Theorem pmatcoe1fsupp 22728

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 4103	. . . . . 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 4268	. . . 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 9386	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin)
7	6	anidms 566	. . . . . 6 ⊢ (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ Fin)
8		snfi 9109	. . . . . . . 8 ⊢ {(coe1‘(𝑀‘𝑢))} ∈ Fin
9	8	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑢 ∈ (𝑁 × 𝑁)) → {(coe1‘(𝑀‘𝑢))} ∈ Fin)
10	9	ralrimiva 3152	. . . . . 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 1133	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑁 × 𝑁) ∈ Fin ∧ ∀𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin))
13		iunfi 9411	. . . 4 ⊢ (((𝑁 × 𝑁) ∈ Fin ∧ ∀𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin) → ∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin)
14		infi 9330	. . . 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 6933	. . . . 5 ⊢ (0g‘𝑅) ∈ V
18	16, 17	eqeltri 2840	. . . 4 ⊢ 0 ∈ V
19	18	a1i 11	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 0 ∈ V)
20		elin 3992	. . . . . 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 5169	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → (𝑣 finSupp 0 ↔ 𝑤 finSupp 0 ))
22	21	elrab 3708	. . . . . . 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 3069	. . . 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 14044	. . . 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 1373	. 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 5737	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ⟨𝑖, 𝑗⟩ ∈ (𝑁 × 𝑁))
31		df-ov 7451	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖𝑀𝑗) = (𝑀‘⟨𝑖, 𝑗⟩)
32	31	fveq2i 6923	. . . . . . . . . . . . . . . . 17 ⊢ (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑀‘⟨𝑖, 𝑗⟩))
33		fvex 6933	. . . . . . . . . . . . . . . . . 18 ⊢ (coe1‘(𝑀‘⟨𝑖, 𝑗⟩)) ∈ V
34	33	snid 4684	. . . . . . . . . . . . . . . . 17 ⊢ (coe1‘(𝑀‘⟨𝑖, 𝑗⟩)) ∈ {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))}
35	32, 34	eqeltri 2840	. . . . . . . . . . . . . . . 16 ⊢ (coe1‘(𝑖𝑀𝑗)) ∈ {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))}
36	35	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (coe1‘(𝑖𝑀𝑗)) ∈ {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))})
37		2fveq3 6925	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = ⟨𝑖, 𝑗⟩ → (coe1‘(𝑀‘𝑢)) = (coe1‘(𝑀‘⟨𝑖, 𝑗⟩)))
38	37	sneqd 4660	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = ⟨𝑖, 𝑗⟩ → {(coe1‘(𝑀‘𝑢))} = {(coe1‘(𝑀‘⟨𝑖, 𝑗⟩))})
39	38	eliuni 5021	. . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑃) = (Base‘𝑃)
44		pmatcoe1fsupp.b	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 = (Base‘𝐶)
45		simprl 770	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
46		simprr 772	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁)
47	44	eleq2i 2836	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ 𝐵 ↔ 𝑀 ∈ (Base‘𝐶))
48	47	biimpi 216	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ (Base‘𝐶))
49	48	3ad2ant3 1135	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ (Base‘𝐶))
50	49	ad3antrrr 729	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑀 ∈ (Base‘𝐶))
51	50, 44	eleqtrrdi 2855	. . . . . . . . . . . . . . . 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 22453	. . . . . . . . . . . . . . 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 22237	. . . . . . . . . . . . . . 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 5178	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑣 finSupp 0 ↔ 𝑣 finSupp (0g‘𝑅)))
61	60	rabbidv 3451	. . . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . 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 257	. . . . . . . . . . . . 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 4223	. . . . . . . . . . . 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 768	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑥 ∈ ℕ0)
67		fveq1 6919	. . . . . . . . . . . . . . 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 340	. . . . . . . . . . . . 13 ⊢ (𝑤 = (coe1‘(𝑖𝑀𝑗)) → ((𝑠 < 𝑧 → (𝑤‘𝑧) = 0 ) ↔ (𝑠 < 𝑧 → ((coe1‘(𝑖𝑀𝑗))‘𝑧) = 0 )))
70		breq2 5170	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑠 < 𝑧 ↔ 𝑠 < 𝑥))
71		fveqeq2 6929	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (((coe1‘(𝑖𝑀𝑗))‘𝑧) = 0 ↔ ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 ))
72	70, 71	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → ((𝑠 < 𝑧 → ((coe1‘(𝑖𝑀𝑗))‘𝑧) = 0 ) ↔ (𝑠 < 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 )))
73	69, 72	rspc2v 3646	. . . . . . . . . . . 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 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 3152	. . . 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 3174	. 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 206   ∧ wa 395   ∨ wo 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  {crab 3443  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976  {csn 4648  ⟨cop 4654  ∪ ciun 5015   class class class wbr 5166   × cxp 5698  ‘cfv 6573  (class class class)co 7448   ↑_m cmap 8884  Fincfn 9003   finSupp cfsupp 9431   < clt 11324  ℕ₀cn0 12553  Basecbs 17258  0_gc0g 17499  Ringcrg 20260  Poly₁cpl1 22199  coe₁cco1 22200   Mat cmat 22432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-sup 9511  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-hom 17335  df-cco 17336  df-0g 17501  df-prds 17507  df-pws 17509  df-sra 21195  df-rgmod 21196  df-dsmm 21775  df-frlm 21790  df-psr 21952  df-mpl 21954  df-opsr 21956  df-psr1 22202  df-ply1 22204  df-coe1 22205  df-mat 22433

This theorem is referenced by:  decpmataa0  22795  decpmatmulsumfsupp  22800  pmatcollpw2lem  22804  pm2mpmhmlem1  22845