Theorem ply1mulgsumlem2 48304

Description: Lemma 2 for ply1mulgsum 48307. (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
ply1mulgsumlem2	⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅)))

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

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

Proof of Theorem ply1mulgsumlem2

Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ply1mulgsum.p	. . 3 ⊢ 𝑃 = (Poly1‘𝑅)
2		ply1mulgsum.b	. . 3 ⊢ 𝐵 = (Base‘𝑃)
3		ply1mulgsum.a	. . 3 ⊢ 𝐴 = (coe1‘𝐾)
4		ply1mulgsum.c	. . 3 ⊢ 𝐶 = (coe1‘𝐿)
5		ply1mulgsum.x	. . 3 ⊢ 𝑋 = (var1‘𝑅)
6		ply1mulgsum.pm	. . 3 ⊢ × = (.r‘𝑃)
7		ply1mulgsum.sm	. . 3 ⊢ · = ( ·𝑠 ‘𝑃)
8		ply1mulgsum.rm	. . 3 ⊢ ∗ = (.r‘𝑅)
9		ply1mulgsum.m	. . 3 ⊢ 𝑀 = (mulGrp‘𝑃)
10		ply1mulgsum.e	. . 3 ⊢ ↑ = (.g‘𝑀)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ply1mulgsumlem1 48303	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑧 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅))))
12		2nn0 12543	. . . . . . . 8 ⊢ 2 ∈ ℕ0
13	12	a1i 11	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0 → 2 ∈ ℕ0)
14		id 22	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0 → 𝑧 ∈ ℕ0)
15	13, 14	nn0mulcld 12592	. . . . . 6 ⊢ (𝑧 ∈ ℕ0 → (2 · 𝑧) ∈ ℕ0)
16	15	ad2antrr 726	. . . . 5 ⊢ (((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) → (2 · 𝑧) ∈ ℕ0)
17		breq1 5146	. . . . . . . 8 ⊢ (𝑠 = (2 · 𝑧) → (𝑠 < 𝑛 ↔ (2 · 𝑧) < 𝑛))
18	17	imbi1d 341	. . . . . . 7 ⊢ (𝑠 = (2 · 𝑧) → ((𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅)) ↔ ((2 · 𝑧) < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅))))
19	18	ralbidv 3178	. . . . . 6 ⊢ (𝑠 = (2 · 𝑧) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅)) ↔ ∀𝑛 ∈ ℕ0 ((2 · 𝑧) < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅))))
20	19	adantl 481	. . . . 5 ⊢ ((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑠 = (2 · 𝑧)) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅)) ↔ ∀𝑛 ∈ ℕ0 ((2 · 𝑧) < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅))))
21		2re 12340	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 ∈ ℝ
22	21	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ ℕ0 → 2 ∈ ℝ)
23		nn0re 12535	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ ℕ0 → 𝑧 ∈ ℝ)
24	22, 23	remulcld 11291	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ ℕ0 → (2 · 𝑧) ∈ ℝ)
25	24	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (2 · 𝑧) ∈ ℝ)
26		nn0re 12535	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℝ)
27	26	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℝ)
28	27	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑛 ∈ ℝ)
29		elfznn0 13660	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑙 ∈ (0...𝑛) → 𝑙 ∈ ℕ0)
30		nn0re 12535	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑙 ∈ ℕ0 → 𝑙 ∈ ℝ)
31	29, 30	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 ∈ (0...𝑛) → 𝑙 ∈ ℝ)
32	31	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑙 ∈ ℝ)
33	25, 28, 32	ltsub1d 11872	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) < 𝑛 ↔ ((2 · 𝑧) − 𝑙) < (𝑛 − 𝑙)))
34	23	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑧 ∈ ℝ)
35	32, 34, 25	lesub2d 11871	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑙 ≤ 𝑧 ↔ ((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙)))
36	35	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ ((2 · 𝑧) − 𝑙) < (𝑛 − 𝑙)) → (𝑙 ≤ 𝑧 ↔ ((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙)))
37	24, 23	resubcld 11691	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ ℕ0 → ((2 · 𝑧) − 𝑧) ∈ ℝ)
38	37	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑧) ∈ ℝ)
39	24	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → (2 · 𝑧) ∈ ℝ)
40		resubcl 11573	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((2 · 𝑧) ∈ ℝ ∧ 𝑙 ∈ ℝ) → ((2 · 𝑧) − 𝑙) ∈ ℝ)
41	39, 31, 40	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑙) ∈ ℝ)
42		resubcl 11573	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑛 ∈ ℝ ∧ 𝑙 ∈ ℝ) → (𝑛 − 𝑙) ∈ ℝ)
43	27, 31, 42	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑛 − 𝑙) ∈ ℝ)
44		lelttr 11351	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((2 · 𝑧) − 𝑧) ∈ ℝ ∧ ((2 · 𝑧) − 𝑙) ∈ ℝ ∧ (𝑛 − 𝑙) ∈ ℝ) → ((((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) ∧ ((2 · 𝑧) − 𝑙) < (𝑛 − 𝑙)) → ((2 · 𝑧) − 𝑧) < (𝑛 − 𝑙)))
45	38, 41, 43, 44	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) ∧ ((2 · 𝑧) − 𝑙) < (𝑛 − 𝑙)) → ((2 · 𝑧) − 𝑧) < (𝑛 − 𝑙)))
46		nn0cn 12536	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 ∈ ℕ0 → 𝑧 ∈ ℂ)
47		2txmxeqx 12406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 ∈ ℂ → ((2 · 𝑧) − 𝑧) = 𝑧)
48	46, 47	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ ℕ0 → ((2 · 𝑧) − 𝑧) = 𝑧)
49	48	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑧) = 𝑧)
50	49	breq1d 5153	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (((2 · 𝑧) − 𝑧) < (𝑛 − 𝑙) ↔ 𝑧 < (𝑛 − 𝑙)))
51	45, 50	sylibd 239	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) ∧ ((2 · 𝑧) − 𝑙) < (𝑛 − 𝑙)) → 𝑧 < (𝑛 − 𝑙)))
52	51	expcomd 416	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (((2 · 𝑧) − 𝑙) < (𝑛 − 𝑙) → (((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) → 𝑧 < (𝑛 − 𝑙))))
53	52	imp 406	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ ((2 · 𝑧) − 𝑙) < (𝑛 − 𝑙)) → (((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) → 𝑧 < (𝑛 − 𝑙)))
54	36, 53	sylbid 240	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ ((2 · 𝑧) − 𝑙) < (𝑛 − 𝑙)) → (𝑙 ≤ 𝑧 → 𝑧 < (𝑛 − 𝑙)))
55	54	ex 412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (((2 · 𝑧) − 𝑙) < (𝑛 − 𝑙) → (𝑙 ≤ 𝑧 → 𝑧 < (𝑛 − 𝑙))))
56	33, 55	sylbid 240	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) < 𝑛 → (𝑙 ≤ 𝑧 → 𝑧 < (𝑛 − 𝑙))))
57	56	ex 412	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → (𝑙 ∈ (0...𝑛) → ((2 · 𝑧) < 𝑛 → (𝑙 ≤ 𝑧 → 𝑧 < (𝑛 − 𝑙)))))
58	57	com23 86	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → ((2 · 𝑧) < 𝑛 → (𝑙 ∈ (0...𝑛) → (𝑙 ≤ 𝑧 → 𝑧 < (𝑛 − 𝑙)))))
59	58	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℕ0 → (𝑛 ∈ ℕ0 → ((2 · 𝑧) < 𝑛 → (𝑙 ∈ (0...𝑛) → (𝑙 ≤ 𝑧 → 𝑧 < (𝑛 − 𝑙))))))
60	59	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) → (𝑛 ∈ ℕ0 → ((2 · 𝑧) < 𝑛 → (𝑙 ∈ (0...𝑛) → (𝑙 ≤ 𝑧 → 𝑧 < (𝑛 − 𝑙))))))
61	60	imp41 425	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛)) → (𝑙 ≤ 𝑧 → 𝑧 < (𝑛 − 𝑙)))
62	61	impcom 407	. . . . . . . . . . . . . 14 ⊢ ((𝑙 ≤ 𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → 𝑧 < (𝑛 − 𝑙))
63		fznn0sub2 13675	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 ∈ (0...𝑛) → (𝑛 − 𝑙) ∈ (0...𝑛))
64		elfznn0 13660	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 − 𝑙) ∈ (0...𝑛) → (𝑛 − 𝑙) ∈ ℕ0)
65		breq2 5147	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = (𝑛 − 𝑙) → (𝑧 < 𝑥 ↔ 𝑧 < (𝑛 − 𝑙)))
66		fveqeq2 6915	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = (𝑛 − 𝑙) → ((𝐴‘𝑥) = (0g‘𝑅) ↔ (𝐴‘(𝑛 − 𝑙)) = (0g‘𝑅)))
67		fveqeq2 6915	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = (𝑛 − 𝑙) → ((𝐶‘𝑥) = (0g‘𝑅) ↔ (𝐶‘(𝑛 − 𝑙)) = (0g‘𝑅)))
68	66, 67	anbi12d 632	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = (𝑛 − 𝑙) → (((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)) ↔ ((𝐴‘(𝑛 − 𝑙)) = (0g‘𝑅) ∧ (𝐶‘(𝑛 − 𝑙)) = (0g‘𝑅))))
69	65, 68	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = (𝑛 − 𝑙) → ((𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅))) ↔ (𝑧 < (𝑛 − 𝑙) → ((𝐴‘(𝑛 − 𝑙)) = (0g‘𝑅) ∧ (𝐶‘(𝑛 − 𝑙)) = (0g‘𝑅)))))
70	69	rspcva 3620	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 − 𝑙) ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) → (𝑧 < (𝑛 − 𝑙) → ((𝐴‘(𝑛 − 𝑙)) = (0g‘𝑅) ∧ (𝐶‘(𝑛 − 𝑙)) = (0g‘𝑅))))
71		simpr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴‘(𝑛 − 𝑙)) = (0g‘𝑅) ∧ (𝐶‘(𝑛 − 𝑙)) = (0g‘𝑅)) → (𝐶‘(𝑛 − 𝑙)) = (0g‘𝑅))
72	70, 71	syl6 35	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 − 𝑙) ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) → (𝑧 < (𝑛 − 𝑙) → (𝐶‘(𝑛 − 𝑙)) = (0g‘𝑅)))
73	72	ex 412	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 − 𝑙) ∈ ℕ0 → (∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅))) → (𝑧 < (𝑛 − 𝑙) → (𝐶‘(𝑛 − 𝑙)) = (0g‘𝑅))))
74	63, 64, 73	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 ∈ (0...𝑛) → (∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅))) → (𝑧 < (𝑛 − 𝑙) → (𝐶‘(𝑛 − 𝑙)) = (0g‘𝑅))))
75	74	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅))) → (𝑙 ∈ (0...𝑛) → (𝑧 < (𝑛 − 𝑙) → (𝐶‘(𝑛 − 𝑙)) = (0g‘𝑅))))
76	75	ad4antlr 733	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → (𝑙 ∈ (0...𝑛) → (𝑧 < (𝑛 − 𝑙) → (𝐶‘(𝑛 − 𝑙)) = (0g‘𝑅))))
77	76	imp 406	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛)) → (𝑧 < (𝑛 − 𝑙) → (𝐶‘(𝑛 − 𝑙)) = (0g‘𝑅)))
78	77	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑙 ≤ 𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → (𝑧 < (𝑛 − 𝑙) → (𝐶‘(𝑛 − 𝑙)) = (0g‘𝑅)))
79	62, 78	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝑙 ≤ 𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → (𝐶‘(𝑛 − 𝑙)) = (0g‘𝑅))
80	79	oveq2d 7447	. . . . . . . . . . . 12 ⊢ ((𝑙 ≤ 𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))) = ((𝐴‘𝑙) ∗ (0g‘𝑅)))
81		simplr1 1216	. . . . . . . . . . . . . . 15 ⊢ ((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
82	81	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ ((((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛)) → 𝑅 ∈ Ring)
83	82	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑙 ≤ 𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → 𝑅 ∈ Ring)
84		simplr2 1217	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝐾 ∈ 𝐵)
85	84	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → 𝐾 ∈ 𝐵)
86	85, 29	anim12i 613	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛)) → (𝐾 ∈ 𝐵 ∧ 𝑙 ∈ ℕ0))
87	86	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑙 ≤ 𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → (𝐾 ∈ 𝐵 ∧ 𝑙 ∈ ℕ0))
88		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑅) = (Base‘𝑅)
89	3, 2, 1, 88	coe1fvalcl 22214	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑙 ∈ ℕ0) → (𝐴‘𝑙) ∈ (Base‘𝑅))
90	87, 89	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑙 ≤ 𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → (𝐴‘𝑙) ∈ (Base‘𝑅))
91		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑅) = (0g‘𝑅)
92	88, 8, 91	ringrz 20291	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ (𝐴‘𝑙) ∈ (Base‘𝑅)) → ((𝐴‘𝑙) ∗ (0g‘𝑅)) = (0g‘𝑅))
93	83, 90, 92	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝑙 ≤ 𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → ((𝐴‘𝑙) ∗ (0g‘𝑅)) = (0g‘𝑅))
94	80, 93	eqtrd 2777	. . . . . . . . . . 11 ⊢ ((𝑙 ≤ 𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))) = (0g‘𝑅))
95		ltnle 11340	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ ℝ ∧ 𝑙 ∈ ℝ) → (𝑧 < 𝑙 ↔ ¬ 𝑙 ≤ 𝑧))
96	23, 30, 95	syl2an 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ ℕ0 ∧ 𝑙 ∈ ℕ0) → (𝑧 < 𝑙 ↔ ¬ 𝑙 ≤ 𝑧))
97	96	bicomd 223	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ ℕ0 ∧ 𝑙 ∈ ℕ0) → (¬ 𝑙 ≤ 𝑧 ↔ 𝑧 < 𝑙))
98	97	expcom 413	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 ∈ ℕ0 → (𝑧 ∈ ℕ0 → (¬ 𝑙 ≤ 𝑧 ↔ 𝑧 < 𝑙)))
99	98, 29	syl11 33	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℕ0 → (𝑙 ∈ (0...𝑛) → (¬ 𝑙 ≤ 𝑧 ↔ 𝑧 < 𝑙)))
100	99	ad4antr 732	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → (𝑙 ∈ (0...𝑛) → (¬ 𝑙 ≤ 𝑧 ↔ 𝑧 < 𝑙)))
101	100	imp 406	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛)) → (¬ 𝑙 ≤ 𝑧 ↔ 𝑧 < 𝑙))
102		breq2 5147	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑙 → (𝑧 < 𝑥 ↔ 𝑧 < 𝑙))
103		fveqeq2 6915	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑙 → ((𝐴‘𝑥) = (0g‘𝑅) ↔ (𝐴‘𝑙) = (0g‘𝑅)))
104		fveqeq2 6915	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑙 → ((𝐶‘𝑥) = (0g‘𝑅) ↔ (𝐶‘𝑙) = (0g‘𝑅)))
105	103, 104	anbi12d 632	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑙 → (((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)) ↔ ((𝐴‘𝑙) = (0g‘𝑅) ∧ (𝐶‘𝑙) = (0g‘𝑅))))
106	102, 105	imbi12d 344	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑙 → ((𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅))) ↔ (𝑧 < 𝑙 → ((𝐴‘𝑙) = (0g‘𝑅) ∧ (𝐶‘𝑙) = (0g‘𝑅)))))
107	106	rspcva 3620	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑙 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) → (𝑧 < 𝑙 → ((𝐴‘𝑙) = (0g‘𝑅) ∧ (𝐶‘𝑙) = (0g‘𝑅))))
108		simpl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴‘𝑙) = (0g‘𝑅) ∧ (𝐶‘𝑙) = (0g‘𝑅)) → (𝐴‘𝑙) = (0g‘𝑅))
109	107, 108	syl6 35	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑙 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) → (𝑧 < 𝑙 → (𝐴‘𝑙) = (0g‘𝑅)))
110	109	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 ∈ ℕ0 → (∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅))) → (𝑧 < 𝑙 → (𝐴‘𝑙) = (0g‘𝑅))))
111	110, 29	syl11 33	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅))) → (𝑙 ∈ (0...𝑛) → (𝑧 < 𝑙 → (𝐴‘𝑙) = (0g‘𝑅))))
112	111	ad4antlr 733	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → (𝑙 ∈ (0...𝑛) → (𝑧 < 𝑙 → (𝐴‘𝑙) = (0g‘𝑅))))
113	112	imp 406	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛)) → (𝑧 < 𝑙 → (𝐴‘𝑙) = (0g‘𝑅)))
114	101, 113	sylbid 240	. . . . . . . . . . . . . 14 ⊢ ((((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛)) → (¬ 𝑙 ≤ 𝑧 → (𝐴‘𝑙) = (0g‘𝑅)))
115	114	impcom 407	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑙 ≤ 𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → (𝐴‘𝑙) = (0g‘𝑅))
116	115	oveq1d 7446	. . . . . . . . . . . 12 ⊢ ((¬ 𝑙 ≤ 𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))) = ((0g‘𝑅) ∗ (𝐶‘(𝑛 − 𝑙))))
117	82	adantl 481	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑙 ≤ 𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → 𝑅 ∈ Ring)
118		simplr3 1218	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝐿 ∈ 𝐵)
119	118	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → 𝐿 ∈ 𝐵)
120		fznn0sub 13596	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ (0...𝑛) → (𝑛 − 𝑙) ∈ ℕ0)
121	119, 120	anim12i 613	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛)) → (𝐿 ∈ 𝐵 ∧ (𝑛 − 𝑙) ∈ ℕ0))
122	121	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑙 ≤ 𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → (𝐿 ∈ 𝐵 ∧ (𝑛 − 𝑙) ∈ ℕ0))
123	4, 2, 1, 88	coe1fvalcl 22214	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ∈ 𝐵 ∧ (𝑛 − 𝑙) ∈ ℕ0) → (𝐶‘(𝑛 − 𝑙)) ∈ (Base‘𝑅))
124	122, 123	syl 17	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑙 ≤ 𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → (𝐶‘(𝑛 − 𝑙)) ∈ (Base‘𝑅))
125	88, 8, 91	ringlz 20290	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ (𝐶‘(𝑛 − 𝑙)) ∈ (Base‘𝑅)) → ((0g‘𝑅) ∗ (𝐶‘(𝑛 − 𝑙))) = (0g‘𝑅))
126	117, 124, 125	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((¬ 𝑙 ≤ 𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → ((0g‘𝑅) ∗ (𝐶‘(𝑛 − 𝑙))) = (0g‘𝑅))
127	116, 126	eqtrd 2777	. . . . . . . . . . 11 ⊢ ((¬ 𝑙 ≤ 𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))) = (0g‘𝑅))
128	94, 127	pm2.61ian 812	. . . . . . . . . 10 ⊢ ((((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛)) → ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))) = (0g‘𝑅))
129	128	mpteq2dva 5242	. . . . . . . . 9 ⊢ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙)))) = (𝑙 ∈ (0...𝑛) ↦ (0g‘𝑅)))
130	129	oveq2d 7447	. . . . . . . 8 ⊢ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ (0g‘𝑅))))
131		ringmnd 20240	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
132	131	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → 𝑅 ∈ Mnd)
133		ovex 7464	. . . . . . . . . . 11 ⊢ (0...𝑛) ∈ V
134	132, 133	jctir 520	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (𝑅 ∈ Mnd ∧ (0...𝑛) ∈ V))
135	134	ad3antlr 731	. . . . . . . . 9 ⊢ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → (𝑅 ∈ Mnd ∧ (0...𝑛) ∈ V))
136	91	gsumz 18849	. . . . . . . . 9 ⊢ ((𝑅 ∈ Mnd ∧ (0...𝑛) ∈ V) → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ (0g‘𝑅))) = (0g‘𝑅))
137	135, 136	syl 17	. . . . . . . 8 ⊢ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ (0g‘𝑅))) = (0g‘𝑅))
138	130, 137	eqtrd 2777	. . . . . . 7 ⊢ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅))
139	138	ex 412	. . . . . 6 ⊢ ((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((2 · 𝑧) < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅)))
140	139	ralrimiva 3146	. . . . 5 ⊢ (((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) → ∀𝑛 ∈ ℕ0 ((2 · 𝑧) < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅)))
141	16, 20, 140	rspcedvd 3624	. . . 4 ⊢ (((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅)))
142	141	ex 412	. . 3 ⊢ ((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) → ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅))))
143	142	rexlimiva 3147	. 2 ⊢ (∃𝑧 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅))) → ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅))))
144	11, 143	mpcom 38	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  Vcvv 3480   class class class wbr 5143   ↦ cmpt 5225  ‘cfv 6561  (class class class)co 7431  ℂcc 11153  ℝcr 11154  0cc0 11155   · cmul 11160   < clt 11295   ≤ cle 11296   − cmin 11492  2c2 12321  ℕ₀cn0 12526  ...cfz 13547  Basecbs 17247  ._rcmulr 17298   ·_𝑠 cvsca 17301  0_gc0g 17484   Σ_g cgsu 17485  Mndcmnd 18747  ._gcmg 19085  mulGrpcmgp 20137  Ringcrg 20230  var₁cv1 22177  Poly₁cpl1 22178  coe₁cco1 22179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-seq 14043  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-tset 17316  df-ple 17317  df-0g 17486  df-gsum 17487  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-psr 21929  df-mpl 21931  df-opsr 21933  df-psr1 22181  df-ply1 22183  df-coe1 22184

This theorem is referenced by:  ply1mulgsumlem3  48305  ply1mulgsumlem4  48306