Theorem pmatcoe1fsupp 22691

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 4018	. . . . . 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 880	. . . 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 4183	. . . 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 17	. . 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 9227	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin)
7	6	anidms 571	. . . . . 6 ⊢ (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ Fin)
8		snfi 8987	. . . . . . . 8 ⊢ {(coe1‘(𝑀‘𝑢))} ∈ Fin
9	8	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑢 ∈ (𝑁 × 𝑁)) → {(coe1‘(𝑀‘𝑢))} ∈ Fin)
10	9	ralrimiva 3132	. . . . . 6 ⊢ (𝑁 ∈ Fin → ∀𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin)
11	7, 10	jca 516	. . . . 5 ⊢ (𝑁 ∈ Fin → ((𝑁 × 𝑁) ∈ Fin ∧ ∀𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin))
12	11	3ad2ant1 1139	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑁 × 𝑁) ∈ Fin ∧ ∀𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin))
13		iunfi 9250	. . . 4 ⊢ (((𝑁 × 𝑁) ∈ Fin ∧ ∀𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin) → ∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin)
14		infi 9177	. . . 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 18	. . 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 6847	. . . . 5 ⊢ (0g‘𝑅) ∈ V
18	16, 17	eqeltri 2836	. . . 4 ⊢ 0 ∈ V
19	18	a1i 11	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 0 ∈ V)
20		elin 3906	. . . . . 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 5082	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → (𝑣 finSupp 0 ↔ 𝑤 finSupp 0 ))
22	21	elrab 3636	. . . . . . 7 ⊢ (𝑤 ∈ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 } ↔ (𝑤 ∈ ((Base‘𝑅) ↑m ℕ0) ∧ 𝑤 finSupp 0 ))
23	22	simprbi 498	. . . . . 6 ⊢ (𝑤 ∈ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 } → 𝑤 finSupp 0 )
24	20, 23	simplbiim 509	. . . . 5 ⊢ (𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 }) → 𝑤 finSupp 0 )
25	24	rgen 3056	. . . 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 13953	. . . 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 407	. . 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 1381	. 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 5662	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ⟨𝑖, 𝑗⟩ ∈ (𝑁 × 𝑁))
31		df-ov 7366	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖𝑀𝑗) = (𝑀‘⟨𝑖, 𝑗⟩)
32	31	fveq2i 6837	. . . . . . . . . . . . . . . . 17 ⊢ (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑀‘⟨𝑖, 𝑗⟩))
33		fvex 6847	. . . . . . . . . . . . . . . . . 18 ⊢ (coe1‘(𝑀‘⟨𝑖, 𝑗⟩)) ∈ V
34	33	snid 4601	. . . . . . . . . . . . . . . . 17 ⊢ (coe1‘(𝑀‘⟨𝑖, 𝑗⟩)) ∈ {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))}
35	32, 34	eqeltri 2836	. . . . . . . . . . . . . . . 16 ⊢ (coe1‘(𝑖𝑀𝑗)) ∈ {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))}
36	35	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (coe1‘(𝑖𝑀𝑗)) ∈ {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))})
37		2fveq3 6839	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = ⟨𝑖, 𝑗⟩ → (coe1‘(𝑀‘𝑢)) = (coe1‘(𝑀‘⟨𝑖, 𝑗⟩)))
38	37	sneqd 4574	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = ⟨𝑖, 𝑗⟩ → {(coe1‘(𝑀‘𝑢))} = {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))})
39	38	eliuni 4934	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑖, 𝑗⟩ ∈ (𝑁 × 𝑁) ∧ (coe1‘(𝑖𝑀𝑗)) ∈ {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))}) → (coe1‘(𝑖𝑀𝑗)) ∈ ∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))})
40	30, 36, 39	syl2anc 590	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (coe1‘(𝑖𝑀𝑗)) ∈ ∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))})
41	40	adantl 482	. . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑃) = (Base‘𝑃)
44		pmatcoe1fsupp.b	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 = (Base‘𝐶)
45		simprl 776	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
46		simprr 778	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁)
47	44	eleq2i 2832	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ 𝐵 ↔ 𝑀 ∈ (Base‘𝐶))
48	47	biimpi 217	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ (Base‘𝐶))
49	48	3ad2ant3 1141	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ (Base‘𝐶))
50	49	ad3antrrr 736	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑀 ∈ (Base‘𝐶))
51	50, 44	eleqtrrdi 2851	. . . . . . . . . . . . . . . 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 22416	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝑀𝑗) ∈ (Base‘𝑃))
53		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗))
54		pmatcoe1fsupp.p	. . . . . . . . . . . . . . . 16 ⊢ 𝑃 = (Poly1‘𝑅)
55		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝑅) = (0g‘𝑅)
56		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑅) = (Base‘𝑅)
57	53, 43, 54, 55, 56	coe1fsupp 22206	. . . . . . . . . . . . . . 15 ⊢ ((𝑖𝑀𝑗) ∈ (Base‘𝑃) → (coe1‘(𝑖𝑀𝑗)) ∈ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp (0g‘𝑅)})
58	52, 57	syl 17	. . . . . . . . . . . . . 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 5091	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑣 finSupp 0 ↔ 𝑣 finSupp (0g‘𝑅)))
61	60	rabbidv 3399	. . . . . . . . . . . . . . . 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 2826	. . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . 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 258	. . . . . . . . . . . . 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 4136	. . . . . . . . . . . 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 774	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑥 ∈ ℕ0)
67		fveq1 6833	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (coe1‘(𝑖𝑀𝑗)) → (𝑤‘𝑧) = ((coe1‘(𝑖𝑀𝑗))‘𝑧))
68	67	eqeq1d 2742	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (coe1‘(𝑖𝑀𝑗)) → ((𝑤‘𝑧) = 0 ↔ ((coe1‘(𝑖𝑀𝑗))‘𝑧) = 0 ))
69	68	imbi2d 341	. . . . . . . . . . . . 13 ⊢ (𝑤 = (coe1‘(𝑖𝑀𝑗)) → ((𝑠 < 𝑧 → (𝑤‘𝑧) = 0 ) ↔ (𝑠 < 𝑧 → ((coe1‘(𝑖𝑀𝑗))‘𝑧) = 0 )))
70		breq2 5083	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑠 < 𝑧 ↔ 𝑠 < 𝑥))
71		fveqeq2 6843	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (((coe1‘(𝑖𝑀𝑗))‘𝑧) = 0 ↔ ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 ))
72	70, 71	imbi12d 345	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → ((𝑠 < 𝑧 → ((coe1‘(𝑖𝑀𝑗))‘𝑧) = 0 ) ↔ (𝑠 < 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 )))
73	69, 72	rspc2v 3578	. . . . . . . . . . . 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 590	. . . . . . . . . . 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 413	. . . . . . . . . 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 86	. . . . . . . . 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 452	. . . . . . . 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 407	. . . . . . 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 86	. . . . . 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 3184	. . . . 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 3132	. . . 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 413	. . 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 3153	. 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 15	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 ))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064  {crab 3392  Vcvv 3432   ∩ cin 3889   ⊆ wss 3890  {csn 4562  ⟨cop 4568  ∪ ciun 4928   class class class wbr 5079   × cxp 5623  ‘cfv 6492  (class class class)co 7363   ↑_m cmap 8770  Fincfn 8890   finSupp cfsupp 9271   < clt 11177  ℕ₀cn0 12435  Basecbs 17177  0_gc0g 17400  Ringcrg 20212  Poly₁cpl1 22169  coe₁cco1 22170   Mat cmat 22397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-ot 4571  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  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-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-of 7627  df-om 7814  df-1st 7938  df-2nd 7939  df-supp 8108  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-er 8640  df-map 8772  df-ixp 8843  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fsupp 9272  df-sup 9352  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-3 12243  df-4 12244  df-5 12245  df-6 12246  df-7 12247  df-8 12248  df-9 12249  df-n0 12436  df-z 12523  df-dec 12643  df-uz 12787  df-fz 13460  df-struct 17115  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17178  df-ress 17199  df-plusg 17231  df-mulr 17232  df-sca 17234  df-vsca 17235  df-ip 17236  df-tset 17237  df-ple 17238  df-ds 17240  df-hom 17242  df-cco 17243  df-0g 17402  df-prds 17408  df-pws 17410  df-sra 21170  df-rgmod 21171  df-dsmm 21714  df-frlm 21729  df-psr 21891  df-mpl 21893  df-opsr 21895  df-psr1 22172  df-ply1 22174  df-coe1 22175  df-mat 22398

This theorem is referenced by:  decpmataa0  22758  decpmatmulsumfsupp  22763  pmatcollpw2lem  22767  pm2mpmhmlem1  22808