Theorem ply1mulgsumlem1 48397

Description: Lemma 1 for ply1mulgsum 48401. (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 2730	. . . 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 1134	. 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 1135	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑎 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)))
10		nn0addcl 12408	. . . . . . . . . . . . . . . 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 5092	. . . . . . . . . . . . . . . . 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 3153	. . . . . . . . . . . . . . 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 3090	. . . . . . . . . . . . . . . 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 12383	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑎 ∈ ℕ0 → 𝑎 ∈ ℂ)
19	18	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0) → 𝑎 ∈ ℂ)
20		nn0cn 12383	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑏 ∈ ℕ0 → 𝑏 ∈ ℂ)
21	20	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0) → 𝑏 ∈ ℂ)
22	19, 21	addcomd 11307	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
23	22	3adant3 1132	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
24	23	breq1d 5099	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 ↔ (𝑏 + 𝑎) < 𝑛))
25		nn0sumltlt 48360	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → ((𝑏 + 𝑎) < 𝑛 → 𝑎 < 𝑛))
26	24, 25	sylbid 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 → 𝑎 < 𝑛))
27	26	3expia 1121	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 48360	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 → 𝑏 < 𝑛))
35	34	3expia 1121	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 609	. . . . . . . . . . . . . . . . . . . . 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 3142	. . . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . . 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 3577	. . . . . . . . . . . . 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 3123	. . . . 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 3123	. . 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2110  ∀wral 3045  ∃wrex 3054   class class class wbr 5089  ‘cfv 6477  (class class class)co 7341  ℂcc 10996   + caddc 11001   < clt 11138  ℕ₀cn0 12373  Basecbs 17112  ._rcmulr 17154   ·_𝑠 cvsca 17157  0_gc0g 17335  ._gcmg 18972  mulGrpcmgp 20051  Ringcrg 20144  var₁cv1 22081  Poly₁cpl1 22082  coe₁cco1 22083

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 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-cnex 11054  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074  ax-pre-mulgt0 11075

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-tp 4579  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7605  df-om 7792  df-1st 7916  df-2nd 7917  df-supp 8086  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-er 8617  df-map 8747  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-fsupp 9241  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-nn 12118  df-2 12180  df-3 12181  df-4 12182  df-5 12183  df-6 12184  df-7 12185  df-8 12186  df-9 12187  df-n0 12374  df-z 12461  df-dec 12581  df-uz 12725  df-fz 13400  df-struct 17050  df-sets 17067  df-slot 17085  df-ndx 17097  df-base 17113  df-ress 17134  df-plusg 17166  df-mulr 17167  df-sca 17169  df-vsca 17170  df-tset 17172  df-ple 17173  df-psr 21839  df-mpl 21841  df-opsr 21843  df-psr1 22085  df-ply1 22087  df-coe1 22088

This theorem is referenced by:  ply1mulgsumlem2  48398