingccrg 20186  var₁cv1 22133  Poly₁cpl1 22134  coe₁cco1 22135   Mat cmat 22368   matToPolyMat cmat2pmat 22665   decompPMat cdecpmat 22723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-of 7634  df-om 7821  df-1st 7945  df-2nd 7946  df-supp 8115  df-cur 8221  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-er 8647  df-map 8779  df-ixp 8850  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-fsupp 9279  df-sup 9359  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-nn 12160  df-2 12222  df-3 12223  df-4 12224  df-5 12225  df-6 12226  df-7 12227  df-8 12228  df-9 12229  df-n0 12416  df-z 12503  df-dec 12622  df-uz 12766  df-fz 13438  df-struct 17088  df-sets 17105  df-slot 17123  df-ndx 17135  df-base 17151  df-ress 17172  df-plusg 17204  df-mulr 17205  df-sca 17207  df-vsca 17208  df-ip 17209  df-tset 17210  df-ple 17211  df-ds 17213  df-hom 17215  df-cco 17216  df-0g 17375  df-prds 17381  df-pws 17383  df-sra 21142  df-rgmod 21143  df-dsmm 21704  df-frlm 21719  df-psr 21882  df-opsr 21886  df-psr1 22137  df-ply1 22139  df-coe1 22140  df-mat 22369  df-decpmat 22724

This theorem is referenced by:  pmatcollpw3  22745  pmatcollpw3fi  22746