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Theorem ply1mulgsumlem2 48232
Description: Lemma 2 for ply1mulgsum 48235. (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 48231 . 2 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑧 ∈ ℕ0𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))))
12 2nn0 12540 . . . . . . . 8 2 ∈ ℕ0
1312a1i 11 . . . . . . 7 (𝑧 ∈ ℕ0 → 2 ∈ ℕ0)
14 id 22 . . . . . . 7 (𝑧 ∈ ℕ0𝑧 ∈ ℕ0)
1513, 14nn0mulcld 12589 . . . . . 6 (𝑧 ∈ ℕ0 → (2 · 𝑧) ∈ ℕ0)
1615ad2antrr 726 . . . . 5 (((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) → (2 · 𝑧) ∈ ℕ0)
17 breq1 5150 . . . . . . . 8 (𝑠 = (2 · 𝑧) → (𝑠 < 𝑛 ↔ (2 · 𝑧) < 𝑛))
1817imbi1d 341 . . . . . . 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 3175 . . . . . 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 481 . . . . 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 12337 . . . . . . . . . . . . . . . . . . . . . . . . 25 2 ∈ ℝ
2221a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 ∈ ℕ0 → 2 ∈ ℝ)
23 nn0re 12532 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 ∈ ℕ0𝑧 ∈ ℝ)
2422, 23remulcld 11288 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ ℕ0 → (2 · 𝑧) ∈ ℝ)
2524ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (2 · 𝑧) ∈ ℝ)
26 nn0re 12532 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ ℕ0𝑛 ∈ ℝ)
2726adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) → 𝑛 ∈ ℝ)
2827adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑛 ∈ ℝ)
29 elfznn0 13656 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑙 ∈ (0...𝑛) → 𝑙 ∈ ℕ0)
30 nn0re 12532 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑙 ∈ ℕ0𝑙 ∈ ℝ)
3129, 30syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 ∈ (0...𝑛) → 𝑙 ∈ ℝ)
3231adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑙 ∈ ℝ)
3325, 28, 32ltsub1d 11869 . . . . . . . . . . . . . . . . . . . . 21 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) < 𝑛 ↔ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)))
3423ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑧 ∈ ℝ)
3532, 34, 25lesub2d 11868 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑙𝑧 ↔ ((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙)))
3635adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → (𝑙𝑧 ↔ ((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙)))
3724, 23resubcld 11688 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ ℕ0 → ((2 · 𝑧) − 𝑧) ∈ ℝ)
3837ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑧) ∈ ℝ)
3924adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) → (2 · 𝑧) ∈ ℝ)
40 resubcl 11570 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((2 · 𝑧) ∈ ℝ ∧ 𝑙 ∈ ℝ) → ((2 · 𝑧) − 𝑙) ∈ ℝ)
4139, 31, 40syl2an 596 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑙) ∈ ℝ)
42 resubcl 11570 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑛 ∈ ℝ ∧ 𝑙 ∈ ℝ) → (𝑛𝑙) ∈ ℝ)
4327, 31, 42syl2an 596 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑛𝑙) ∈ ℝ)
44 lelttr 11348 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → ((2 · 𝑧) − 𝑧) < (𝑛𝑙)))
46 nn0cn 12533 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 ∈ ℕ0𝑧 ∈ ℂ)
47 2txmxeqx 12403 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 ∈ ℂ → ((2 · 𝑧) − 𝑧) = 𝑧)
4846, 47syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ ℕ0 → ((2 · 𝑧) − 𝑧) = 𝑧)
4948ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑧) = 𝑧)
5049breq1d 5157 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (((2 · 𝑧) − 𝑧) < (𝑛𝑙) ↔ 𝑧 < (𝑛𝑙)))
5145, 50sylibd 239 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → 𝑧 < (𝑛𝑙)))
5251expcomd 416 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (((2 · 𝑧) − 𝑙) < (𝑛𝑙) → (((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) → 𝑧 < (𝑛𝑙))))
5352imp 406 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → (((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) → 𝑧 < (𝑛𝑙)))
5436, 53sylbid 240 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → (𝑙𝑧𝑧 < (𝑛𝑙)))
5554ex 412 . . . . . . . . . . . . . . . . . . . . 21 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (((2 · 𝑧) − 𝑙) < (𝑛𝑙) → (𝑙𝑧𝑧 < (𝑛𝑙))))
5633, 55sylbid 240 . . . . . . . . . . . . . . . . . . . 20 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) < 𝑛 → (𝑙𝑧𝑧 < (𝑛𝑙))))
5756ex 412 . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℕ0 → (𝑛 ∈ ℕ0 → ((2 · 𝑧) < 𝑛 → (𝑙 ∈ (0...𝑛) → (𝑙𝑧𝑧 < (𝑛𝑙))))))
6059ad2antrr 726 . . . . . . . . . . . . . . . 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 425 . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . 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 13671 . . . . . . . . . . . . . . . . . . 19 (𝑙 ∈ (0...𝑛) → (𝑛𝑙) ∈ (0...𝑛))
64 elfznn0 13656 . . . . . . . . . . . . . . . . . . 19 ((𝑛𝑙) ∈ (0...𝑛) → (𝑛𝑙) ∈ ℕ0)
65 breq2 5151 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = (𝑛𝑙) → (𝑧 < 𝑥𝑧 < (𝑛𝑙)))
66 fveqeq2 6915 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = (𝑛𝑙) → ((𝐴𝑥) = (0g𝑅) ↔ (𝐴‘(𝑛𝑙)) = (0g𝑅)))
67 fveqeq2 6915 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = (𝑛𝑙) → ((𝐶𝑥) = (0g𝑅) ↔ (𝐶‘(𝑛𝑙)) = (0g𝑅)))
6866, 67anbi12d 632 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = (𝑛𝑙) → (((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)) ↔ ((𝐴‘(𝑛𝑙)) = (0g𝑅) ∧ (𝐶‘(𝑛𝑙)) = (0g𝑅))))
6965, 68imbi12d 344 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = (𝑛𝑙) → ((𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) ↔ (𝑧 < (𝑛𝑙) → ((𝐴‘(𝑛𝑙)) = (0g𝑅) ∧ (𝐶‘(𝑛𝑙)) = (0g𝑅)))))
7069rspcva 3619 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛𝑙) ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) → (𝑧 < (𝑛𝑙) → ((𝐴‘(𝑛𝑙)) = (0g𝑅) ∧ (𝐶‘(𝑛𝑙)) = (0g𝑅))))
71 simpr 484 . . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . 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 1214 . . . . . . . . . . . . . . 15 ((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
8281ad2antrr 726 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 1215 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝐾𝐵)
8584adantr 480 . . . . . . . . . . . . . . . 16 (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → 𝐾𝐵)
8685, 29anim12i 613 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 2734 . . . . . . . . . . . . . . 15 (Base‘𝑅) = (Base‘𝑅)
893, 2, 1, 88coe1fvalcl 22229 . . . . . . . . . . . . . 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 2734 . . . . . . . . . . . . . 14 (0g𝑅) = (0g𝑅)
9288, 8, 91ringrz 20307 . . . . . . . . . . . . 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 2774 . . . . . . . . . . 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 11337 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧 ∈ ℝ ∧ 𝑙 ∈ ℝ) → (𝑧 < 𝑙 ↔ ¬ 𝑙𝑧))
9623, 30, 95syl2an 596 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 ∈ ℕ0𝑙 ∈ ℕ0) → (𝑧 < 𝑙 ↔ ¬ 𝑙𝑧))
9796bicomd 223 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ ℕ0𝑙 ∈ ℕ0) → (¬ 𝑙𝑧𝑧 < 𝑙))
9897expcom 413 . . . . . . . . . . . . . . . . . 18 (𝑙 ∈ ℕ0 → (𝑧 ∈ ℕ0 → (¬ 𝑙𝑧𝑧 < 𝑙)))
9998, 29syl11 33 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℕ0 → (𝑙 ∈ (0...𝑛) → (¬ 𝑙𝑧𝑧 < 𝑙)))
10099ad4antr 732 . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . 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 5151 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑙 → (𝑧 < 𝑥𝑧 < 𝑙))
103 fveqeq2 6915 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑙 → ((𝐴𝑥) = (0g𝑅) ↔ (𝐴𝑙) = (0g𝑅)))
104 fveqeq2 6915 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑙 → ((𝐶𝑥) = (0g𝑅) ↔ (𝐶𝑙) = (0g𝑅)))
105103, 104anbi12d 632 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑙 → (((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)) ↔ ((𝐴𝑙) = (0g𝑅) ∧ (𝐶𝑙) = (0g𝑅))))
106102, 105imbi12d 344 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑙 → ((𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) ↔ (𝑧 < 𝑙 → ((𝐴𝑙) = (0g𝑅) ∧ (𝐶𝑙) = (0g𝑅)))))
107106rspcva 3619 . . . . . . . . . . . . . . . . . . . 20 ((𝑙 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) → (𝑧 < 𝑙 → ((𝐴𝑙) = (0g𝑅) ∧ (𝐶𝑙) = (0g𝑅))))
108 simpl 482 . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . 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 240 . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . 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 7445 . . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 1216 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝐿𝐵)
119118adantr 480 . . . . . . . . . . . . . . . 16 (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → 𝐿𝐵)
120 fznn0sub 13592 . . . . . . . . . . . . . . . 16 (𝑙 ∈ (0...𝑛) → (𝑛𝑙) ∈ ℕ0)
121119, 120anim12i 613 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 22229 . . . . . . . . . . . . . 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 20306 . . . . . . . . . . . . 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 2774 . . . . . . . . . . 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 812 . . . . . . . . . 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 5247 . . . . . . . . 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 7446 . . . . . . . 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 20260 . . . . . . . . . . . 12 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
1321313ad2ant1 1132 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → 𝑅 ∈ Mnd)
133 ovex 7463 . . . . . . . . . . 11 (0...𝑛) ∈ V
134132, 133jctir 520 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → (𝑅 ∈ Mnd ∧ (0...𝑛) ∈ V))
135134ad3antlr 731 . . . . . . . . 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 18861 . . . . . . . . 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 2774 . . . . . . 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 412 . . . . . 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 3143 . . . . 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 3623 . . . 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 412 . . 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 3144 . 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 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  wrex 3067  Vcvv 3477   class class class wbr 5147  cmpt 5230  cfv 6562  (class class class)co 7430  cc 11150  cr 11151  0cc0 11152   · cmul 11157   < clt 11292  cle 11293  cmin 11489  2c2 12318  0cn0 12523  ...cfz 13543  Basecbs 17244  .rcmulr 17298   ·𝑠 cvsca 17301  0gc0g 17485   Σg cgsu 17486  Mndcmnd 18759  .gcmg 19097  mulGrpcmgp 20151  Ringcrg 20250  var1cv1 22192  Poly1cpl1 22193  coe1cco1 22194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-seq 14039  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-tset 17316  df-ple 17317  df-0g 17487  df-gsum 17488  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-grp 18966  df-minusg 18967  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-psr 21946  df-mpl 21948  df-opsr 21950  df-psr1 22196  df-ply1 22198  df-coe1 22199
This theorem is referenced by:  ply1mulgsumlem3  48233  ply1mulgsumlem4  48234
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