Theorem pmatcoe1fsupp 22867

Description: For a polynomial matrix there is an upper bound for the coefficients of all the polynomials being not 0. (Contributed by AV, 3-Oct-2019.) (Proof shortened by AV, 28-Nov-2019.)

Hypotheses

Ref	Expression
pmatcoe1fsupp.p	⊢ 𝑃 = (Poly1‘𝑅)
pmatcoe1fsupp.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
pmatcoe1fsupp.b	⊢ 𝐵 = (Base‘𝐶)
pmatcoe1fsupp.0	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
pmatcoe1fsupp	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 ))

Distinct variable groups:   𝐵,𝑖,𝑗,𝑠,𝑥   𝑖,𝑀,𝑗,𝑠,𝑥   𝑖,𝑁,𝑗,𝑠,𝑥   𝑅,𝑖,𝑗,𝑠,𝑥   0 ,𝑖,𝑗,𝑠,𝑥

Allowed substitution hints:   𝐶(𝑥, 𝑖, 𝑗, 𝑠)   𝑃(𝑥, 𝑖, 𝑗, 𝑠)

Proof of Theorem pmatcoe1fsupp

Dummy variables 𝑣 𝑢 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 4034	. . . . . 6 ⊢ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 } ⊆ ((Base‘𝑅) ↑m ℕ0)
2	1	a1i 11	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 } ⊆ ((Base‘𝑅) ↑m ℕ0))
3	2	olcd 887	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ⊆ ((Base‘𝑅) ↑m ℕ0) ∨ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 } ⊆ ((Base‘𝑅) ↑m ℕ0)))
4		inss 4201	. . . 4 ⊢ ((∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ⊆ ((Base‘𝑅) ↑m ℕ0) ∨ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 } ⊆ ((Base‘𝑅) ↑m ℕ0)) → (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 }) ⊆ ((Base‘𝑅) ↑m ℕ0))
5	3, 4	syl 18	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 }) ⊆ ((Base‘𝑅) ↑m ℕ0))
6		xpfi 9275	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin)
7	6	anidms 576	. . . . . 6 ⊢ (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ Fin)
8		snfi 9036	. . . . . . . 8 ⊢ {(coe1‘(𝑀‘𝑢))} ∈ Fin
9	8	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑢 ∈ (𝑁 × 𝑁)) → {(coe1‘(𝑀‘𝑢))} ∈ Fin)
10	9	ralrimiva 3157	. . . . . 6 ⊢ (𝑁 ∈ Fin → ∀𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin)
11	7, 10	jca 520	. . . . 5 ⊢ (𝑁 ∈ Fin → ((𝑁 × 𝑁) ∈ Fin ∧ ∀𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin))
12	11	3ad2ant1 1151	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑁 × 𝑁) ∈ Fin ∧ ∀𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin))
13		iunfi 9296	. . . 4 ⊢ (((𝑁 × 𝑁) ∈ Fin ∧ ∀𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin) → ∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin)
14		infi 9226	. . . 4 ⊢ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin → (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 }) ∈ Fin)
15	12, 13, 14	3syl 19	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 }) ∈ Fin)
16		pmatcoe1fsupp.0	. . . . 5 ⊢ 0 = (0g‘𝑅)
17		fvex 6894	. . . . 5 ⊢ (0g‘𝑅) ∈ V
18	16, 17	eqeltri 2859	. . . 4 ⊢ 0 ∈ V
19	18	a1i 11	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 0 ∈ V)
20		elin 3921	. . . . . 6 ⊢ (𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 }) ↔ (𝑤 ∈ ∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∧ 𝑤 ∈ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 }))
21		breq1 5112	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → (𝑣 finSupp 0 ↔ 𝑤 finSupp 0 ))
22	21	elrab 3650	. . . . . . 7 ⊢ (𝑤 ∈ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 } ↔ (𝑤 ∈ ((Base‘𝑅) ↑m ℕ0) ∧ 𝑤 finSupp 0 ))
23	22	simprbi 502	. . . . . 6 ⊢ (𝑤 ∈ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 } → 𝑤 finSupp 0 )
24	20, 23	simplbiim 513	. . . . 5 ⊢ (𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 }) → 𝑤 finSupp 0 )
25	24	rgen 3081	. . . 4 ⊢ ∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 })𝑤 finSupp 0
26	25	a1i 11	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 })𝑤 finSupp 0 )
27		fsuppmapnn0fiub0 14034	. . . 4 ⊢ (((∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 }) ⊆ ((Base‘𝑅) ↑m ℕ0) ∧ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 }) ∈ Fin ∧ 0 ∈ V) → (∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 })𝑤 finSupp 0 → ∃𝑠 ∈ ℕ0 ∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0 (𝑠 < 𝑧 → (𝑤‘𝑧) = 0 )))
28	27	imp 411	. . 3 ⊢ ((((∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 }) ⊆ ((Base‘𝑅) ↑m ℕ0) ∧ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 }) ∈ Fin ∧ 0 ∈ V) ∧ ∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 })𝑤 finSupp 0 ) → ∃𝑠 ∈ ℕ0 ∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0 (𝑠 < 𝑧 → (𝑤‘𝑧) = 0 ))
29	5, 15, 19, 26, 28	syl31anc 1400	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0 (𝑠 < 𝑧 → (𝑤‘𝑧) = 0 ))
30		opelxpi 5698	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ⟨𝑖, 𝑗⟩ ∈ (𝑁 × 𝑁))
31		df-ov 7413	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖𝑀𝑗) = (𝑀‘⟨𝑖, 𝑗⟩)
32	31	fveq2i 6884	. . . . . . . . . . . . . . . . 17 ⊢ (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑀‘⟨𝑖, 𝑗⟩))
33		fvex 6894	. . . . . . . . . . . . . . . . . 18 ⊢ (coe1‘(𝑀‘⟨𝑖, 𝑗⟩)) ∈ V
34	33	snid 4628	. . . . . . . . . . . . . . . . 17 ⊢ (coe1‘(𝑀‘⟨𝑖, 𝑗⟩)) ∈ {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))}
35	32, 34	eqeltri 2859	. . . . . . . . . . . . . . . 16 ⊢ (coe1‘(𝑖𝑀𝑗)) ∈ {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))}
36	35	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (coe1‘(𝑖𝑀𝑗)) ∈ {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))})
37		2fveq3 6886	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = ⟨𝑖, 𝑗⟩ → (coe1‘(𝑀‘𝑢)) = (coe1‘(𝑀‘⟨𝑖, 𝑗⟩)))
38	37	sneqd 4601	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = ⟨𝑖, 𝑗⟩ → {(coe1‘(𝑀‘𝑢))} = {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))})
39	38	eliuni 4962	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑖, 𝑗⟩ ∈ (𝑁 × 𝑁) ∧ (coe1‘(𝑖𝑀𝑗)) ∈ {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))}) → (coe1‘(𝑖𝑀𝑗)) ∈ ∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))})
40	30, 36, 39	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (coe1‘(𝑖𝑀𝑗)) ∈ ∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))})
41	40	adantl 486	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (coe1‘(𝑖𝑀𝑗)) ∈ ∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))})
42		pmatcoe1fsupp.c	. . . . . . . . . . . . . . . 16 ⊢ 𝐶 = (𝑁 Mat 𝑃)
43		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑃) = (Base‘𝑃)
44		pmatcoe1fsupp.b	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 = (Base‘𝐶)
45		simprl 782	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
46		simprr 784	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁)
47	44	eleq2i 2855	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ 𝐵 ↔ 𝑀 ∈ (Base‘𝐶))
48	47	biimpi 219	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ (Base‘𝐶))
49	48	3ad2ant3 1153	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ (Base‘𝐶))
50	49	ad3antrrr 742	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑀 ∈ (Base‘𝐶))
51	50, 44	eleqtrrdi 2874	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑀 ∈ 𝐵)
52	42, 43, 44, 45, 46, 51	matecld 22592	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝑀𝑗) ∈ (Base‘𝑃))
53		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗))
54		pmatcoe1fsupp.p	. . . . . . . . . . . . . . . 16 ⊢ 𝑃 = (Poly1‘𝑅)
55		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝑅) = (0g‘𝑅)
56		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑅) = (Base‘𝑅)
57	53, 43, 54, 55, 56	coe1fsupp 22383	. . . . . . . . . . . . . . 15 ⊢ ((𝑖𝑀𝑗) ∈ (Base‘𝑃) → (coe1‘(𝑖𝑀𝑗)) ∈ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp (0g‘𝑅)})
58	52, 57	syl 18	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (coe1‘(𝑖𝑀𝑗)) ∈ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp (0g‘𝑅)})
59	16	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 0 = (0g‘𝑅))
60	59	breq2d 5121	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑣 finSupp 0 ↔ 𝑣 finSupp (0g‘𝑅)))
61	60	rabbidv 3423	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 } = {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp (0g‘𝑅)})
62	61	eleq2d 2849	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((coe1‘(𝑖𝑀𝑗)) ∈ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 } ↔ (coe1‘(𝑖𝑀𝑗)) ∈ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp (0g‘𝑅)}))
63	62	ad3antrrr 742	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ((coe1‘(𝑖𝑀𝑗)) ∈ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 } ↔ (coe1‘(𝑖𝑀𝑗)) ∈ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp (0g‘𝑅)}))
64	58, 63	mpbird 260	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (coe1‘(𝑖𝑀𝑗)) ∈ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 })
65	41, 64	elind 4153	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (coe1‘(𝑖𝑀𝑗)) ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 }))
66		simplr 780	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑥 ∈ ℕ0)
67		fveq1 6880	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (coe1‘(𝑖𝑀𝑗)) → (𝑤‘𝑧) = ((coe1‘(𝑖𝑀𝑗))‘𝑧))
68	67	eqeq1d 2765	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (coe1‘(𝑖𝑀𝑗)) → ((𝑤‘𝑧) = 0 ↔ ((coe1‘(𝑖𝑀𝑗))‘𝑧) = 0 ))
69	68	imbi2d 343	. . . . . . . . . . . . 13 ⊢ (𝑤 = (coe1‘(𝑖𝑀𝑗)) → ((𝑠 < 𝑧 → (𝑤‘𝑧) = 0 ) ↔ (𝑠 < 𝑧 → ((coe1‘(𝑖𝑀𝑗))‘𝑧) = 0 )))
70		breq2 5113	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑠 < 𝑧 ↔ 𝑠 < 𝑥))
71		fveqeq2 6890	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (((coe1‘(𝑖𝑀𝑗))‘𝑧) = 0 ↔ ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 ))
72	70, 71	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → ((𝑠 < 𝑧 → ((coe1‘(𝑖𝑀𝑗))‘𝑧) = 0 ) ↔ (𝑠 < 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 )))
73	69, 72	rspc2v 3592	. . . . . . . . . . . 12 ⊢ (((coe1‘(𝑖𝑀𝑗)) ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 }) ∧ 𝑥 ∈ ℕ0) → (∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0 (𝑠 < 𝑧 → (𝑤‘𝑧) = 0 ) → (𝑠 < 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 )))
74	65, 66, 73	syl2anc 595	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0 (𝑠 < 𝑧 → (𝑤‘𝑧) = 0 ) → (𝑠 < 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 )))
75	74	ex 417	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0 (𝑠 < 𝑧 → (𝑤‘𝑧) = 0 ) → (𝑠 < 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 ))))
76	75	com23 87	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0 (𝑠 < 𝑧 → (𝑤‘𝑧) = 0 ) → ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑠 < 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 ))))
77	76	impancom 456	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0 (𝑠 < 𝑧 → (𝑤‘𝑧) = 0 )) → (𝑥 ∈ ℕ0 → ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑠 < 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 ))))
78	77	imp 411	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0 (𝑠 < 𝑧 → (𝑤‘𝑧) = 0 )) ∧ 𝑥 ∈ ℕ0) → ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑠 < 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 )))
79	78	com23 87	. . . . . 6 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0 (𝑠 < 𝑧 → (𝑤‘𝑧) = 0 )) ∧ 𝑥 ∈ ℕ0) → (𝑠 < 𝑥 → ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 )))
80	79	ralrimdvv 3209	. . . . 5 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0 (𝑠 < 𝑧 → (𝑤‘𝑧) = 0 )) ∧ 𝑥 ∈ ℕ0) → (𝑠 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 ))
81	80	ralrimiva 3157	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0 (𝑠 < 𝑧 → (𝑤‘𝑧) = 0 )) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 ))
82	81	ex 417	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) → (∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0 (𝑠 < 𝑧 → (𝑤‘𝑧) = 0 ) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 )))
83	82	reximdva 3178	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∃𝑠 ∈ ℕ0 ∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0 (𝑠 < 𝑧 → (𝑤‘𝑧) = 0 ) → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 )))
84	29, 83	mpd 16	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  {crab 3416  Vcvv 3455   ∩ cin 3904   ⊆ wss 3905  {csn 4589  ⟨cop 4595  ∪ ciun 4956   class class class wbr 5109   × cxp 5659  ‘cfv 6536  (class class class)co 7410   ↑_m cmap 8820  Fincfn 8939   finSupp cfsupp 9317   < clt 11247  ℕ₀cn0 12508  Basecbs 17273  0_gc0g 17496  Ringcrg 20319  Poly₁cpl1 22346  coe₁cco1 22347   Mat cmat 22573

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11160  ax-resscn 11161  ax-1cn 11162  ax-icn 11163  ax-addcl 11164  ax-addrcl 11165  ax-mulcl 11166  ax-mulrcl 11167  ax-mulcom 11168  ax-addass 11169  ax-mulass 11170  ax-distr 11171  ax-i2m1 11172  ax-1ne0 11173  ax-1rid 11174  ax-rnegex 11175  ax-rrecex 11176  ax-cnre 11177  ax-pre-lttri 11178  ax-pre-lttrn 11179  ax-pre-ltadd 11180  ax-pre-mulgt0 11181

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-ot 4598  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-om 7859  df-1st 7982  df-2nd 7983  df-supp 8153  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9318  df-sup 9398  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11447  df-neg 11448  df-nn 12238  df-2 12307  df-3 12308  df-4 12309  df-5 12310  df-6 12311  df-7 12312  df-8 12313  df-9 12314  df-n0 12509  df-z 12596  df-dec 12716  df-uz 12867  df-fz 13540  df-struct 17211  df-sets 17228  df-slot 17246  df-ndx 17258  df-base 17274  df-ress 17295  df-plusg 17327  df-mulr 17328  df-sca 17330  df-vsca 17331  df-ip 17332  df-tset 17333  df-ple 17334  df-ds 17336  df-hom 17338  df-cco 17339  df-0g 17498  df-prds 17504  df-pws 17506  df-sra 21303  df-rgmod 21304  df-dsmm 21891  df-frlm 21906  df-psr 22068  df-mpl 22070  df-opsr 22072  df-psr1 22349  df-ply1 22351  df-coe1 22352  df-mat 22574

This theorem is used by:  decpmataa0  22934  decpmatmulsumfsupp  22939  pmatcollpw2lem  22943  pm2mpmhmlem1  22984