Theorem ply1mulgsumlem1 47377

Description: Lemma 1 for ply1mulgsum 47381. (Contributed by AV, 19-Oct-2019.)

Hypotheses

Ref	Expression
ply1mulgsum.p	⊢ 𝑃 = (Poly1‘𝑅)
ply1mulgsum.b	⊢ 𝐵 = (Base‘𝑃)
ply1mulgsum.a	⊢ 𝐴 = (coe1‘𝐾)
ply1mulgsum.c	⊢ 𝐶 = (coe1‘𝐿)
ply1mulgsum.x	⊢ 𝑋 = (var1‘𝑅)
ply1mulgsum.pm	⊢ × = (.r‘𝑃)
ply1mulgsum.sm	⊢ · = ( ·𝑠 ‘𝑃)
ply1mulgsum.rm	⊢ ∗ = (.r‘𝑅)
ply1mulgsum.m	⊢ 𝑀 = (mulGrp‘𝑃)
ply1mulgsum.e	⊢ ↑ = (.g‘𝑀)

Assertion

Ref	Expression
ply1mulgsumlem1	⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))

Distinct variable groups:   𝐴,𝑛,𝑠   𝐵,𝑛,𝑠   𝐶,𝑛,𝑠   𝑛,𝐾,𝑠   𝑛,𝐿,𝑠   𝑅,𝑛,𝑠

Allowed substitution hints:   𝑃(𝑛,𝑠)   · (𝑛,𝑠)   × (𝑛,𝑠)   ↑ (𝑛,𝑠)   ∗ (𝑛,𝑠)   𝑀(𝑛,𝑠)   𝑋(𝑛,𝑠)

Proof of Theorem ply1mulgsumlem1

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ply1mulgsum.a	. . . 4 ⊢ 𝐴 = (coe1‘𝐾)
2		ply1mulgsum.b	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
3		ply1mulgsum.p	. . . 4 ⊢ 𝑃 = (Poly1‘𝑅)
4		eqid 2727	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
5	1, 2, 3, 4	coe1ae0 22122	. . 3 ⊢ (𝐾 ∈ 𝐵 → ∃𝑏 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)))
6	5	3ad2ant2 1132	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑏 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)))
7		ply1mulgsum.c	. . . . 5 ⊢ 𝐶 = (coe1‘𝐿)
8	7, 2, 3, 4	coe1ae0 22122	. . . 4 ⊢ (𝐿 ∈ 𝐵 → ∃𝑎 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)))
9	8	3ad2ant3 1133	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑎 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)))
10		nn0addcl 12529	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑎 + 𝑏) ∈ ℕ0)
11	10	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) → (𝑎 + 𝑏) ∈ ℕ0)
12	11	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)))) → (𝑎 + 𝑏) ∈ ℕ0)
13		breq1 5145	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (𝑎 + 𝑏) → (𝑠 < 𝑛 ↔ (𝑎 + 𝑏) < 𝑛))
14	13	imbi1d 341	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = (𝑎 + 𝑏) → ((𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))) ↔ ((𝑎 + 𝑏) < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅)))))
15	14	ralbidv 3172	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = (𝑎 + 𝑏) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))) ↔ ∀𝑛 ∈ ℕ0 ((𝑎 + 𝑏) < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅)))))
16	15	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)))) ∧ 𝑠 = (𝑎 + 𝑏)) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))) ↔ ∀𝑛 ∈ ℕ0 ((𝑎 + 𝑏) < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅)))))
17		r19.26 3106	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑛 ∈ ℕ0 ((𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅))) ↔ (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅))))
18		nn0cn 12504	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑎 ∈ ℕ0 → 𝑎 ∈ ℂ)
19	18	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0) → 𝑎 ∈ ℂ)
20		nn0cn 12504	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑏 ∈ ℕ0 → 𝑏 ∈ ℂ)
21	20	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0) → 𝑏 ∈ ℂ)
22	19, 21	addcomd 11438	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
23	22	3adant3 1130	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
24	23	breq1d 5152	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 ↔ (𝑏 + 𝑎) < 𝑛))
25		nn0sumltlt 47337	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → ((𝑏 + 𝑎) < 𝑛 → 𝑎 < 𝑛))
26	24, 25	sylbid 239	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 → 𝑎 < 𝑛))
27	26	3expia 1119	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛 → 𝑎 < 𝑛)))
28	27	ancoms 458	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛 → 𝑎 < 𝑛)))
29	28	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛 → 𝑎 < 𝑛)))
30	29	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 → 𝑎 < 𝑛))
31	30	imim1d 82	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) → ((𝑎 + 𝑏) < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅))))
32	31	com23 86	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 → ((𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) → (𝐶‘𝑛) = (0g‘𝑅))))
33	32	imp 406	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 + 𝑏) < 𝑛) → ((𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) → (𝐶‘𝑛) = (0g‘𝑅)))
34		nn0sumltlt 47337	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 → 𝑏 < 𝑛))
35	34	3expia 1119	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛 → 𝑏 < 𝑛)))
36	35	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛 → 𝑏 < 𝑛)))
37	36	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 → 𝑏 < 𝑛))
38	37	imim1d 82	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)) → ((𝑎 + 𝑏) < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅))))
39	38	com23 86	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 → ((𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)) → (𝐴‘𝑛) = (0g‘𝑅))))
40	39	imp 406	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 + 𝑏) < 𝑛) → ((𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)) → (𝐴‘𝑛) = (0g‘𝑅)))
41	33, 40	anim12d 608	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 + 𝑏) < 𝑛) → (((𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅))) → ((𝐶‘𝑛) = (0g‘𝑅) ∧ (𝐴‘𝑛) = (0g‘𝑅))))
42	41	imp 406	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 + 𝑏) < 𝑛) ∧ ((𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)))) → ((𝐶‘𝑛) = (0g‘𝑅) ∧ (𝐴‘𝑛) = (0g‘𝑅)))
43	42	ancomd 461	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 + 𝑏) < 𝑛) ∧ ((𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)))) → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅)))
44	43	exp31 419	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 → (((𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅))) → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅)))))
45	44	com23 86	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → (((𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅))) → ((𝑎 + 𝑏) < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅)))))
46	45	ralimdva 3162	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) → (∀𝑛 ∈ ℕ0 ((𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅))) → ∀𝑛 ∈ ℕ0 ((𝑎 + 𝑏) < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅)))))
47	17, 46	biimtrrid 242	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) → ((∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅))) → ∀𝑛 ∈ ℕ0 ((𝑎 + 𝑏) < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅)))))
48	47	imp 406	. . . . . . . . . . . . . 14 ⊢ ((((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)))) → ∀𝑛 ∈ ℕ0 ((𝑎 + 𝑏) < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))
49	12, 16, 48	rspcedvd 3609	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)))) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))
50	49	exp31 419	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ((∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅))) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))))
51	50	com23 86	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → ((∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅))) → ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))))
52	51	expd 415	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) → (∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)) → ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅)))))))
53	52	com34 91	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) → ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅)))))))
54	53	impancom 451	. . . . . . . 8 ⊢ ((𝑎 ∈ ℕ0 ∧ ∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅))) → (𝑏 ∈ ℕ0 → ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅)))))))
55	54	com14 96	. . . . . . 7 ⊢ (∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)) → (𝑏 ∈ ℕ0 → ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ((𝑎 ∈ ℕ0 ∧ ∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅))) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅)))))))
56	55	impcom 407	. . . . . 6 ⊢ ((𝑏 ∈ ℕ0 ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅))) → ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ((𝑎 ∈ ℕ0 ∧ ∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅))) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))))
57	56	rexlimiva 3142	. . . . 5 ⊢ (∃𝑏 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)) → ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ((𝑎 ∈ ℕ0 ∧ ∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅))) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))))
58	57	com13 88	. . . 4 ⊢ ((𝑎 ∈ ℕ0 ∧ ∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅))) → ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (∃𝑏 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))))
59	58	rexlimiva 3142	. . 3 ⊢ (∃𝑎 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) → ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (∃𝑏 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))))
60	9, 59	mpcom 38	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (∃𝑏 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅)))))
61	6, 60	mpd 15	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ∀wral 3056  ∃wrex 3065   class class class wbr 5142  ‘cfv 6542  (class class class)co 7414  ℂcc 11128   + caddc 11133   < clt 11270  ℕ₀cn0 12494  Basecbs 17171  ._rcmulr 17225   ·_𝑠 cvsca 17228  0_gc0g 17412  ._gcmg 19014  mulGrpcmgp 20065  Ringcrg 20164  var₁cv1 22082  Poly₁cpl1 22083  coe₁cco1 22084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7679  df-om 7865  df-1st 7987  df-2nd 7988  df-supp 8160  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8838  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-fsupp 9378  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-3 12298  df-4 12299  df-5 12300  df-6 12301  df-7 12302  df-8 12303  df-9 12304  df-n0 12495  df-z 12581  df-dec 12700  df-uz 12845  df-fz 13509  df-struct 17107  df-sets 17124  df-slot 17142  df-ndx 17154  df-base 17172  df-ress 17201  df-plusg 17237  df-mulr 17238  df-sca 17240  df-vsca 17241  df-tset 17243  df-ple 17244  df-psr 21829  df-mpl 21831  df-opsr 21833  df-psr1 22086  df-ply1 22088  df-coe1 22089

This theorem is referenced by:  ply1mulgsumlem2  47378