Theorem pmatcoe1fsupp 21452

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 3969	. . . . . 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 873	. . . 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 4129	. . . 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 8863	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin)
7	6	anidms 570	. . . . . 6 ⊢ (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ Fin)
8		snfi 8642	. . . . . . . 8 ⊢ {(coe1‘(𝑀‘𝑢))} ∈ Fin
9	8	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑢 ∈ (𝑁 × 𝑁)) → {(coe1‘(𝑀‘𝑢))} ∈ Fin)
10	9	ralrimiva 3096	. . . . . 6 ⊢ (𝑁 ∈ Fin → ∀𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin)
11	7, 10	jca 515	. . . . 5 ⊢ (𝑁 ∈ Fin → ((𝑁 × 𝑁) ∈ Fin ∧ ∀𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin))
12	11	3ad2ant1 1134	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑁 × 𝑁) ∈ Fin ∧ ∀𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin))
13		iunfi 8885	. . . 4 ⊢ (((𝑁 × 𝑁) ∈ Fin ∧ ∀𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin) → ∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin)
14		infi 8820	. . . 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 6687	. . . . 5 ⊢ (0g‘𝑅) ∈ V
18	16, 17	eqeltri 2829	. . . 4 ⊢ 0 ∈ V
19	18	a1i 11	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 0 ∈ V)
20		elin 3859	. . . . . 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 5033	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → (𝑣 finSupp 0 ↔ 𝑤 finSupp 0 ))
22	21	elrab 3588	. . . . . . 7 ⊢ (𝑤 ∈ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 } ↔ (𝑤 ∈ ((Base‘𝑅) ↑m ℕ0) ∧ 𝑤 finSupp 0 ))
23	22	simprbi 500	. . . . . 6 ⊢ (𝑤 ∈ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 } → 𝑤 finSupp 0 )
24	20, 23	simplbiim 508	. . . . 5 ⊢ (𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 }) → 𝑤 finSupp 0 )
25	24	rgen 3063	. . . 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 13452	. . . 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 410	. . 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 1374	. 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 5562	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ⟨𝑖, 𝑗⟩ ∈ (𝑁 × 𝑁))
31		df-ov 7173	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖𝑀𝑗) = (𝑀‘⟨𝑖, 𝑗⟩)
32	31	fveq2i 6677	. . . . . . . . . . . . . . . . 17 ⊢ (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑀‘⟨𝑖, 𝑗⟩))
33		fvex 6687	. . . . . . . . . . . . . . . . . 18 ⊢ (coe1‘(𝑀‘⟨𝑖, 𝑗⟩)) ∈ V
34	33	snid 4552	. . . . . . . . . . . . . . . . 17 ⊢ (coe1‘(𝑀‘⟨𝑖, 𝑗⟩)) ∈ {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))}
35	32, 34	eqeltri 2829	. . . . . . . . . . . . . . . 16 ⊢ (coe1‘(𝑖𝑀𝑗)) ∈ {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))}
36	35	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (coe1‘(𝑖𝑀𝑗)) ∈ {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))})
37		2fveq3 6679	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = ⟨𝑖, 𝑗⟩ → (coe1‘(𝑀‘𝑢)) = (coe1‘(𝑀‘⟨𝑖, 𝑗⟩)))
38	37	sneqd 4528	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = ⟨𝑖, 𝑗⟩ → {(coe1‘(𝑀‘𝑢))} = {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))})
39	38	eliuni 4887	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑖, 𝑗⟩ ∈ (𝑁 × 𝑁) ∧ (coe1‘(𝑖𝑀𝑗)) ∈ {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))}) → (coe1‘(𝑖𝑀𝑗)) ∈ ∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))})
40	30, 36, 39	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (coe1‘(𝑖𝑀𝑗)) ∈ ∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))})
41	40	adantl 485	. . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑃) = (Base‘𝑃)
44		pmatcoe1fsupp.b	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 = (Base‘𝐶)
45		simprl 771	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
46		simprr 773	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁)
47	44	eleq2i 2824	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ 𝐵 ↔ 𝑀 ∈ (Base‘𝐶))
48	47	biimpi 219	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ (Base‘𝐶))
49	48	3ad2ant3 1136	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ (Base‘𝐶))
50	49	ad3antrrr 730	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑀 ∈ (Base‘𝐶))
51	50, 44	eleqtrrdi 2844	. . . . . . . . . . . . . . . 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 21177	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝑀𝑗) ∈ (Base‘𝑃))
53		eqid 2738	. . . . . . . . . . . . . . . 16 ⊢ (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗))
54		pmatcoe1fsupp.p	. . . . . . . . . . . . . . . 16 ⊢ 𝑃 = (Poly1‘𝑅)
55		eqid 2738	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝑅) = (0g‘𝑅)
56		eqid 2738	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑅) = (Base‘𝑅)
57	53, 43, 54, 55, 56	coe1fsupp 20989	. . . . . . . . . . . . . . 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 5042	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑣 finSupp 0 ↔ 𝑣 finSupp (0g‘𝑅)))
61	60	rabbidv 3381	. . . . . . . . . . . . . . . 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 2818	. . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . 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 4084	. . . . . . . . . . . 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 769	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑥 ∈ ℕ0)
67		fveq1 6673	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (coe1‘(𝑖𝑀𝑗)) → (𝑤‘𝑧) = ((coe1‘(𝑖𝑀𝑗))‘𝑧))
68	67	eqeq1d 2740	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (coe1‘(𝑖𝑀𝑗)) → ((𝑤‘𝑧) = 0 ↔ ((coe1‘(𝑖𝑀𝑗))‘𝑧) = 0 ))
69	68	imbi2d 344	. . . . . . . . . . . . 13 ⊢ (𝑤 = (coe1‘(𝑖𝑀𝑗)) → ((𝑠 < 𝑧 → (𝑤‘𝑧) = 0 ) ↔ (𝑠 < 𝑧 → ((coe1‘(𝑖𝑀𝑗))‘𝑧) = 0 )))
70		breq2 5034	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑠 < 𝑧 ↔ 𝑠 < 𝑥))
71		fveqeq2 6683	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (((coe1‘(𝑖𝑀𝑗))‘𝑧) = 0 ↔ ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 ))
72	70, 71	imbi12d 348	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → ((𝑠 < 𝑧 → ((coe1‘(𝑖𝑀𝑗))‘𝑧) = 0 ) ↔ (𝑠 < 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 )))
73	69, 72	rspc2v 3536	. . . . . . . . . . . 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 587	. . . . . . . . . . 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 416	. . . . . . . . . 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 455	. . . . . . . 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 410	. . . . . . 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 3105	. . . . 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 3096	. . . 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 416	. . 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 3184	. 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 209   ∧ wa 399   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114  ∀wral 3053  ∃wrex 3054  {crab 3057  Vcvv 3398   ∩ cin 3842   ⊆ wss 3843  {csn 4516  ⟨cop 4522  ∪ ciun 4881   class class class wbr 5030   × cxp 5523  ‘cfv 6339  (class class class)co 7170   ↑_m cmap 8437  Fincfn 8555   finSupp cfsupp 8906   < clt 10753  ℕ₀cn0 11976  Basecbs 16586  0_gc0g 16816  Ringcrg 19416  Poly₁cpl1 20952  coe₁cco1 20953   Mat cmat 21158

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 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-ot 4525  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-of 7425  df-om 7600  df-1st 7714  df-2nd 7715  df-supp 7857  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-map 8439  df-ixp 8508  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-fsupp 8907  df-sup 8979  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-nn 11717  df-2 11779  df-3 11780  df-4 11781  df-5 11782  df-6 11783  df-7 11784  df-8 11785  df-9 11786  df-n0 11977  df-z 12063  df-dec 12180  df-uz 12325  df-fz 12982  df-struct 16588  df-ndx 16589  df-slot 16590  df-base 16592  df-sets 16593  df-ress 16594  df-plusg 16681  df-mulr 16682  df-sca 16684  df-vsca 16685  df-ip 16686  df-tset 16687  df-ple 16688  df-ds 16690  df-hom 16692  df-cco 16693  df-0g 16818  df-prds 16824  df-pws 16826  df-sra 20063  df-rgmod 20064  df-dsmm 20548  df-frlm 20563  df-psr 20722  df-mpl 20724  df-opsr 20726  df-psr1 20955  df-ply1 20957  df-coe1 20958  df-mat 21159

This theorem is referenced by:  decpmataa0  21519  decpmatmulsumfsupp  21524  pmatcollpw2lem  21528  pm2mpmhmlem1  21569