Theorem ply1mulgsumlem2 43023

Description: Lemma 2 for ply1mulgsum 43026. (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 43022	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑧 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅))))
12		2nn0 11638	. . . . . . . 8 ⊢ 2 ∈ ℕ0
13	12	a1i 11	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0 → 2 ∈ ℕ0)
14		id 22	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0 → 𝑧 ∈ ℕ0)
15	13, 14	nn0mulcld 11684	. . . . . 6 ⊢ (𝑧 ∈ ℕ0 → (2 · 𝑧) ∈ ℕ0)
16	15	ad2antrr 719	. . . . 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 4877	. . . . . . . 8 ⊢ (𝑠 = (2 · 𝑧) → (𝑠 < 𝑛 ↔ (2 · 𝑧) < 𝑛))
18	17	imbi1d 333	. . . . . . 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 3196	. . . . . 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 475	. . . . 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 11426	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 ∈ ℝ
22	21	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ ℕ0 → 2 ∈ ℝ)
23		nn0re 11629	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ ℕ0 → 𝑧 ∈ ℝ)
24	22, 23	remulcld 10388	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ ℕ0 → (2 · 𝑧) ∈ ℝ)
25	24	ad2antrr 719	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (2 · 𝑧) ∈ ℝ)
26		nn0re 11629	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℝ)
27	26	adantl 475	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℝ)
28	27	adantr 474	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑛 ∈ ℝ)
29		elfznn0 12728	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑙 ∈ (0...𝑛) → 𝑙 ∈ ℕ0)
30		nn0re 11629	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑙 ∈ ℕ0 → 𝑙 ∈ ℝ)
31	29, 30	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 ∈ (0...𝑛) → 𝑙 ∈ ℝ)
32	31	adantl 475	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑙 ∈ ℝ)
33	25, 28, 32	ltsub1d 10962	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) < 𝑛 ↔ ((2 · 𝑧) − 𝑙) < (𝑛 − 𝑙)))
34	23	ad2antrr 719	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑧 ∈ ℝ)
35	32, 34, 25	lesub2d 10961	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑙 ≤ 𝑧 ↔ ((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙)))
36	35	adantr 474	. . . . . . . . . . . . . . . . . . . . . . 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 10783	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ ℕ0 → ((2 · 𝑧) − 𝑧) ∈ ℝ)
38	37	ad2antrr 719	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑧) ∈ ℝ)
39	24	adantr 474	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → (2 · 𝑧) ∈ ℝ)
40		resubcl 10667	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((2 · 𝑧) ∈ ℝ ∧ 𝑙 ∈ ℝ) → ((2 · 𝑧) − 𝑙) ∈ ℝ)
41	39, 31, 40	syl2an 591	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑙) ∈ ℝ)
42		resubcl 10667	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑛 ∈ ℝ ∧ 𝑙 ∈ ℝ) → (𝑛 − 𝑙) ∈ ℝ)
43	27, 31, 42	syl2an 591	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑛 − 𝑙) ∈ ℝ)
44		lelttr 10448	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1496	. . . . . . . . . . . . . . . . . . . . . . . . . 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 11630	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 ∈ ℕ0 → 𝑧 ∈ ℂ)
47		2txmxeqx 11499	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 ∈ ℂ → ((2 · 𝑧) − 𝑧) = 𝑧)
48	46, 47	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ ℕ0 → ((2 · 𝑧) − 𝑧) = 𝑧)
49	48	ad2antrr 719	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑧) = 𝑧)
50	49	breq1d 4884	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (((2 · 𝑧) − 𝑧) < (𝑛 − 𝑙) ↔ 𝑧 < (𝑛 − 𝑙)))
51	45, 50	sylibd 231	. . . . . . . . . . . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . . . . . . . . . . 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 397	. . . . . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ ((2 · 𝑧) − 𝑙) < (𝑛 − 𝑙)) → (𝑙 ≤ 𝑧 → 𝑧 < (𝑛 − 𝑙)))
55	54	ex 403	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (((2 · 𝑧) − 𝑙) < (𝑛 − 𝑙) → (𝑙 ≤ 𝑧 → 𝑧 < (𝑛 − 𝑙))))
56	33, 55	sylbid 232	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) < 𝑛 → (𝑙 ≤ 𝑧 → 𝑧 < (𝑛 − 𝑙))))
57	56	ex 403	. . . . . . . . . . . . . . . . . . 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 403	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℕ0 → (𝑛 ∈ ℕ0 → ((2 · 𝑧) < 𝑛 → (𝑙 ∈ (0...𝑛) → (𝑙 ≤ 𝑧 → 𝑧 < (𝑛 − 𝑙))))))
60	59	ad2antrr 719	. . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . 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 398	. . . . . . . . . . . . . 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 12742	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 ∈ (0...𝑛) → (𝑛 − 𝑙) ∈ (0...𝑛))
64		elfznn0 12728	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 − 𝑙) ∈ (0...𝑛) → (𝑛 − 𝑙) ∈ ℕ0)
65		breq2 4878	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = (𝑛 − 𝑙) → (𝑧 < 𝑥 ↔ 𝑧 < (𝑛 − 𝑙)))
66		fveqeq2 6443	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = (𝑛 − 𝑙) → ((𝐴‘𝑥) = (0g‘𝑅) ↔ (𝐴‘(𝑛 − 𝑙)) = (0g‘𝑅)))
67		fveqeq2 6443	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = (𝑛 − 𝑙) → ((𝐶‘𝑥) = (0g‘𝑅) ↔ (𝐶‘(𝑛 − 𝑙)) = (0g‘𝑅)))
68	66, 67	anbi12d 626	. . . . . . . . . . . . . . . . . . . . . . 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 336	. . . . . . . . . . . . . . . . . . . . . 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 3525	. . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . . 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 403	. . . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . 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 397	. . . . . . . . . . . . . . 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 475	. . . . . . . . . . . . . 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 6922	. . . . . . . . . . . 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 1281	. . . . . . . . . . . . . . 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 719	. . . . . . . . . . . . . 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 475	. . . . . . . . . . . . 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 1283	. . . . . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . . . 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 608	. . . . . . . . . . . . . . 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 475	. . . . . . . . . . . . . 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 2826	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑅) = (Base‘𝑅)
89	3, 2, 1, 88	coe1fvalcl 19943	. . . . . . . . . . . . . 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 2826	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑅) = (0g‘𝑅)
92	88, 8, 91	ringrz 18943	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ (𝐴‘𝑙) ∈ (Base‘𝑅)) → ((𝐴‘𝑙) ∗ (0g‘𝑅)) = (0g‘𝑅))
93	83, 90, 92	syl2anc 581	. . . . . . . . . . . 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 2862	. . . . . . . . . . 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 10437	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ ℝ ∧ 𝑙 ∈ ℝ) → (𝑧 < 𝑙 ↔ ¬ 𝑙 ≤ 𝑧))
96	23, 30, 95	syl2an 591	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ ℕ0 ∧ 𝑙 ∈ ℕ0) → (𝑧 < 𝑙 ↔ ¬ 𝑙 ≤ 𝑧))
97	96	bicomd 215	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ ℕ0 ∧ 𝑙 ∈ ℕ0) → (¬ 𝑙 ≤ 𝑧 ↔ 𝑧 < 𝑙))
98	97	expcom 404	. . . . . . . . . . . . . . . . . 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 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 < 𝑙)))
101	100	imp 397	. . . . . . . . . . . . . . 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 4878	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑙 → (𝑧 < 𝑥 ↔ 𝑧 < 𝑙))
103		fveqeq2 6443	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑙 → ((𝐴‘𝑥) = (0g‘𝑅) ↔ (𝐴‘𝑙) = (0g‘𝑅)))
104		fveqeq2 6443	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑙 → ((𝐶‘𝑥) = (0g‘𝑅) ↔ (𝐶‘𝑙) = (0g‘𝑅)))
105	103, 104	anbi12d 626	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑙 → (((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)) ↔ ((𝐴‘𝑙) = (0g‘𝑅) ∧ (𝐶‘𝑙) = (0g‘𝑅))))
106	102, 105	imbi12d 336	. . . . . . . . . . . . . . . . . . . . 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 3525	. . . . . . . . . . . . . . . . . . . 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 476	. . . . . . . . . . . . . . . . . . . 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 403	. . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . 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 397	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 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 398	. . . . . . . . . . . . 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 6921	. . . . . . . . . . . 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 475	. . . . . . . . . . . . 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 1285	. . . . . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . . . 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 12667	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ (0...𝑛) → (𝑛 − 𝑙) ∈ ℕ0)
121	119, 120	anim12i 608	. . . . . . . . . . . . . . 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 475	. . . . . . . . . . . . . 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 19943	. . . . . . . . . . . . . 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 18942	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ (𝐶‘(𝑛 − 𝑙)) ∈ (Base‘𝑅)) → ((0g‘𝑅) ∗ (𝐶‘(𝑛 − 𝑙))) = (0g‘𝑅))
126	117, 124, 125	syl2anc 581	. . . . . . . . . . . 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 2862	. . . . . . . . . . 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 848	. . . . . . . . . 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 4968	. . . . . . . . 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 6922	. . . . . . . 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 18911	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
132	131	3ad2ant1 1169	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → 𝑅 ∈ Mnd)
133		ovex 6938	. . . . . . . . . . 11 ⊢ (0...𝑛) ∈ V
134	132, 133	jctir 518	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (𝑅 ∈ Mnd ∧ (0...𝑛) ∈ V))
135	134	ad3antlr 724	. . . . . . . . 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 17728	. . . . . . . . 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 2862	. . . . . . 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 403	. . . . . 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 3176	. . . . 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 3534	. . . 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 403	. . 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 3238	. 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 198   ∧ wa 386   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166  ∀wral 3118  ∃wrex 3119  Vcvv 3415   class class class wbr 4874   ↦ cmpt 4953  ‘cfv 6124  (class class class)co 6906  ℂcc 10251  ℝcr 10252  0cc0 10253   · cmul 10258   < clt 10392   ≤ cle 10393   − cmin 10586  2c2 11407  ℕ₀cn0 11619  ...cfz 12620  Basecbs 16223  ._rcmulr 16307   ·_𝑠 cvsca 16310  0_gc0g 16454   Σ_g cgsu 16455  Mndcmnd 17648  ._gcmg 17895  mulGrpcmgp 18844  Ringcrg 18902  var₁cv1 19907  Poly₁cpl1 19908  coe₁cco1 19909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-cnex 10309  ax-resscn 10310  ax-1cn 10311  ax-icn 10312  ax-addcl 10313  ax-addrcl 10314  ax-mulcl 10315  ax-mulrcl 10316  ax-mulcom 10317  ax-addass 10318  ax-mulass 10319  ax-distr 10320  ax-i2m1 10321  ax-1ne0 10322  ax-1rid 10323  ax-rnegex 10324  ax-rrecex 10325  ax-cnre 10326  ax-pre-lttri 10327  ax-pre-lttrn 10328  ax-pre-ltadd 10329  ax-pre-mulgt0 10330

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-nel 3104  df-ral 3123  df-rex 3124  df-reu 3125  df-rmo 3126  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-int 4699  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-riota 6867  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-of 7158  df-om 7328  df-1st 7429  df-2nd 7430  df-supp 7561  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-1o 7827  df-oadd 7831  df-er 8010  df-map 8125  df-en 8224  df-dom 8225  df-sdom 8226  df-fin 8227  df-fsupp 8546  df-pnf 10394  df-mnf 10395  df-xr 10396  df-ltxr 10397  df-le 10398  df-sub 10588  df-neg 10589  df-nn 11352  df-2 11415  df-3 11416  df-4 11417  df-5 11418  df-6 11419  df-7 11420  df-8 11421  df-9 11422  df-n0 11620  df-z 11706  df-dec 11823  df-uz 11970  df-fz 12621  df-seq 13097  df-struct 16225  df-ndx 16226  df-slot 16227  df-base 16229  df-sets 16230  df-ress 16231  df-plusg 16319  df-mulr 16320  df-sca 16322  df-vsca 16323  df-tset 16325  df-ple 16326  df-0g 16456  df-gsum 16457  df-mgm 17596  df-sgrp 17638  df-mnd 17649  df-grp 17780  df-minusg 17781  df-mgp 18845  df-ring 18904  df-psr 19718  df-mpl 19720  df-opsr 19722  df-psr1 19911  df-ply1 19913  df-coe1 19914

This theorem is referenced by:  ply1mulgsumlem3  43024  ply1mulgsumlem4  43025