Theorem ply1mulgsumlem1 48999

Description: Lemma 1 for ply1mulgsum 49003. (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 2763	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
5	1, 2, 3, 4	coe1ae0 22285	. . 3 ⊢ (𝐾 ∈ 𝐵 → ∃𝑏 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)))
6	5	3ad2ant2 1148	. 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 22285	. . . 4 ⊢ (𝐿 ∈ 𝐵 → ∃𝑎 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)))
9	8	3ad2ant3 1149	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑎 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)))
10		nn0addcl 12526	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑎 + 𝑏) ∈ ℕ0)
11	10	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) → (𝑎 + 𝑏) ∈ ℕ0)
12	11	adantr 484	. . . . . . . . . . . . . 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 5104	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (𝑎 + 𝑏) → (𝑠 < 𝑛 ↔ (𝑎 + 𝑏) < 𝑛))
14	13	imbi1d 343	. . . . . . . . . . . . . . . 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 3186	. . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . 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 3123	. . . . . . . . . . . . . . . 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 12501	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑎 ∈ ℕ0 → 𝑎 ∈ ℂ)
19	18	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0) → 𝑎 ∈ ℂ)
20		nn0cn 12501	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑏 ∈ ℕ0 → 𝑏 ∈ ℂ)
21	20	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0) → 𝑏 ∈ ℂ)
22	19, 21	addcomd 11396	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
23	22	3adant3 1146	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
24	23	breq1d 5111	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 ↔ (𝑏 + 𝑎) < 𝑛))
25		nn0sumltlt 48963	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → ((𝑏 + 𝑎) < 𝑛 → 𝑎 < 𝑛))
26	24, 25	sylbid 242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 → 𝑎 < 𝑛))
27	26	3expia 1135	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝑎 ∈ ℕ0) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛 → 𝑎 < 𝑛)))
28	27	ancoms 462	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛 → 𝑎 < 𝑛)))
29	28	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛 → 𝑎 < 𝑛)))
30	29	imp 410	. . . . . . . . . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . . . . . . 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 48963	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 → 𝑏 < 𝑛))
35	34	3expia 1135	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛 → 𝑏 < 𝑛)))
36	35	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛 → 𝑏 < 𝑛)))
37	36	imp 410	. . . . . . . . . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . . . . . . 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 618	. . . . . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . . . . . . 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 423	. . . . . . . . . . . . . . . . . 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 3175	. . . . . . . . . . . . . . . 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 245	. . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . 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 3584	. . . . . . . . . . . . 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 423	. . . . . . . . . . . 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 419	. . . . . . . . . 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 455	. . . . . . . 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 411	. . . . . 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 3156	. . . . 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 3156	. . 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 208   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143  ∀wral 3077  ∃wrex 3087   class class class wbr 5101  ‘cfv 6521  (class class class)co 7396  ℂcc 11082   + caddc 11087   < clt 11227  ℕ₀cn0 12491  Basecbs 17255  ._rcmulr 17297   ·_𝑠 cvsca 17300  0_gc0g 17478  ._gcmg 19119  mulGrpcmgp 20196  Ringcrg 20293  var₁cv1 22245  Poly₁cpl1 22246  coe₁cco1 22247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718  ax-cnex 11140  ax-resscn 11141  ax-1cn 11142  ax-icn 11143  ax-addcl 11144  ax-addrcl 11145  ax-mulcl 11146  ax-mulrcl 11147  ax-mulcom 11148  ax-addass 11149  ax-mulass 11150  ax-distr 11151  ax-i2m1 11152  ax-1ne0 11153  ax-1rid 11154  ax-rnegex 11155  ax-rrecex 11156  ax-cnre 11157  ax-pre-lttri 11158  ax-pre-lttrn 11159  ax-pre-ltadd 11160  ax-pre-mulgt0 11161

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7660  df-om 7847  df-1st 7970  df-2nd 7971  df-supp 8141  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-er 8678  df-map 8810  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fsupp 9306  df-pnf 11229  df-mnf 11230  df-xr 11231  df-ltxr 11232  df-le 11233  df-sub 11427  df-neg 11428  df-nn 12221  df-2 12290  df-3 12291  df-4 12292  df-5 12293  df-6 12294  df-7 12295  df-8 12296  df-9 12297  df-n0 12492  df-z 12579  df-dec 12699  df-uz 12850  df-fz 13523  df-struct 17193  df-sets 17210  df-slot 17228  df-ndx 17240  df-base 17256  df-ress 17277  df-plusg 17309  df-mulr 17310  df-sca 17312  df-vsca 17313  df-tset 17315  df-ple 17316  df-psr 21968  df-mpl 21970  df-opsr 21972  df-psr1 22249  df-ply1 22251  df-coe1 22252

This theorem is referenced by:  ply1mulgsumlem2  49000