Theorem pmatcollpw2lem 23100

Description: Lemma for pmatcollpw2 23101. (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 8090	. . . . . . 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 3158	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ ℕ0 (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))) ∈ V)
5		eqid 2760	. . . . . 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 6679	. . . . 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 12615	. . . . 5 ⊢ ℕ0 ∈ V
9	8	a1i 11	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ℕ0 ∈ V)
10		fvexd 6900	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (0g‘𝐶) ∈ V)
11		suppvalfn 8188	. . . 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 2760	. . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅)
17	13, 14, 15, 16	pmatcoe1fsupp 23024	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g‘𝑅)))
18		oveq1 7427	. . . . . . . . . . . . . . . . 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 22575	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
22	21	3ad2ant2 1152	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑅 = (Scalar‘𝑃))
23	22	fveq2d 6889	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (0g‘𝑅) = (0g‘(Scalar‘𝑃)))
24		eqidd 2761	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑥 ↑ 𝑋) = (𝑥 ↑ 𝑋))
25	20, 23, 24	oveq123d 7441	. . . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . . . . . 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 6889	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (0g‘(Scalar‘𝑃)) = (0g‘𝑅))
30	29	oveq1d 7435	. . . . . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
34		pmatcollpw1.e	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
35		eqid 2760	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Base‘𝑃) = (Base‘𝑃)
36	13, 32, 33, 34, 35	ply1moncl 22595	. . . . . . . . . . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
42	13, 35, 41, 16	ply10s0 22580	. . . . . . . . . . . . . . . . . . 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 2799	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → ((0g‘𝑅) × (𝑥 ↑ 𝑋)) = (0g‘𝑃))
45	18, 44	sylan9eqr 2817	. . . . . . . . . . . . . . . 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 3209	. . . . . . . . . . . . 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 3174	. . . . . . . . . . 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 3177	. . . . . . . . . 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 23089	. . . . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . . 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 2762	. . . . . . . . . . . . 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 3224	. . . . . . . . . . . 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 3183	. . . . . . . . . 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 3186	. . . . . . . . 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 2760	. . . . . . . . . . . . 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 3108	. . . . . . . . . . . 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 3228	. . . . . . . 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 7492	. . . . . . . . . 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 3099	. . . . . . . 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 3100	. . . . . . 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 2761	. . . . . . . . . . 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 7428	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑥 → (𝑀 decompPMat 𝑛) = (𝑀 decompPMat 𝑥))
82	81	oveqd 7437	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑥 → (𝑖(𝑀 decompPMat 𝑛)𝑗) = (𝑖(𝑀 decompPMat 𝑥)𝑗))
83		oveq1 7427	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑥 → (𝑛 ↑ 𝑋) = (𝑥 ↑ 𝑋))
84	82, 83	oveq12d 7438	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑥 → ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)) = ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)))
85	84	mpoeq3dv 7499	. . . . . . . . . . . 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 8090	. . . . . . . . . . . 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 7001	. . . . . . . . . 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 22570	. . . . . . . . . . . . . 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 2760	. . . . . . . . . . . 12 ⊢ (0g‘𝑃) = (0g‘𝑃)
99	14, 98	mat0op 22739	. . . . . . . . . . 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 2776	. . . . . . . . 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 3183	. . . . . . 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 3186	. . . . . 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 2959	. . . . . . . 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 3108	. . . . . 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 3109	. . . . 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 14132	. . . 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 2860	. 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 6578	. . 3 ⊢ Fun (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))
115	8	mptex 7229	. . 3 ⊢ (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) ∈ V
116		funisfsupp 9362	. . 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 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   class class class wbr 5102   ↦ cmpt 5185  Fun wfun 6532   Fn wfn 6533  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422   supp csupp 8179  Fincfn 8976   finSupp cfsupp 9356   < clt 11346  ℕ₀cn0 12609  Basecbs 17392  Scalarcsca 17436   ·_𝑠 cvsca 17437  0_gc0g 17615  ._gcmg 19282  mulGrpcmgp 20365  Ringcrg 20464  var₁cv1 22499  Poly₁cpl1 22500  coe₁cco1 22501   Mat cmat 22727   decompPMat cdecpmat 23085

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-ofr 7694  df-om 7878  df-1st 8001  df-2nd 8002  df-supp 8180  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-2o 8480  df-er 8720  df-map 8852  df-pm 8853  df-ixp 8929  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-fsupp 9357  df-sup 9437  df-oi 9507  df-card 10023  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-nn 12339  df-2 12408  df-3 12409  df-4 12410  df-5 12411  df-6 12412  df-7 12413  df-8 12414  df-9 12415  df-n0 12610  df-z 12697  df-dec 12818  df-uz 12969  df-fz 13643  df-fzo 13792  df-seq 14148  df-hash 14478  df-struct 17330  df-sets 17347  df-slot 17365  df-ndx 17377  df-base 17393  df-ress 17414  df-plusg 17446  df-mulr 17447  df-sca 17449  df-vsca 17450  df-ip 17451  df-tset 17452  df-ple 17453  df-ds 17455  df-hom 17457  df-cco 17458  df-0g 17617  df-gsum 17618  df-prds 17623  df-pws 17625  df-mre 17761  df-mrc 17762  df-acs 17764  df-mgm 18821  df-sgrp 18913  df-mnd 18929  df-mhm 18983  df-submnd 18984  df-grp 19152  df-minusg 19153  df-sbg 19154  df-mulg 19283  df-subg 19338  df-ghm 19433  df-cntz 19536  df-cmn 20001  df-abl 20002  df-mgp 20366  df-rng 20380  df-ur 20413  df-ring 20466  df-subrng 20803  df-subrg 20827  df-lmod 21142  df-lss 21212  df-sra 21453  df-rgmod 21454  df-dsmm 22043  df-frlm 22058  df-psr 22222  df-mvr 22223  df-mpl 22224  df-opsr 22226  df-psr1 22503  df-vr1 22504  df-ply1 22505  df-coe1 22506  df-mat 22728  df-decpmat 23086

This theorem is used by:  pmatcollpw2  23101