Theorem ply1mulgsumlem1 48511

Description: Lemma 1 for ply1mulgsum 48515. (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 2733	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
5	1, 2, 3, 4	coe1ae0 22130	. . 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 22130	. . . 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 12423	. . . . . . . . . . . . . . . 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 5096	. . . . . . . . . . . . . . . . 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 3156	. . . . . . . . . . . . . . 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 3093	. . . . . . . . . . . . . . . 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 12398	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑎 ∈ ℕ0 → 𝑎 ∈ ℂ)
19	18	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0) → 𝑎 ∈ ℂ)
20		nn0cn 12398	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑏 ∈ ℕ0 → 𝑏 ∈ ℂ)
21	20	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0) → 𝑏 ∈ ℂ)
22	19, 21	addcomd 11322	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 ↔ (𝑏 + 𝑎) < 𝑛))
25		nn0sumltlt 48474	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 48474	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3145	. . . . . . . . . . . . . . . 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 3575	. . . . . . . . . . . . 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 3126	. . . . 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 3126	. . 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 2113  ∀wral 3048  ∃wrex 3057   class class class wbr 5093  ‘cfv 6486  (class class class)co 7352  ℂcc 11011   + caddc 11016   < clt 11153  ℕ₀cn0 12388  Basecbs 17122  ._rcmulr 17164   ·_𝑠 cvsca 17167  0_gc0g 17345  ._gcmg 18982  mulGrpcmgp 20060  Ringcrg 20153  var₁cv1 22089  Poly₁cpl1 22090  coe₁cco1 22091

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

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 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-of 7616  df-om 7803  df-1st 7927  df-2nd 7928  df-supp 8097  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-er 8628  df-map 8758  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-fsupp 9253  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-2 12195  df-3 12196  df-4 12197  df-5 12198  df-6 12199  df-7 12200  df-8 12201  df-9 12202  df-n0 12389  df-z 12476  df-dec 12595  df-uz 12739  df-fz 13410  df-struct 17060  df-sets 17077  df-slot 17095  df-ndx 17107  df-base 17123  df-ress 17144  df-plusg 17176  df-mulr 17177  df-sca 17179  df-vsca 17180  df-tset 17182  df-ple 17183  df-psr 21848  df-mpl 21850  df-opsr 21852  df-psr1 22093  df-ply1 22095  df-coe1 22096

This theorem is referenced by:  ply1mulgsumlem2  48512