Theorem pmatcollpw2lem 22752

Description: Lemma for pmatcollpw2 22753. (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 1137	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin)
2		mpoexga 8023	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))) ∈ V)
3	1, 1, 2	syl2anc 585	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))) ∈ V)
4	3	ralrimivw 3134	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ ℕ0 (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))) ∈ V)
5		eqid 2737	. . . . . 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 6632	. . . . 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 17	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) Fn ℕ0)
8		nn0ex 12434	. . . . 5 ⊢ ℕ0 ∈ V
9	8	a1i 11	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ℕ0 ∈ V)
10		fvexd 6849	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (0g‘𝐶) ∈ V)
11		suppvalfn 8111	. . . 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 1374	. . 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 2737	. . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅)
17	13, 14, 15, 16	pmatcoe1fsupp 22676	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g‘𝑅)))
18		oveq1 7367	. . . . . . . . . . . . . . . . 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 22226	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
22	21	3ad2ant2 1135	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑅 = (Scalar‘𝑃))
23	22	fveq2d 6838	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (0g‘𝑅) = (0g‘(Scalar‘𝑃)))
24		eqidd 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑥 ↑ 𝑋) = (𝑥 ↑ 𝑋))
25	20, 23, 24	oveq123d 7381	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((0g‘𝑅) × (𝑥 ↑ 𝑋)) = ((0g‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝑥 ↑ 𝑋)))
26	25	ad3antrrr 731	. . . . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑃) = 𝑅)
28	27	ad3antrrr 731	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (Scalar‘𝑃) = 𝑅)
29	28	fveq2d 6838	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (0g‘(Scalar‘𝑃)) = (0g‘𝑅))
30	29	oveq1d 7375	. . . . . . . . . . . . . . . . . 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 1194	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑅 ∈ Ring)
32		pmatcollpw1.x	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑋 = (var1‘𝑅)
33		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
34		pmatcollpw1.e	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
35		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Base‘𝑃) = (Base‘𝑃)
36	13, 32, 33, 34, 35	ply1moncl 22246	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ ℕ0) → (𝑥 ↑ 𝑋) ∈ (Base‘𝑃))
37	36	3ad2antl2 1188	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑥 ↑ 𝑋) ∈ (Base‘𝑃))
38	31, 37	jca 511	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑅 ∈ Ring ∧ (𝑥 ↑ 𝑋) ∈ (Base‘𝑃)))
39	38	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁) → (𝑅 ∈ Ring ∧ (𝑥 ↑ 𝑋) ∈ (Base‘𝑃)))
40	39	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑅 ∈ Ring ∧ (𝑥 ↑ 𝑋) ∈ (Base‘𝑃)))
41		eqid 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
42	13, 35, 41, 16	ply10s0 22231	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ↑ 𝑋) ∈ (Base‘𝑃)) → ((0g‘𝑅)( ·𝑠 ‘𝑃)(𝑥 ↑ 𝑋)) = (0g‘𝑃))
43	40, 42	syl 17	. . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → ((0g‘𝑅) × (𝑥 ↑ 𝑋)) = (0g‘𝑃))
45	18, 44	sylan9eqr 2794	. . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . 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 466	. . . . . . . . . . . . . 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 3185	. . . . . . . . . . . . 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 57	. . . . . . . . . . . 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 3150	. . . . . . . . . . 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 3153	. . . . . . . . . 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 15	. . . . . . . . 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 1195	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑀 ∈ 𝐵)
54		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0)
55	31, 53, 54	3jca 1129	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑥 ∈ ℕ0))
56	13, 14, 15	decpmate 22741	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑀 decompPMat 𝑥)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝑥))
57	55, 56	sylan 581	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑀 decompPMat 𝑥)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝑥))
58	57	oveq1d 7375	. . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . 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 3200	. . . . . . . . . . . 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 340	. . . . . . . . . . 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 3159	. . . . . . . . . 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 3162	. . . . . . . . 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 257	. . . . . . . 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 2737	. . . . . . . . . . . . 13 ⊢ 𝑁 = 𝑁
66	65	biantrur 530	. . . . . . . . . . . 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 530	. . . . . . . . . . . . . 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 224	. . . . . . . . . . . . 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 3084	. . . . . . . . . . . 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 277	. . . . . . . . . . 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 340	. . . . . . . . 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 3204	. . . . . . . 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 257	. . . . . . 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 7432	. . . . . . . . . 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 16	. . . . . . . . 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 3075	. . . . . . . 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 3076	. . . . . . 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 17	. . . . . 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 2738	. . . . . . . . . . 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 7368	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑥 → (𝑀 decompPMat 𝑛) = (𝑀 decompPMat 𝑥))
82	81	oveqd 7377	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑥 → (𝑖(𝑀 decompPMat 𝑛)𝑗) = (𝑖(𝑀 decompPMat 𝑥)𝑗))
83		oveq1 7367	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑥 → (𝑛 ↑ 𝑋) = (𝑥 ↑ 𝑋))
84	82, 83	oveq12d 7378	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑥 → ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)) = ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)))
85	84	mpoeq3dv 7439	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 22	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ Fin → 𝑁 ∈ Fin)
88	87	ancri 549	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ Fin → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
89	88	3ad2ant1 1134	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
90	89	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
91		mpoexga 8023	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋))) ∈ V)
92	90, 91	syl 17	. . . . . . . . . . 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 6949	. . . . . . . . . 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 22221	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
95	94	anim2i 618	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
96	95	3adant3 1133	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
97	96	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
98		eqid 2737	. . . . . . . . . . . 12 ⊢ (0g‘𝑃) = (0g‘𝑃)
99	14, 98	mat0op 22394	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → (0g‘𝐶) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑃)))
100	97, 99	syl 17	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → (0g‘𝐶) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑃)))
101	93, 100	eqeq12d 2753	. . . . . . . . 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 340	. . . . . . . 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 3159	. . . . . . 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 3162	. . . . . 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 257	. . . . 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 2937	. . . . . . . 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 336	. . . . . . 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 3084	. . . . . 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 3085	. . . . 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 234	. . . 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 13939	. . . 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 234	. . 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 2837	. 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 6530	. . 3 ⊢ Fun (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))
115	8	mptex 7171	. . 3 ⊢ (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) ∈ V
116		funisfsupp 9273	. . 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 1468	. 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 257	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) finSupp (0g‘𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3390  Vcvv 3430   class class class wbr 5086   ↦ cmpt 5167  Fun wfun 6486   Fn wfn 6487  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362   supp csupp 8103  Fincfn 8886   finSupp cfsupp 9267   < clt 11170  ℕ₀cn0 12428  Basecbs 17170  Scalarcsca 17214   ·_𝑠 cvsca 17215  0_gc0g 17393  ._gcmg 19034  mulGrpcmgp 20112  Ringcrg 20205  var₁cv1 22149  Poly₁cpl1 22150  coe₁cco1 22151   Mat cmat 22382   decompPMat cdecpmat 22737

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

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-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-ofr 7625  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8104  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-er 8636  df-map 8768  df-pm 8769  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9268  df-sup 9348  df-oi 9418  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-fzo 13600  df-seq 13955  df-hash 14284  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-sca 17227  df-vsca 17228  df-ip 17229  df-tset 17230  df-ple 17231  df-ds 17233  df-hom 17235  df-cco 17236  df-0g 17395  df-gsum 17396  df-prds 17401  df-pws 17403  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18742  df-submnd 18743  df-grp 18903  df-minusg 18904  df-sbg 18905  df-mulg 19035  df-subg 19090  df-ghm 19179  df-cntz 19283  df-cmn 19748  df-abl 19749  df-mgp 20113  df-rng 20125  df-ur 20154  df-ring 20207  df-subrng 20514  df-subrg 20538  df-lmod 20848  df-lss 20918  df-sra 21160  df-rgmod 21161  df-dsmm 21722  df-frlm 21737  df-psr 21899  df-mvr 21900  df-mpl 21901  df-opsr 21903  df-psr1 22153  df-vr1 22154  df-ply1 22155  df-coe1 22156  df-mat 22383  df-decpmat 22738

This theorem is referenced by:  pmatcollpw2  22753