Theorem pmatcollpw2lem 22971

Description: Lemma for pmatcollpw2 22972. (Contributed by AV, 3-Oct-2019.) (Revised by AV, 3-Dec-2019.)

Hypotheses

Ref	Expression
pmatcollpw1.p	⊢ 𝑃 = (Poly1‘𝑅)
pmatcollpw1.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
pmatcollpw1.b	⊢ 𝐵 = (Base‘𝐶)
pmatcollpw1.m	⊢ × = ( ·𝑠 ‘𝑃)
pmatcollpw1.e	⊢ ↑ = (.g‘(mulGrp‘𝑃))
pmatcollpw1.x	⊢ 𝑋 = (var1‘𝑅)

Assertion

Ref	Expression
pmatcollpw2lem	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) finSupp (0g‘𝐶))

Distinct variable groups:   𝐵,𝑛   𝑛,𝑀   𝑛,𝑁   𝑅,𝑛   𝑛,𝑋   × ,𝑛   ↑ ,𝑛   𝑃,𝑛   𝐵,𝑖,𝑗   𝑖,𝑀,𝑗   𝑖,𝑁,𝑗   𝑃,𝑖,𝑗,𝑛   𝑅,𝑖,𝑗   𝑖,𝑋,𝑗   × ,𝑖,𝑗   ↑ ,𝑖,𝑗

Allowed substitution hints:   𝐶(𝑖, 𝑗, 𝑛)

Proof of Theorem pmatcollpw2lem

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1154	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin)
2		mpoexga 8083	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))) ∈ V)
3	1, 1, 2	syl2anc 596	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))) ∈ V)
4	3	ralrimivw 3164	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ ℕ0 (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))) ∈ V)
5		eqid 2766	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) = (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))
6	5	fnmpt 6682	. . . . 5 ⊢ (∀𝑛 ∈ ℕ0 (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))) ∈ V → (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) Fn ℕ0)
7	4, 6	syl 18	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) Fn ℕ0)
8		nn0ex 12528	. . . . 5 ⊢ ℕ0 ∈ V
9	8	a1i 11	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ℕ0 ∈ V)
10		fvexd 6903	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (0g‘𝐶) ∈ V)
11		suppvalfn 8173	. . . 4 ⊢ (((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) Fn ℕ0 ∧ ℕ0 ∈ V ∧ (0g‘𝐶) ∈ V) → ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) supp (0g‘𝐶)) = {𝑥 ∈ ℕ0 ∣ ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))‘𝑥) ≠ (0g‘𝐶)})
12	7, 9, 10, 11	syl3anc 1398	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) supp (0g‘𝐶)) = {𝑥 ∈ ℕ0 ∣ ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))‘𝑥) ≠ (0g‘𝐶)})
13		pmatcollpw1.p	. . . . . . . . . . 11 ⊢ 𝑃 = (Poly1‘𝑅)
14		pmatcollpw1.c	. . . . . . . . . . 11 ⊢ 𝐶 = (𝑁 Mat 𝑃)
15		pmatcollpw1.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐶)
16		eqid 2766	. . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅)
17	13, 14, 15, 16	pmatcoe1fsupp 22895	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g‘𝑅)))
18		oveq1 7430	. . . . . . . . . . . . . . . . 17 ⊢ (((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g‘𝑅) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 ↑ 𝑋)) = ((0g‘𝑅) × (𝑥 ↑ 𝑋)))
19		pmatcollpw1.m	. . . . . . . . . . . . . . . . . . . . 21 ⊢ × = ( ·𝑠 ‘𝑃)
20	19	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → × = ( ·𝑠 ‘𝑃))
21	13	ply1sca 22449	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
22	21	3ad2ant2 1152	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑅 = (Scalar‘𝑃))
23	22	fveq2d 6892	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (0g‘𝑅) = (0g‘(Scalar‘𝑃)))
24		eqidd 2767	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑥 ↑ 𝑋) = (𝑥 ↑ 𝑋))
25	20, 23, 24	oveq123d 7444	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((0g‘𝑅) × (𝑥 ↑ 𝑋)) = ((0g‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝑥 ↑ 𝑋)))
26	25	ad3antrrr 743	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → ((0g‘𝑅) × (𝑥 ↑ 𝑋)) = ((0g‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝑥 ↑ 𝑋)))
27	22	eqcomd 2772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑃) = 𝑅)
28	27	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (Scalar‘𝑃) = 𝑅)
29	28	fveq2d 6892	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (0g‘(Scalar‘𝑃)) = (0g‘𝑅))
30	29	oveq1d 7438	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → ((0g‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝑥 ↑ 𝑋)) = ((0g‘𝑅)( ·𝑠 ‘𝑃)(𝑥 ↑ 𝑋)))
31		simpl2 1211	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑅 ∈ Ring)
32		pmatcollpw1.x	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑋 = (var1‘𝑅)
33		eqid 2766	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
34		pmatcollpw1.e	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
35		eqid 2766	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Base‘𝑃) = (Base‘𝑃)
36	13, 32, 33, 34, 35	ply1moncl 22469	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ ℕ0) → (𝑥 ↑ 𝑋) ∈ (Base‘𝑃))
37	36	3ad2antl2 1205	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑥 ↑ 𝑋) ∈ (Base‘𝑃))
38	31, 37	jca 521	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑅 ∈ Ring ∧ (𝑥 ↑ 𝑋) ∈ (Base‘𝑃)))
39	38	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁) → (𝑅 ∈ Ring ∧ (𝑥 ↑ 𝑋) ∈ (Base‘𝑃)))
40	39	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑅 ∈ Ring ∧ (𝑥 ↑ 𝑋) ∈ (Base‘𝑃)))
41		eqid 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
42	13, 35, 41, 16	ply10s0 22454	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ↑ 𝑋) ∈ (Base‘𝑃)) → ((0g‘𝑅)( ·𝑠 ‘𝑃)(𝑥 ↑ 𝑋)) = (0g‘𝑃))
43	40, 42	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → ((0g‘𝑅)( ·𝑠 ‘𝑃)(𝑥 ↑ 𝑋)) = (0g‘𝑃))
44	26, 30, 43	3eqtrd 2805	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → ((0g‘𝑅) × (𝑥 ↑ 𝑋)) = (0g‘𝑃))
45	18, 44	sylan9eqr 2823	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) ∧ ((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g‘𝑅)) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 ↑ 𝑋)) = (0g‘𝑃))
46	45	ex 418	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g‘𝑅) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 ↑ 𝑋)) = (0g‘𝑃)))
47	46	anasss 472	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g‘𝑅) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 ↑ 𝑋)) = (0g‘𝑃)))
48	47	ralimdvva 3215	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g‘𝑅) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 ↑ 𝑋)) = (0g‘𝑃)))
49	48	imim2d 58	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → ((𝑦 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g‘𝑅)) → (𝑦 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 ↑ 𝑋)) = (0g‘𝑃))))
50	49	ralimdva 3180	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g‘𝑅)) → ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 ↑ 𝑋)) = (0g‘𝑃))))
51	50	reximdv 3183	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g‘𝑅)) → ∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 ↑ 𝑋)) = (0g‘𝑃))))
52	17, 51	mpd 16	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 ↑ 𝑋)) = (0g‘𝑃)))
53		simpl3 1212	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑀 ∈ 𝐵)
54		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0)
55	31, 53, 54	3jca 1146	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑥 ∈ ℕ0))
56	13, 14, 15	decpmate 22960	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑀 decompPMat 𝑥)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝑥))
57	55, 56	sylan 592	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑀 decompPMat 𝑥)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝑥))
58	57	oveq1d 7438	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 ↑ 𝑋)))
59	58	eqeq1d 2768	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃) ↔ (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 ↑ 𝑋)) = (0g‘𝑃)))
60	59	2ralbidva 3230	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 ↑ 𝑋)) = (0g‘𝑃)))
61	60	imbi2d 343	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → ((𝑦 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃)) ↔ (𝑦 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 ↑ 𝑋)) = (0g‘𝑃))))
62	61	ralbidva 3189	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃)) ↔ ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 ↑ 𝑋)) = (0g‘𝑃))))
63	62	rexbidv 3192	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃)) ↔ ∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 ↑ 𝑋)) = (0g‘𝑃))))
64	52, 63	mpbird 260	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃)))
65		eqid 2766	. . . . . . . . . . . . 13 ⊢ 𝑁 = 𝑁
66	65	biantrur 540	. . . . . . . . . . . 12 ⊢ (∀𝑖 ∈ 𝑁 (𝑁 = 𝑁 ∧ ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃)) ↔ (𝑁 = 𝑁 ∧ ∀𝑖 ∈ 𝑁 (𝑁 = 𝑁 ∧ ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃))))
67	65	biantrur 540	. . . . . . . . . . . . . 14 ⊢ (∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃) ↔ (𝑁 = 𝑁 ∧ ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃)))
68	67	bicomi 227	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 𝑁 ∧ ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃)) ↔ ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃))
69	68	ralbii 3114	. . . . . . . . . . . 12 ⊢ (∀𝑖 ∈ 𝑁 (𝑁 = 𝑁 ∧ ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃)) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃))
70	66, 69	bitr3i 280	. . . . . . . . . . 11 ⊢ ((𝑁 = 𝑁 ∧ ∀𝑖 ∈ 𝑁 (𝑁 = 𝑁 ∧ ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃))) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃))
71	70	a1i 11	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑁 = 𝑁 ∧ ∀𝑖 ∈ 𝑁 (𝑁 = 𝑁 ∧ ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃))) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃)))
72	71	imbi2d 343	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑦 < 𝑥 → (𝑁 = 𝑁 ∧ ∀𝑖 ∈ 𝑁 (𝑁 = 𝑁 ∧ ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃)))) ↔ (𝑦 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃))))
73	72	rexralbidv 3234	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → (𝑁 = 𝑁 ∧ ∀𝑖 ∈ 𝑁 (𝑁 = 𝑁 ∧ ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃)))) ↔ ∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃))))
74	64, 73	mpbird 260	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → (𝑁 = 𝑁 ∧ ∀𝑖 ∈ 𝑁 (𝑁 = 𝑁 ∧ ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃)))))
75		mpoeq123 7495	. . . . . . . . . 10 ⊢ ((𝑁 = 𝑁 ∧ ∀𝑖 ∈ 𝑁 (𝑁 = 𝑁 ∧ ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃))) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑃)))
76	75	imim2i 17	. . . . . . . . 9 ⊢ ((𝑦 < 𝑥 → (𝑁 = 𝑁 ∧ ∀𝑖 ∈ 𝑁 (𝑁 = 𝑁 ∧ ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃)))) → (𝑦 < 𝑥 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑃))))
77	76	ralimi 3105	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → (𝑁 = 𝑁 ∧ ∀𝑖 ∈ 𝑁 (𝑁 = 𝑁 ∧ ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃)))) → ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑃))))
78	77	reximi 3106	. . . . . . 7 ⊢ (∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → (𝑁 = 𝑁 ∧ ∀𝑖 ∈ 𝑁 (𝑁 = 𝑁 ∧ ∀𝑗 ∈ 𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)) = (0g‘𝑃)))) → ∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑃))))
79	74, 78	syl 18	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑃))))
80		eqidd 2767	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) = (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))))
81		oveq2 7431	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑥 → (𝑀 decompPMat 𝑛) = (𝑀 decompPMat 𝑥))
82	81	oveqd 7440	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑥 → (𝑖(𝑀 decompPMat 𝑛)𝑗) = (𝑖(𝑀 decompPMat 𝑥)𝑗))
83		oveq1 7430	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑥 → (𝑛 ↑ 𝑋) = (𝑥 ↑ 𝑋))
84	82, 83	oveq12d 7441	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑥 → ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)) = ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)))
85	84	mpoeq3dv 7502	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑥 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋))))
86	85	adantl 487	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑛 = 𝑥) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋))))
87		id 23	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ Fin → 𝑁 ∈ Fin)
88	87	ancri 559	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ Fin → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
89	88	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
90	89	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
91		mpoexga 8083	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋))) ∈ V)
92	90, 91	syl 18	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋))) ∈ V)
93	80, 86, 54, 92	fvmptd 7004	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))‘𝑥) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋))))
94	13	ply1ring 22444	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
95	94	anim2i 629	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
96	95	3adant3 1150	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
97	96	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
98		eqid 2766	. . . . . . . . . . . 12 ⊢ (0g‘𝑃) = (0g‘𝑃)
99	14, 98	mat0op 22613	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → (0g‘𝐶) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑃)))
100	97, 99	syl 18	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → (0g‘𝐶) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑃)))
101	93, 100	eqeq12d 2782	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → (((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))‘𝑥) = (0g‘𝐶) ↔ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑃))))
102	101	imbi2d 343	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → ((𝑦 < 𝑥 → ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))‘𝑥) = (0g‘𝐶)) ↔ (𝑦 < 𝑥 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑃)))))
103	102	ralbidva 3189	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))‘𝑥) = (0g‘𝐶)) ↔ ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑃)))))
104	103	rexbidv 3192	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))‘𝑥) = (0g‘𝐶)) ↔ ∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑃)))))
105	79, 104	mpbird 260	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))‘𝑥) = (0g‘𝐶)))
106		nne 2965	. . . . . . . 8 ⊢ (¬ ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))‘𝑥) ≠ (0g‘𝐶) ↔ ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))‘𝑥) = (0g‘𝐶))
107	106	imbi2i 339	. . . . . . 7 ⊢ ((𝑦 < 𝑥 → ¬ ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))‘𝑥) ≠ (0g‘𝐶)) ↔ (𝑦 < 𝑥 → ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))‘𝑥) = (0g‘𝐶)))
108	107	ralbii 3114	. . . . . 6 ⊢ (∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ¬ ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))‘𝑥) ≠ (0g‘𝐶)) ↔ ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))‘𝑥) = (0g‘𝐶)))
109	108	rexbii 3115	. . . . 5 ⊢ (∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ¬ ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))‘𝑥) ≠ (0g‘𝐶)) ↔ ∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))‘𝑥) = (0g‘𝐶)))
110	105, 109	sylibr 237	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ¬ ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))‘𝑥) ≠ (0g‘𝐶)))
111		rabssnn0fi 14042	. . . 4 ⊢ ({𝑥 ∈ ℕ0 ∣ ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))‘𝑥) ≠ (0g‘𝐶)} ∈ Fin ↔ ∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ¬ ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))‘𝑥) ≠ (0g‘𝐶)))
112	110, 111	sylibr 237	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → {𝑥 ∈ ℕ0 ∣ ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))‘𝑥) ≠ (0g‘𝐶)} ∈ Fin)
113	12, 112	eqeltrd 2866	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) supp (0g‘𝐶)) ∈ Fin)
114		funmpt 6581	. . 3 ⊢ Fun (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))
115	8	mptex 7228	. . 3 ⊢ (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) ∈ V
116		funisfsupp 9337	. . 3 ⊢ ((Fun (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) ∧ (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) ∈ V ∧ (0g‘𝐶) ∈ V) → ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) finSupp (0g‘𝐶) ↔ ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) supp (0g‘𝐶)) ∈ Fin))
117	114, 115, 10, 116	mp3an12i 1494	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) finSupp (0g‘𝐶) ↔ ((𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) supp (0g‘𝐶)) ∈ Fin))
118	113, 117	mpbird 260	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) finSupp (0g‘𝐶))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2961  ∀wral 3082  ∃wrex 3092  {crab 3419  Vcvv 3458   class class class wbr 5114   ↦ cmpt 5197  Fun wfun 6537   Fn wfn 6538  ‘cfv 6543  (class class class)co 7423   ∈ cmpo 7425   supp csupp 8165  Fincfn 8952   finSupp cfsupp 9331   < clt 11261  ℕ₀cn0 12522  Basecbs 17294  Scalarcsca 17338   ·_𝑠 cvsca 17339  0_gc0g 17517  ._gcmg 19164  mulGrpcmgp 20247  Ringcrg 20346  var₁cv1 22373  Poly₁cpl1 22374  coe₁cco1 22375   Mat cmat 22601   decompPMat cdecpmat 22956

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

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 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-ot 4603  df-uni 4878  df-int 4918  df-iun 4963  df-iin 4964  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  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 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-of 7687  df-ofr 7688  df-om 7872  df-1st 7995  df-2nd 7996  df-supp 8166  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-er 8703  df-map 8835  df-pm 8836  df-ixp 8905  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-fsupp 9332  df-sup 9412  df-oi 9482  df-card 9944  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-6 12325  df-7 12326  df-8 12327  df-9 12328  df-n0 12523  df-z 12610  df-dec 12730  df-uz 12881  df-fz 13554  df-fzo 13702  df-seq 14058  df-hash 14387  df-struct 17232  df-sets 17249  df-slot 17267  df-ndx 17279  df-base 17295  df-ress 17316  df-plusg 17348  df-mulr 17349  df-sca 17351  df-vsca 17352  df-ip 17353  df-tset 17354  df-ple 17355  df-ds 17357  df-hom 17359  df-cco 17360  df-0g 17519  df-gsum 17520  df-prds 17525  df-pws 17527  df-mre 17663  df-mrc 17664  df-acs 17666  df-mgm 18723  df-sgrp 18806  df-mnd 18822  df-mhm 18872  df-submnd 18873  df-grp 19034  df-minusg 19035  df-sbg 19036  df-mulg 19165  df-subg 19220  df-ghm 19315  df-cntz 19418  df-cmn 19883  df-abl 19884  df-mgp 20248  df-rng 20262  df-ur 20295  df-ring 20348  df-subrng 20682  df-subrg 20706  df-lmod 21020  df-lss 21090  df-sra 21331  df-rgmod 21332  df-dsmm 21919  df-frlm 21934  df-psr 22096  df-mvr 22097  df-mpl 22098  df-opsr 22100  df-psr1 22377  df-vr1 22378  df-ply1 22379  df-coe1 22380  df-mat 22602  df-decpmat 22957

This theorem is used by:  pmatcollpw2  22972