Theorem pmatcollpw2lem 21382

Description: Lemma for pmatcollpw2 21383. (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 1133	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin)
2		mpoexga 7758	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))) ∈ V)
3	1, 1, 2	syl2anc 587	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))) ∈ V)
4	3	ralrimivw 3150	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ ℕ0 (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))) ∈ V)
5		eqid 2798	. . . . . 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 6460	. . . . 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 11891	. . . . 5 ⊢ ℕ0 ∈ V
9	8	a1i 11	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ℕ0 ∈ V)
10		fvexd 6660	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (0g‘𝐶) ∈ V)
11		suppvalfn 7820	. . . 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 1368	. . 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 2798	. . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅)
17	13, 14, 15, 16	pmatcoe1fsupp 21306	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g‘𝑅)))
18		oveq1 7142	. . . . . . . . . . . . . . . . 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 20882	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
22	21	3ad2ant2 1131	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑅 = (Scalar‘𝑃))
23	22	fveq2d 6649	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (0g‘𝑅) = (0g‘(Scalar‘𝑃)))
24		eqidd 2799	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑥 ↑ 𝑋) = (𝑥 ↑ 𝑋))
25	20, 23, 24	oveq123d 7156	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((0g‘𝑅) × (𝑥 ↑ 𝑋)) = ((0g‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝑥 ↑ 𝑋)))
26	25	ad3antrrr 729	. . . . . . . . . . . . . . . . . 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 2804	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑃) = 𝑅)
28	27	ad3antrrr 729	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (Scalar‘𝑃) = 𝑅)
29	28	fveq2d 6649	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (0g‘(Scalar‘𝑃)) = (0g‘𝑅))
30	29	oveq1d 7150	. . . . . . . . . . . . . . . . . 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 1189	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑅 ∈ Ring)
32		pmatcollpw1.x	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑋 = (var1‘𝑅)
33		eqid 2798	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
34		pmatcollpw1.e	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
35		eqid 2798	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Base‘𝑃) = (Base‘𝑃)
36	13, 32, 33, 34, 35	ply1moncl 20900	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ ℕ0) → (𝑥 ↑ 𝑋) ∈ (Base‘𝑃))
37	36	3ad2antl2 1183	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑥 ↑ 𝑋) ∈ (Base‘𝑃))
38	31, 37	jca 515	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑅 ∈ Ring ∧ (𝑥 ↑ 𝑋) ∈ (Base‘𝑃)))
39	38	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁) → (𝑅 ∈ Ring ∧ (𝑥 ↑ 𝑋) ∈ (Base‘𝑃)))
40	39	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑅 ∈ Ring ∧ (𝑥 ↑ 𝑋) ∈ (Base‘𝑃)))
41		eqid 2798	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
42	13, 35, 41, 16	ply10s0 20885	. . . . . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → ((0g‘𝑅) × (𝑥 ↑ 𝑋)) = (0g‘𝑃))
45	18, 44	sylan9eqr 2855	. . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . 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 470	. . . . . . . . . . . . . 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 3146	. . . . . . . . . . . . 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 3144	. . . . . . . . . . 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 3232	. . . . . . . . . 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 1190	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑀 ∈ 𝐵)
54		simpr 488	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0)
55	31, 53, 54	3jca 1125	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑥 ∈ ℕ0))
56	13, 14, 15	decpmate 21371	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑀 decompPMat 𝑥)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝑥))
57	55, 56	sylan 583	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑀 decompPMat 𝑥)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝑥))
58	57	oveq1d 7150	. . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . 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 3163	. . . . . . . . . . . 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 344	. . . . . . . . . . 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 3161	. . . . . . . . . 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 3256	. . . . . . . . 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 2798	. . . . . . . . . . . . 13 ⊢ 𝑁 = 𝑁
66	65	biantrur 534	. . . . . . . . . . . 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 534	. . . . . . . . . . . . . 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 3133	. . . . . . . . . . . 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 344	. . . . . . . . 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 3260	. . . . . . . 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 7205	. . . . . . . . . 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 3128	. . . . . . . 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 3206	. . . . . . 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 2799	. . . . . . . . . . 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 7143	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑥 → (𝑀 decompPMat 𝑛) = (𝑀 decompPMat 𝑥))
82	81	oveqd 7152	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑥 → (𝑖(𝑀 decompPMat 𝑛)𝑗) = (𝑖(𝑀 decompPMat 𝑥)𝑗))
83		oveq1 7142	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑥 → (𝑛 ↑ 𝑋) = (𝑥 ↑ 𝑋))
84	82, 83	oveq12d 7153	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑥 → ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)) = ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 ↑ 𝑋)))
85	84	mpoeq3dv 7212	. . . . . . . . . . . 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 485	. . . . . . . . . . 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 553	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ Fin → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
89	88	3ad2ant1 1130	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
90	89	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
91		mpoexga 7758	. . . . . . . . . . . 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 6752	. . . . . . . . . 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 20877	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
95	94	anim2i 619	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
96	95	3adant3 1129	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
97	96	adantr 484	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
98		eqid 2798	. . . . . . . . . . . 12 ⊢ (0g‘𝑃) = (0g‘𝑃)
99	14, 98	mat0op 21024	. . . . . . . . . . 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 2814	. . . . . . . . 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 344	. . . . . . . 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 3161	. . . . . . 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 3256	. . . . . 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 2991	. . . . . . . 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 3133	. . . . . 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 3210	. . . . 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 13349	. . . 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 2890	. 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 6362	. . 3 ⊢ Fun (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))
115	8	mptex 6963	. . 3 ⊢ (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) ∈ V
116		funisfsupp 8822	. . 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 1462	. 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))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441   class class class wbr 5030   ↦ cmpt 5110  Fun wfun 6318   Fn wfn 6319  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137   supp csupp 7813  Fincfn 8492   finSupp cfsupp 8817   < clt 10664  ℕ₀cn0 11885  Basecbs 16475  Scalarcsca 16560   ·_𝑠 cvsca 16561  0_gc0g 16705  ._gcmg 18216  mulGrpcmgp 19232  Ringcrg 19290  var₁cv1 20805  Poly₁cpl1 20806  coe₁cco1 20807   Mat cmat 21012   decompPMat cdecpmat 21367

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-ot 4534  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-ofr 7390  df-om 7561  df-1st 7671  df-2nd 7672  df-supp 7814  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fsupp 8818  df-sup 8890  df-oi 8958  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12886  df-fzo 13029  df-seq 13365  df-hash 13687  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-hom 16581  df-cco 16582  df-0g 16707  df-gsum 16708  df-prds 16713  df-pws 16715  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-mhm 17948  df-submnd 17949  df-grp 18098  df-minusg 18099  df-sbg 18100  df-mulg 18217  df-subg 18268  df-ghm 18348  df-cntz 18439  df-cmn 18900  df-abl 18901  df-mgp 19233  df-ur 19245  df-ring 19292  df-subrg 19526  df-lmod 19629  df-lss 19697  df-sra 19937  df-rgmod 19938  df-dsmm 20421  df-frlm 20436  df-psr 20594  df-mvr 20595  df-mpl 20596  df-opsr 20598  df-psr1 20809  df-vr1 20810  df-ply1 20811  df-coe1 20812  df-mat 21013  df-decpmat 21368

This theorem is referenced by:  pmatcollpw2  21383