Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ply1mulgsumlem2 Structured version   Visualization version   GIF version

Theorem ply1mulgsumlem2 44270
Description: Lemma 2 for ply1mulgsum 44273. (Contributed by AV, 19-Oct-2019.)
Hypotheses
Ref Expression
ply1mulgsum.p 𝑃 = (Poly1𝑅)
ply1mulgsum.b 𝐵 = (Base‘𝑃)
ply1mulgsum.a 𝐴 = (coe1𝐾)
ply1mulgsum.c 𝐶 = (coe1𝐿)
ply1mulgsum.x 𝑋 = (var1𝑅)
ply1mulgsum.pm × = (.r𝑃)
ply1mulgsum.sm · = ( ·𝑠𝑃)
ply1mulgsum.rm = (.r𝑅)
ply1mulgsum.m 𝑀 = (mulGrp‘𝑃)
ply1mulgsum.e = (.g𝑀)
Assertion
Ref Expression
ply1mulgsumlem2 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴𝑙) (𝐶‘(𝑛𝑙))))) = (0g𝑅)))
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.
StepHypRef Expression
1 ply1mulgsum.p . . 3 𝑃 = (Poly1𝑅)
2 ply1mulgsum.b . . 3 𝐵 = (Base‘𝑃)
3 ply1mulgsum.a . . 3 𝐴 = (coe1𝐾)
4 ply1mulgsum.c . . 3 𝐶 = (coe1𝐿)
5 ply1mulgsum.x . . 3 𝑋 = (var1𝑅)
6 ply1mulgsum.pm . . 3 × = (.r𝑃)
7 ply1mulgsum.sm . . 3 · = ( ·𝑠𝑃)
8 ply1mulgsum.rm . . 3 = (.r𝑅)
9 ply1mulgsum.m . . 3 𝑀 = (mulGrp‘𝑃)
10 ply1mulgsum.e . . 3 = (.g𝑀)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10ply1mulgsumlem1 44269 . 2 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑧 ∈ ℕ0𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))))
12 2nn0 11906 . . . . . . . 8 2 ∈ ℕ0
1312a1i 11 . . . . . . 7 (𝑧 ∈ ℕ0 → 2 ∈ ℕ0)
14 id 22 . . . . . . 7 (𝑧 ∈ ℕ0𝑧 ∈ ℕ0)
1513, 14nn0mulcld 11952 . . . . . 6 (𝑧 ∈ ℕ0 → (2 · 𝑧) ∈ ℕ0)
1615ad2antrr 722 . . . . 5 (((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) → (2 · 𝑧) ∈ ℕ0)
17 breq1 5065 . . . . . . . 8 (𝑠 = (2 · 𝑧) → (𝑠 < 𝑛 ↔ (2 · 𝑧) < 𝑛))
1817imbi1d 343 . . . . . . 7 (𝑠 = (2 · 𝑧) → ((𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴𝑙) (𝐶‘(𝑛𝑙))))) = (0g𝑅)) ↔ ((2 · 𝑧) < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴𝑙) (𝐶‘(𝑛𝑙))))) = (0g𝑅))))
1918ralbidv 3201 . . . . . 6 (𝑠 = (2 · 𝑧) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴𝑙) (𝐶‘(𝑛𝑙))))) = (0g𝑅)) ↔ ∀𝑛 ∈ ℕ0 ((2 · 𝑧) < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴𝑙) (𝐶‘(𝑛𝑙))))) = (0g𝑅))))
2019adantl 482 . . . . 5 ((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑠 = (2 · 𝑧)) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴𝑙) (𝐶‘(𝑛𝑙))))) = (0g𝑅)) ↔ ∀𝑛 ∈ ℕ0 ((2 · 𝑧) < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴𝑙) (𝐶‘(𝑛𝑙))))) = (0g𝑅))))
21 2re 11703 . . . . . . . . . . . . . . . . . . . . . . . . 25 2 ∈ ℝ
2221a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 ∈ ℕ0 → 2 ∈ ℝ)
23 nn0re 11898 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 ∈ ℕ0𝑧 ∈ ℝ)
2422, 23remulcld 10663 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ ℕ0 → (2 · 𝑧) ∈ ℝ)
2524ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (2 · 𝑧) ∈ ℝ)
26 nn0re 11898 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ ℕ0𝑛 ∈ ℝ)
2726adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) → 𝑛 ∈ ℝ)
2827adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑛 ∈ ℝ)
29 elfznn0 12993 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑙 ∈ (0...𝑛) → 𝑙 ∈ ℕ0)
30 nn0re 11898 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑙 ∈ ℕ0𝑙 ∈ ℝ)
3129, 30syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 ∈ (0...𝑛) → 𝑙 ∈ ℝ)
3231adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑙 ∈ ℝ)
3325, 28, 32ltsub1d 11241 . . . . . . . . . . . . . . . . . . . . 21 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) < 𝑛 ↔ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)))
3423ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑧 ∈ ℝ)
3532, 34, 25lesub2d 11240 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑙𝑧 ↔ ((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙)))
3635adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → (𝑙𝑧 ↔ ((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙)))
3724, 23resubcld 11060 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ ℕ0 → ((2 · 𝑧) − 𝑧) ∈ ℝ)
3837ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑧) ∈ ℝ)
3924adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) → (2 · 𝑧) ∈ ℝ)
40 resubcl 10942 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((2 · 𝑧) ∈ ℝ ∧ 𝑙 ∈ ℝ) → ((2 · 𝑧) − 𝑙) ∈ ℝ)
4139, 31, 40syl2an 595 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑙) ∈ ℝ)
42 resubcl 10942 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑛 ∈ ℝ ∧ 𝑙 ∈ ℝ) → (𝑛𝑙) ∈ ℝ)
4327, 31, 42syl2an 595 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑛𝑙) ∈ ℝ)
44 lelttr 10723 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((2 · 𝑧) − 𝑧) ∈ ℝ ∧ ((2 · 𝑧) − 𝑙) ∈ ℝ ∧ (𝑛𝑙) ∈ ℝ) → ((((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → ((2 · 𝑧) − 𝑧) < (𝑛𝑙)))
4538, 41, 43, 44syl3anc 1365 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → ((2 · 𝑧) − 𝑧) < (𝑛𝑙)))
46 nn0cn 11899 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 ∈ ℕ0𝑧 ∈ ℂ)
47 2txmxeqx 11769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 ∈ ℂ → ((2 · 𝑧) − 𝑧) = 𝑧)
4846, 47syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ ℕ0 → ((2 · 𝑧) − 𝑧) = 𝑧)
4948ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑧) = 𝑧)
5049breq1d 5072 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (((2 · 𝑧) − 𝑧) < (𝑛𝑙) ↔ 𝑧 < (𝑛𝑙)))
5145, 50sylibd 240 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → 𝑧 < (𝑛𝑙)))
5251expcomd 417 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (((2 · 𝑧) − 𝑙) < (𝑛𝑙) → (((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) → 𝑧 < (𝑛𝑙))))
5352imp 407 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → (((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) → 𝑧 < (𝑛𝑙)))
5436, 53sylbid 241 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → (𝑙𝑧𝑧 < (𝑛𝑙)))
5554ex 413 . . . . . . . . . . . . . . . . . . . . 21 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (((2 · 𝑧) − 𝑙) < (𝑛𝑙) → (𝑙𝑧𝑧 < (𝑛𝑙))))
5633, 55sylbid 241 . . . . . . . . . . . . . . . . . . . 20 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) < 𝑛 → (𝑙𝑧𝑧 < (𝑛𝑙))))
5756ex 413 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) → (𝑙 ∈ (0...𝑛) → ((2 · 𝑧) < 𝑛 → (𝑙𝑧𝑧 < (𝑛𝑙)))))
5857com23 86 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) → ((2 · 𝑧) < 𝑛 → (𝑙 ∈ (0...𝑛) → (𝑙𝑧𝑧 < (𝑛𝑙)))))
5958ex 413 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℕ0 → (𝑛 ∈ ℕ0 → ((2 · 𝑧) < 𝑛 → (𝑙 ∈ (0...𝑛) → (𝑙𝑧𝑧 < (𝑛𝑙))))))
6059ad2antrr 722 . . . . . . . . . . . . . . . 16 (((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) → (𝑛 ∈ ℕ0 → ((2 · 𝑧) < 𝑛 → (𝑙 ∈ (0...𝑛) → (𝑙𝑧𝑧 < (𝑛𝑙))))))
6160imp41 426 . . . . . . . . . . . . . . 15 ((((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛)) → (𝑙𝑧𝑧 < (𝑛𝑙)))
6261impcom 408 . . . . . . . . . . . . . 14 ((𝑙𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → 𝑧 < (𝑛𝑙))
63 fznn0sub2 13007 . . . . . . . . . . . . . . . . . . 19 (𝑙 ∈ (0...𝑛) → (𝑛𝑙) ∈ (0...𝑛))
64 elfznn0 12993 . . . . . . . . . . . . . . . . . . 19 ((𝑛𝑙) ∈ (0...𝑛) → (𝑛𝑙) ∈ ℕ0)
65 breq2 5066 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = (𝑛𝑙) → (𝑧 < 𝑥𝑧 < (𝑛𝑙)))
66 fveqeq2 6675 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = (𝑛𝑙) → ((𝐴𝑥) = (0g𝑅) ↔ (𝐴‘(𝑛𝑙)) = (0g𝑅)))
67 fveqeq2 6675 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = (𝑛𝑙) → ((𝐶𝑥) = (0g𝑅) ↔ (𝐶‘(𝑛𝑙)) = (0g𝑅)))
6866, 67anbi12d 630 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = (𝑛𝑙) → (((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)) ↔ ((𝐴‘(𝑛𝑙)) = (0g𝑅) ∧ (𝐶‘(𝑛𝑙)) = (0g𝑅))))
6965, 68imbi12d 346 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = (𝑛𝑙) → ((𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) ↔ (𝑧 < (𝑛𝑙) → ((𝐴‘(𝑛𝑙)) = (0g𝑅) ∧ (𝐶‘(𝑛𝑙)) = (0g𝑅)))))
7069rspcva 3624 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛𝑙) ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) → (𝑧 < (𝑛𝑙) → ((𝐴‘(𝑛𝑙)) = (0g𝑅) ∧ (𝐶‘(𝑛𝑙)) = (0g𝑅))))
71 simpr 485 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴‘(𝑛𝑙)) = (0g𝑅) ∧ (𝐶‘(𝑛𝑙)) = (0g𝑅)) → (𝐶‘(𝑛𝑙)) = (0g𝑅))
7270, 71syl6 35 . . . . . . . . . . . . . . . . . . . 20 (((𝑛𝑙) ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) → (𝑧 < (𝑛𝑙) → (𝐶‘(𝑛𝑙)) = (0g𝑅)))
7372ex 413 . . . . . . . . . . . . . . . . . . 19 ((𝑛𝑙) ∈ ℕ0 → (∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) → (𝑧 < (𝑛𝑙) → (𝐶‘(𝑛𝑙)) = (0g𝑅))))
7463, 64, 733syl 18 . . . . . . . . . . . . . . . . . 18 (𝑙 ∈ (0...𝑛) → (∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) → (𝑧 < (𝑛𝑙) → (𝐶‘(𝑛𝑙)) = (0g𝑅))))
7574com12 32 . . . . . . . . . . . . . . . . 17 (∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) → (𝑙 ∈ (0...𝑛) → (𝑧 < (𝑛𝑙) → (𝐶‘(𝑛𝑙)) = (0g𝑅))))
7675ad4antlr 729 . . . . . . . . . . . . . . . 16 (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → (𝑙 ∈ (0...𝑛) → (𝑧 < (𝑛𝑙) → (𝐶‘(𝑛𝑙)) = (0g𝑅))))
7776imp 407 . . . . . . . . . . . . . . 15 ((((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛)) → (𝑧 < (𝑛𝑙) → (𝐶‘(𝑛𝑙)) = (0g𝑅)))
7877adantl 482 . . . . . . . . . . . . . 14 ((𝑙𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → (𝑧 < (𝑛𝑙) → (𝐶‘(𝑛𝑙)) = (0g𝑅)))
7962, 78mpd 15 . . . . . . . . . . . . 13 ((𝑙𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → (𝐶‘(𝑛𝑙)) = (0g𝑅))
8079oveq2d 7167 . . . . . . . . . . . 12 ((𝑙𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → ((𝐴𝑙) (𝐶‘(𝑛𝑙))) = ((𝐴𝑙) (0g𝑅)))
81 simplr1 1209 . . . . . . . . . . . . . . 15 ((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
8281ad2antrr 722 . . . . . . . . . . . . . 14 ((((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛)) → 𝑅 ∈ Ring)
8382adantl 482 . . . . . . . . . . . . 13 ((𝑙𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → 𝑅 ∈ Ring)
84 simplr2 1210 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝐾𝐵)
8584adantr 481 . . . . . . . . . . . . . . . 16 (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → 𝐾𝐵)
8685, 29anim12i 612 . . . . . . . . . . . . . . 15 ((((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛)) → (𝐾𝐵𝑙 ∈ ℕ0))
8786adantl 482 . . . . . . . . . . . . . 14 ((𝑙𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → (𝐾𝐵𝑙 ∈ ℕ0))
88 eqid 2824 . . . . . . . . . . . . . . 15 (Base‘𝑅) = (Base‘𝑅)
893, 2, 1, 88coe1fvalcl 20297 . . . . . . . . . . . . . 14 ((𝐾𝐵𝑙 ∈ ℕ0) → (𝐴𝑙) ∈ (Base‘𝑅))
9087, 89syl 17 . . . . . . . . . . . . 13 ((𝑙𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → (𝐴𝑙) ∈ (Base‘𝑅))
91 eqid 2824 . . . . . . . . . . . . . 14 (0g𝑅) = (0g𝑅)
9288, 8, 91ringrz 19260 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ (𝐴𝑙) ∈ (Base‘𝑅)) → ((𝐴𝑙) (0g𝑅)) = (0g𝑅))
9383, 90, 92syl2anc 584 . . . . . . . . . . . 12 ((𝑙𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → ((𝐴𝑙) (0g𝑅)) = (0g𝑅))
9480, 93eqtrd 2860 . . . . . . . . . . 11 ((𝑙𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → ((𝐴𝑙) (𝐶‘(𝑛𝑙))) = (0g𝑅))
95 ltnle 10712 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧 ∈ ℝ ∧ 𝑙 ∈ ℝ) → (𝑧 < 𝑙 ↔ ¬ 𝑙𝑧))
9623, 30, 95syl2an 595 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 ∈ ℕ0𝑙 ∈ ℕ0) → (𝑧 < 𝑙 ↔ ¬ 𝑙𝑧))
9796bicomd 224 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ ℕ0𝑙 ∈ ℕ0) → (¬ 𝑙𝑧𝑧 < 𝑙))
9897expcom 414 . . . . . . . . . . . . . . . . . 18 (𝑙 ∈ ℕ0 → (𝑧 ∈ ℕ0 → (¬ 𝑙𝑧𝑧 < 𝑙)))
9998, 29syl11 33 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℕ0 → (𝑙 ∈ (0...𝑛) → (¬ 𝑙𝑧𝑧 < 𝑙)))
10099ad4antr 728 . . . . . . . . . . . . . . . 16 (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → (𝑙 ∈ (0...𝑛) → (¬ 𝑙𝑧𝑧 < 𝑙)))
101100imp 407 . . . . . . . . . . . . . . 15 ((((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛)) → (¬ 𝑙𝑧𝑧 < 𝑙))
102 breq2 5066 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑙 → (𝑧 < 𝑥𝑧 < 𝑙))
103 fveqeq2 6675 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑙 → ((𝐴𝑥) = (0g𝑅) ↔ (𝐴𝑙) = (0g𝑅)))
104 fveqeq2 6675 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑙 → ((𝐶𝑥) = (0g𝑅) ↔ (𝐶𝑙) = (0g𝑅)))
105103, 104anbi12d 630 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑙 → (((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)) ↔ ((𝐴𝑙) = (0g𝑅) ∧ (𝐶𝑙) = (0g𝑅))))
106102, 105imbi12d 346 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑙 → ((𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) ↔ (𝑧 < 𝑙 → ((𝐴𝑙) = (0g𝑅) ∧ (𝐶𝑙) = (0g𝑅)))))
107106rspcva 3624 . . . . . . . . . . . . . . . . . . . 20 ((𝑙 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) → (𝑧 < 𝑙 → ((𝐴𝑙) = (0g𝑅) ∧ (𝐶𝑙) = (0g𝑅))))
108 simpl 483 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑙) = (0g𝑅) ∧ (𝐶𝑙) = (0g𝑅)) → (𝐴𝑙) = (0g𝑅))
109107, 108syl6 35 . . . . . . . . . . . . . . . . . . 19 ((𝑙 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) → (𝑧 < 𝑙 → (𝐴𝑙) = (0g𝑅)))
110109ex 413 . . . . . . . . . . . . . . . . . 18 (𝑙 ∈ ℕ0 → (∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) → (𝑧 < 𝑙 → (𝐴𝑙) = (0g𝑅))))
111110, 29syl11 33 . . . . . . . . . . . . . . . . 17 (∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) → (𝑙 ∈ (0...𝑛) → (𝑧 < 𝑙 → (𝐴𝑙) = (0g𝑅))))
112111ad4antlr 729 . . . . . . . . . . . . . . . 16 (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → (𝑙 ∈ (0...𝑛) → (𝑧 < 𝑙 → (𝐴𝑙) = (0g𝑅))))
113112imp 407 . . . . . . . . . . . . . . 15 ((((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛)) → (𝑧 < 𝑙 → (𝐴𝑙) = (0g𝑅)))
114101, 113sylbid 241 . . . . . . . . . . . . . 14 ((((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛)) → (¬ 𝑙𝑧 → (𝐴𝑙) = (0g𝑅)))
115114impcom 408 . . . . . . . . . . . . 13 ((¬ 𝑙𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → (𝐴𝑙) = (0g𝑅))
116115oveq1d 7166 . . . . . . . . . . . 12 ((¬ 𝑙𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → ((𝐴𝑙) (𝐶‘(𝑛𝑙))) = ((0g𝑅) (𝐶‘(𝑛𝑙))))
11782adantl 482 . . . . . . . . . . . . 13 ((¬ 𝑙𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → 𝑅 ∈ Ring)
118 simplr3 1211 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝐿𝐵)
119118adantr 481 . . . . . . . . . . . . . . . 16 (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → 𝐿𝐵)
120 fznn0sub 12932 . . . . . . . . . . . . . . . 16 (𝑙 ∈ (0...𝑛) → (𝑛𝑙) ∈ ℕ0)
121119, 120anim12i 612 . . . . . . . . . . . . . . 15 ((((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛)) → (𝐿𝐵 ∧ (𝑛𝑙) ∈ ℕ0))
122121adantl 482 . . . . . . . . . . . . . 14 ((¬ 𝑙𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → (𝐿𝐵 ∧ (𝑛𝑙) ∈ ℕ0))
1234, 2, 1, 88coe1fvalcl 20297 . . . . . . . . . . . . . 14 ((𝐿𝐵 ∧ (𝑛𝑙) ∈ ℕ0) → (𝐶‘(𝑛𝑙)) ∈ (Base‘𝑅))
124122, 123syl 17 . . . . . . . . . . . . 13 ((¬ 𝑙𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → (𝐶‘(𝑛𝑙)) ∈ (Base‘𝑅))
12588, 8, 91ringlz 19259 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ (𝐶‘(𝑛𝑙)) ∈ (Base‘𝑅)) → ((0g𝑅) (𝐶‘(𝑛𝑙))) = (0g𝑅))
126117, 124, 125syl2anc 584 . . . . . . . . . . . 12 ((¬ 𝑙𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → ((0g𝑅) (𝐶‘(𝑛𝑙))) = (0g𝑅))
127116, 126eqtrd 2860 . . . . . . . . . . 11 ((¬ 𝑙𝑧 ∧ (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛))) → ((𝐴𝑙) (𝐶‘(𝑛𝑙))) = (0g𝑅))
12894, 127pm2.61ian 808 . . . . . . . . . 10 ((((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛)) → ((𝐴𝑙) (𝐶‘(𝑛𝑙))) = (0g𝑅))
129128mpteq2dva 5157 . . . . . . . . 9 (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → (𝑙 ∈ (0...𝑛) ↦ ((𝐴𝑙) (𝐶‘(𝑛𝑙)))) = (𝑙 ∈ (0...𝑛) ↦ (0g𝑅)))
130129oveq2d 7167 . . . . . . . 8 (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴𝑙) (𝐶‘(𝑛𝑙))))) = (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ (0g𝑅))))
131 ringmnd 19228 . . . . . . . . . . . 12 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
1321313ad2ant1 1127 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → 𝑅 ∈ Mnd)
133 ovex 7184 . . . . . . . . . . 11 (0...𝑛) ∈ V
134132, 133jctir 521 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → (𝑅 ∈ Mnd ∧ (0...𝑛) ∈ V))
135134ad3antlr 727 . . . . . . . . 9 (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → (𝑅 ∈ Mnd ∧ (0...𝑛) ∈ V))
13691gsumz 17985 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ (0...𝑛) ∈ V) → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ (0g𝑅))) = (0g𝑅))
137135, 136syl 17 . . . . . . . 8 (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ (0g𝑅))) = (0g𝑅))
138130, 137eqtrd 2860 . . . . . . 7 (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴𝑙) (𝐶‘(𝑛𝑙))))) = (0g𝑅))
139138ex 413 . . . . . 6 ((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → ((2 · 𝑧) < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴𝑙) (𝐶‘(𝑛𝑙))))) = (0g𝑅)))
140139ralrimiva 3186 . . . . 5 (((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) → ∀𝑛 ∈ ℕ0 ((2 · 𝑧) < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴𝑙) (𝐶‘(𝑛𝑙))))) = (0g𝑅)))
14116, 20, 140rspcedvd 3629 . . . 4 (((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴𝑙) (𝐶‘(𝑛𝑙))))) = (0g𝑅)))
142141ex 413 . . 3 ((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) → ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴𝑙) (𝐶‘(𝑛𝑙))))) = (0g𝑅))))
143142rexlimiva 3285 . 2 (∃𝑧 ∈ ℕ0𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) → ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴𝑙) (𝐶‘(𝑛𝑙))))) = (0g𝑅))))
14411, 143mpcom 38 1 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴𝑙) (𝐶‘(𝑛𝑙))))) = (0g𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2106  wral 3142  wrex 3143  Vcvv 3499   class class class wbr 5062  cmpt 5142  cfv 6351  (class class class)co 7151  cc 10527  cr 10528  0cc0 10529   · cmul 10534   < clt 10667  cle 10668  cmin 10862  2c2 11684  0cn0 11889  ...cfz 12885  Basecbs 16475  .rcmulr 16558   ·𝑠 cvsca 16561  0gc0g 16705   Σg cgsu 16706  Mndcmnd 17902  .gcmg 18156  mulGrpcmgp 19161  Ringcrg 19219  var1cv1 20261  Poly1cpl1 20262  coe1cco1 20263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7402  df-om 7572  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8826  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12886  df-seq 13363  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-tset 16576  df-ple 16577  df-0g 16707  df-gsum 16708  df-mgm 17844  df-sgrp 17892  df-mnd 17903  df-grp 18038  df-minusg 18039  df-mgp 19162  df-ring 19221  df-psr 20057  df-mpl 20059  df-opsr 20061  df-psr1 20265  df-ply1 20267  df-coe1 20268
This theorem is referenced by:  ply1mulgsumlem3  44271  ply1mulgsumlem4  44272
  Copyright terms: Public domain W3C validator