Theorem ply1mulgsumlem2 42175

Description: Lemma 2 for ply1mulgsum 42178. (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 42174	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑧 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅))))
12		2nn0 11309	. . . . . . . 8 ⊢ 2 ∈ ℕ0
13	12	a1i 11	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0 → 2 ∈ ℕ0)
14		id 22	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0 → 𝑧 ∈ ℕ0)
15	13, 14	nn0mulcld 11356	. . . . . 6 ⊢ (𝑧 ∈ ℕ0 → (2 · 𝑧) ∈ ℕ0)
16	15	ad2antrr 762	. . . . 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 4656	. . . . . . . 8 ⊢ (𝑠 = (2 · 𝑧) → (𝑠 < 𝑛 ↔ (2 · 𝑧) < 𝑛))
18	17	imbi1d 331	. . . . . . 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 2986	. . . . . 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 482	. . . . 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 11090	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 ∈ ℝ
22	21	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ ℕ0 → 2 ∈ ℝ)
23		nn0re 11301	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ ℕ0 → 𝑧 ∈ ℝ)
24	22, 23	remulcld 10070	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ ℕ0 → (2 · 𝑧) ∈ ℝ)
25	24	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (2 · 𝑧) ∈ ℝ)
26		nn0re 11301	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℝ)
27	26	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℝ)
28	27	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑛 ∈ ℝ)
29		elfznn0 12433	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑙 ∈ (0...𝑛) → 𝑙 ∈ ℕ0)
30		nn0re 11301	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑙 ∈ ℕ0 → 𝑙 ∈ ℝ)
31	29, 30	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 ∈ (0...𝑛) → 𝑙 ∈ ℝ)
32	31	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑙 ∈ ℝ)
33	25, 28, 32	ltsub1d 10636	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) < 𝑛 ↔ ((2 · 𝑧) − 𝑙) < (𝑛 − 𝑙)))
34	23	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑧 ∈ ℝ)
35	32, 34, 25	lesub2d 10635	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑙 ≤ 𝑧 ↔ ((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙)))
36	35	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 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 10458	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ ℕ0 → ((2 · 𝑧) − 𝑧) ∈ ℝ)
38	37	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑧) ∈ ℝ)
39	24	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → (2 · 𝑧) ∈ ℝ)
40		resubcl 10345	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((2 · 𝑧) ∈ ℝ ∧ 𝑙 ∈ ℝ) → ((2 · 𝑧) − 𝑙) ∈ ℝ)
41	39, 31, 40	syl2an 494	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑙) ∈ ℝ)
42		resubcl 10345	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑛 ∈ ℝ ∧ 𝑙 ∈ ℝ) → (𝑛 − 𝑙) ∈ ℝ)
43	27, 31, 42	syl2an 494	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑛 − 𝑙) ∈ ℝ)
44		lelttr 10128	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . . . . . . . . . . . . . . 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 11302	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 ∈ ℕ0 → 𝑧 ∈ ℂ)
47		2txmxeqx 11149	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 ∈ ℂ → ((2 · 𝑧) − 𝑧) = 𝑧)
48	46, 47	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ ℕ0 → ((2 · 𝑧) − 𝑧) = 𝑧)
49	48	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑧) = 𝑧)
50	49	breq1d 4663	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (((2 · 𝑧) − 𝑧) < (𝑛 − 𝑙) ↔ 𝑧 < (𝑛 − 𝑙)))
51	45, 50	sylibd 229	. . . . . . . . . . . . . . . . . . . . . . . . 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 454	. . . . . . . . . . . . . . . . . . . . . . . 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 445	. . . . . . . . . . . . . . . . . . . . . . 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 230	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ ((2 · 𝑧) − 𝑙) < (𝑛 − 𝑙)) → (𝑙 ≤ 𝑧 → 𝑧 < (𝑛 − 𝑙)))
55	54	ex 450	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (((2 · 𝑧) − 𝑙) < (𝑛 − 𝑙) → (𝑙 ≤ 𝑧 → 𝑧 < (𝑛 − 𝑙))))
56	33, 55	sylbid 230	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑧 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) < 𝑛 → (𝑙 ≤ 𝑧 → 𝑧 < (𝑛 − 𝑙))))
57	56	ex 450	. . . . . . . . . . . . . . . . . . 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 450	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℕ0 → (𝑛 ∈ ℕ0 → ((2 · 𝑧) < 𝑛 → (𝑙 ∈ (0...𝑛) → (𝑙 ≤ 𝑧 → 𝑧 < (𝑛 − 𝑙))))))
60	59	ad2antrr 762	. . . . . . . . . . . . . . . 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 619	. . . . . . . . . . . . . . 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 446	. . . . . . . . . . . . . 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 12446	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 ∈ (0...𝑛) → (𝑛 − 𝑙) ∈ (0...𝑛))
64		elfznn0 12433	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 − 𝑙) ∈ (0...𝑛) → (𝑛 − 𝑙) ∈ ℕ0)
65		breq2 4657	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = (𝑛 − 𝑙) → (𝑧 < 𝑥 ↔ 𝑧 < (𝑛 − 𝑙)))
66		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = (𝑛 − 𝑙) → (𝐴‘𝑥) = (𝐴‘(𝑛 − 𝑙)))
67	66	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = (𝑛 − 𝑙) → ((𝐴‘𝑥) = (0g‘𝑅) ↔ (𝐴‘(𝑛 − 𝑙)) = (0g‘𝑅)))
68		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = (𝑛 − 𝑙) → (𝐶‘𝑥) = (𝐶‘(𝑛 − 𝑙)))
69	68	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = (𝑛 − 𝑙) → ((𝐶‘𝑥) = (0g‘𝑅) ↔ (𝐶‘(𝑛 − 𝑙)) = (0g‘𝑅)))
70	67, 69	anbi12d 747	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = (𝑛 − 𝑙) → (((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)) ↔ ((𝐴‘(𝑛 − 𝑙)) = (0g‘𝑅) ∧ (𝐶‘(𝑛 − 𝑙)) = (0g‘𝑅))))
71	65, 70	imbi12d 334	. . . . . . . . . . . . . . . . . . . . . 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)))))
72	71	rspcva 3307	. . . . . . . . . . . . . . . . . . . . 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))))
73		simpr 477	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴‘(𝑛 − 𝑙)) = (0g‘𝑅) ∧ (𝐶‘(𝑛 − 𝑙)) = (0g‘𝑅)) → (𝐶‘(𝑛 − 𝑙)) = (0g‘𝑅))
74	72, 73	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)))
75	74	ex 450	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 − 𝑙) ∈ ℕ0 → (∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅))) → (𝑧 < (𝑛 − 𝑙) → (𝐶‘(𝑛 − 𝑙)) = (0g‘𝑅))))
76	63, 64, 75	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))))
77	76	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))))
78	77	ad4antlr 769	. . . . . . . . . . . . . . . 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))))
79	78	imp 445	. . . . . . . . . . . . . . 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)))
80	79	adantl 482	. . . . . . . . . . . . . 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)))
81	62, 80	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))
82	81	oveq2d 6666	. . . . . . . . . . . 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)))
83		simplr1 1103	. . . . . . . . . . . . . . 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)
84	83	ad2antrr 762	. . . . . . . . . . . . . 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)
85	84	adantl 482	. . . . . . . . . . . . 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)
86		simplr2 1104	. . . . . . . . . . . . . . . . 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)
87	86	adantr 481	. . . . . . . . . . . . . . . 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)
88	87, 29	anim12i 590	. . . . . . . . . . . . . . 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))
89	88	adantl 482	. . . . . . . . . . . . . 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))
90		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑅) = (Base‘𝑅)
91	3, 2, 1, 90	coe1fvalcl 19582	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑙 ∈ ℕ0) → (𝐴‘𝑙) ∈ (Base‘𝑅))
92	89, 91	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))
93		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑅) = (0g‘𝑅)
94	90, 8, 93	ringrz 18588	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ (𝐴‘𝑙) ∈ (Base‘𝑅)) → ((𝐴‘𝑙) ∗ (0g‘𝑅)) = (0g‘𝑅))
95	85, 92, 94	syl2anc 693	. . . . . . . . . . . 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))
96	82, 95	eqtrd 2656	. . . . . . . . . . 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))
97		ltnle 10117	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ ℝ ∧ 𝑙 ∈ ℝ) → (𝑧 < 𝑙 ↔ ¬ 𝑙 ≤ 𝑧))
98	23, 30, 97	syl2an 494	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ ℕ0 ∧ 𝑙 ∈ ℕ0) → (𝑧 < 𝑙 ↔ ¬ 𝑙 ≤ 𝑧))
99	98	bicomd 213	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ ℕ0 ∧ 𝑙 ∈ ℕ0) → (¬ 𝑙 ≤ 𝑧 ↔ 𝑧 < 𝑙))
100	99	expcom 451	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 ∈ ℕ0 → (𝑧 ∈ ℕ0 → (¬ 𝑙 ≤ 𝑧 ↔ 𝑧 < 𝑙)))
101	100, 29	syl11 33	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℕ0 → (𝑙 ∈ (0...𝑛) → (¬ 𝑙 ≤ 𝑧 ↔ 𝑧 < 𝑙)))
102	101	ad4antr 768	. . . . . . . . . . . . . . . 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)))
103	102	imp 445	. . . . . . . . . . . . . . 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))
104		breq2 4657	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑙 → (𝑧 < 𝑥 ↔ 𝑧 < 𝑙))
105		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑙 → (𝐴‘𝑥) = (𝐴‘𝑙))
106	105	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑙 → ((𝐴‘𝑥) = (0g‘𝑅) ↔ (𝐴‘𝑙) = (0g‘𝑅)))
107		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑙 → (𝐶‘𝑥) = (𝐶‘𝑙))
108	107	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑙 → ((𝐶‘𝑥) = (0g‘𝑅) ↔ (𝐶‘𝑙) = (0g‘𝑅)))
109	106, 108	anbi12d 747	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑙 → (((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)) ↔ ((𝐴‘𝑙) = (0g‘𝑅) ∧ (𝐶‘𝑙) = (0g‘𝑅))))
110	104, 109	imbi12d 334	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑙 → ((𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅))) ↔ (𝑧 < 𝑙 → ((𝐴‘𝑙) = (0g‘𝑅) ∧ (𝐶‘𝑙) = (0g‘𝑅)))))
111	110	rspcva 3307	. . . . . . . . . . . . . . . . . . . 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))))
112		simpl 473	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴‘𝑙) = (0g‘𝑅) ∧ (𝐶‘𝑙) = (0g‘𝑅)) → (𝐴‘𝑙) = (0g‘𝑅))
113	111, 112	syl6 35	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑙 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅)))) → (𝑧 < 𝑙 → (𝐴‘𝑙) = (0g‘𝑅)))
114	113	ex 450	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 ∈ ℕ0 → (∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴‘𝑥) = (0g‘𝑅) ∧ (𝐶‘𝑥) = (0g‘𝑅))) → (𝑧 < 𝑙 → (𝐴‘𝑙) = (0g‘𝑅))))
115	114, 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))))
116	115	ad4antlr 769	. . . . . . . . . . . . . . . 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))))
117	116	imp 445	. . . . . . . . . . . . . . 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)))
118	103, 117	sylbid 230	. . . . . . . . . . . . . 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)))
119	118	impcom 446	. . . . . . . . . . . . 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))
120	119	oveq1d 6665	. . . . . . . . . . . 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))))
121	84	adantl 482	. . . . . . . . . . . . 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)
122		simplr3 1105	. . . . . . . . . . . . . . . . 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)
123	122	adantr 481	. . . . . . . . . . . . . . . 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)
124		fznn0sub 12373	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ (0...𝑛) → (𝑛 − 𝑙) ∈ ℕ0)
125	123, 124	anim12i 590	. . . . . . . . . . . . . . 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))
126	125	adantl 482	. . . . . . . . . . . . . 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))
127	4, 2, 1, 90	coe1fvalcl 19582	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ∈ 𝐵 ∧ (𝑛 − 𝑙) ∈ ℕ0) → (𝐶‘(𝑛 − 𝑙)) ∈ (Base‘𝑅))
128	126, 127	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))
129	90, 8, 93	ringlz 18587	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ (𝐶‘(𝑛 − 𝑙)) ∈ (Base‘𝑅)) → ((0g‘𝑅) ∗ (𝐶‘(𝑛 − 𝑙))) = (0g‘𝑅))
130	121, 128, 129	syl2anc 693	. . . . . . . . . . . 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))
131	120, 130	eqtrd 2656	. . . . . . . . . . 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))
132	96, 131	pm2.61ian 831	. . . . . . . . . 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))
133	132	mpteq2dva 4744	. . . . . . . . 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)))
134	133	oveq2d 6666	. . . . . . . 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))))
135		ringmnd 18556	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
136	135	3ad2ant1 1082	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → 𝑅 ∈ Mnd)
137		ovex 6678	. . . . . . . . . . 11 ⊢ (0...𝑛) ∈ V
138	136, 137	jctir 561	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (𝑅 ∈ Mnd ∧ (0...𝑛) ∈ V))
139	138	ad3antlr 767	. . . . . . . . 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))
140	93	gsumz 17374	. . . . . . . . 9 ⊢ ((𝑅 ∈ Mnd ∧ (0...𝑛) ∈ V) → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ (0g‘𝑅))) = (0g‘𝑅))
141	139, 140	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))
142	134, 141	eqtrd 2656	. . . . . . 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))
143	142	ex 450	. . . . . 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)))
144	143	ralrimiva 2966	. . . . 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)))
145	16, 20, 144	rspcedvd 3317	. . . 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)))
146	145	ex 450	. . 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))))
147	146	rexlimiva 3028	. 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))))
148	11, 147	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   class class class wbr 4653   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  ℝcr 9935  0cc0 9936   · cmul 9941   < clt 10074   ≤ cle 10075   − cmin 10266  2c2 11070  ℕ₀cn0 11292  ...cfz 12326  Basecbs 15857  ._rcmulr 15942   ·_𝑠 cvsca 15945  0_gc0g 16100   Σ_g cgsu 16101  Mndcmnd 17294  ._gcmg 17540  mulGrpcmgp 18489  Ringcrg 18547  var₁cv1 19546  Poly₁cpl1 19547  coe₁cco1 19548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-seq 12802  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-tset 15960  df-ple 15961  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-mgp 18490  df-ring 18549  df-psr 19356  df-mpl 19358  df-opsr 19360  df-psr1 19550  df-ply1 19552  df-coe1 19553

This theorem is referenced by:  ply1mulgsumlem3  42176  ply1mulgsumlem4  42177