Theorem pmatcoe1fsupp 22424

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 4078	. . . . . 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 871	. . . 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 4239	. . . 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 9320	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin)
7	6	anidms 566	. . . . . 6 ⊢ (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ Fin)
8		snfi 9047	. . . . . . . 8 ⊢ {(coe1‘(𝑀‘𝑢))} ∈ Fin
9	8	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑢 ∈ (𝑁 × 𝑁)) → {(coe1‘(𝑀‘𝑢))} ∈ Fin)
10	9	ralrimiva 3145	. . . . . 6 ⊢ (𝑁 ∈ Fin → ∀𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin)
11	7, 10	jca 511	. . . . 5 ⊢ (𝑁 ∈ Fin → ((𝑁 × 𝑁) ∈ Fin ∧ ∀𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin))
12	11	3ad2ant1 1132	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑁 × 𝑁) ∈ Fin ∧ ∀𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin))
13		iunfi 9343	. . . 4 ⊢ (((𝑁 × 𝑁) ∈ Fin ∧ ∀𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin) → ∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin)
14		infi 9271	. . . 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 6905	. . . . 5 ⊢ (0g‘𝑅) ∈ V
18	16, 17	eqeltri 2828	. . . 4 ⊢ 0 ∈ V
19	18	a1i 11	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 0 ∈ V)
20		elin 3965	. . . . . 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 5152	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → (𝑣 finSupp 0 ↔ 𝑤 finSupp 0 ))
22	21	elrab 3684	. . . . . . 7 ⊢ (𝑤 ∈ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 } ↔ (𝑤 ∈ ((Base‘𝑅) ↑m ℕ0) ∧ 𝑤 finSupp 0 ))
23	22	simprbi 496	. . . . . 6 ⊢ (𝑤 ∈ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 } → 𝑤 finSupp 0 )
24	20, 23	simplbiim 504	. . . . 5 ⊢ (𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑣 finSupp 0 }) → 𝑤 finSupp 0 )
25	24	rgen 3062	. . . 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 13963	. . . 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 406	. . 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 1372	. 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 5714	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ⟨𝑖, 𝑗⟩ ∈ (𝑁 × 𝑁))
31		df-ov 7415	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖𝑀𝑗) = (𝑀‘⟨𝑖, 𝑗⟩)
32	31	fveq2i 6895	. . . . . . . . . . . . . . . . 17 ⊢ (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑀‘⟨𝑖, 𝑗⟩))
33		fvex 6905	. . . . . . . . . . . . . . . . . 18 ⊢ (coe1‘(𝑀‘⟨𝑖, 𝑗⟩)) ∈ V
34	33	snid 4665	. . . . . . . . . . . . . . . . 17 ⊢ (coe1‘(𝑀‘⟨𝑖, 𝑗⟩)) ∈ {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))}
35	32, 34	eqeltri 2828	. . . . . . . . . . . . . . . 16 ⊢ (coe1‘(𝑖𝑀𝑗)) ∈ {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))}
36	35	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (coe1‘(𝑖𝑀𝑗)) ∈ {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))})
37		2fveq3 6897	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = ⟨𝑖, 𝑗⟩ → (coe1‘(𝑀‘𝑢)) = (coe1‘(𝑀‘⟨𝑖, 𝑗⟩)))
38	37	sneqd 4641	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = ⟨𝑖, 𝑗⟩ → {(coe1‘(𝑀‘𝑢))} = {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))})
39	38	eliuni 5004	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑖, 𝑗⟩ ∈ (𝑁 × 𝑁) ∧ (coe1‘(𝑖𝑀𝑗)) ∈ {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))}) → (coe1‘(𝑖𝑀𝑗)) ∈ ∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))})
40	30, 36, 39	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (coe1‘(𝑖𝑀𝑗)) ∈ ∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))})
41	40	adantl 481	. . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑃) = (Base‘𝑃)
44		pmatcoe1fsupp.b	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 = (Base‘𝐶)
45		simprl 768	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
46		simprr 770	. . . . . . . . . . . . . . . 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 215	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ (Base‘𝐶))
49	48	3ad2ant3 1134	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ (Base‘𝐶))
50	49	ad3antrrr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑀 ∈ (Base‘𝐶))
51	50, 44	eleqtrrdi 2843	. . . . . . . . . . . . . . . 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 22149	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝑀𝑗) ∈ (Base‘𝑃))
53		eqid 2731	. . . . . . . . . . . . . . . 16 ⊢ (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗))
54		pmatcoe1fsupp.p	. . . . . . . . . . . . . . . 16 ⊢ 𝑃 = (Poly1‘𝑅)
55		eqid 2731	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝑅) = (0g‘𝑅)
56		eqid 2731	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑅) = (Base‘𝑅)
57	53, 43, 54, 55, 56	coe1fsupp 21958	. . . . . . . . . . . . . . 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 5161	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑣 finSupp 0 ↔ 𝑣 finSupp (0g‘𝑅)))
61	60	rabbidv 3439	. . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . 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 256	. . . . . . . . . . . . 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 4195	. . . . . . . . . . . 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 766	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑥 ∈ ℕ0)
67		fveq1 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (coe1‘(𝑖𝑀𝑗)) → (𝑤‘𝑧) = ((coe1‘(𝑖𝑀𝑗))‘𝑧))
68	67	eqeq1d 2733	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (coe1‘(𝑖𝑀𝑗)) → ((𝑤‘𝑧) = 0 ↔ ((coe1‘(𝑖𝑀𝑗))‘𝑧) = 0 ))
69	68	imbi2d 339	. . . . . . . . . . . . 13 ⊢ (𝑤 = (coe1‘(𝑖𝑀𝑗)) → ((𝑠 < 𝑧 → (𝑤‘𝑧) = 0 ) ↔ (𝑠 < 𝑧 → ((coe1‘(𝑖𝑀𝑗))‘𝑧) = 0 )))
70		breq2 5153	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑠 < 𝑧 ↔ 𝑠 < 𝑥))
71		fveqeq2 6901	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (((coe1‘(𝑖𝑀𝑗))‘𝑧) = 0 ↔ ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 ))
72	70, 71	imbi12d 343	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → ((𝑠 < 𝑧 → ((coe1‘(𝑖𝑀𝑗))‘𝑧) = 0 ) ↔ (𝑠 < 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 )))
73	69, 72	rspc2v 3623	. . . . . . . . . . . 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 583	. . . . . . . . . . 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 412	. . . . . . . . . 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 451	. . . . . . . 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 406	. . . . . . 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 3200	. . . . 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 3145	. . . 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 412	. . 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 3167	. 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 205   ∧ wa 395   ∨ wo 844   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3060  ∃wrex 3069  {crab 3431  Vcvv 3473   ∩ cin 3948   ⊆ wss 3949  {csn 4629  ⟨cop 4635  ∪ ciun 4998   class class class wbr 5149   × cxp 5675  ‘cfv 6544  (class class class)co 7412   ↑_m cmap 8823  Fincfn 8942   finSupp cfsupp 9364   < clt 11253  ℕ₀cn0 12477  Basecbs 17149  0_gc0g 17390  Ringcrg 20128  Poly₁cpl1 21921  coe₁cco1 21922   Mat cmat 22128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-ot 4638  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7673  df-om 7859  df-1st 7978  df-2nd 7979  df-supp 8150  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-er 8706  df-map 8825  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-fsupp 9365  df-sup 9440  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12478  df-z 12564  df-dec 12683  df-uz 12828  df-fz 13490  df-struct 17085  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-ress 17179  df-plusg 17215  df-mulr 17216  df-sca 17218  df-vsca 17219  df-ip 17220  df-tset 17221  df-ple 17222  df-ds 17224  df-hom 17226  df-cco 17227  df-0g 17392  df-prds 17398  df-pws 17400  df-sra 20931  df-rgmod 20932  df-dsmm 21507  df-frlm 21522  df-psr 21682  df-mpl 21684  df-opsr 21686  df-psr1 21924  df-ply1 21926  df-coe1 21927  df-mat 22129

This theorem is referenced by:  decpmataa0  22491  decpmatmulsumfsupp  22496  pmatcollpw2lem  22500  pm2mpmhmlem1  22541