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Theorem ply1mulgsumlem2 42776
Description: Lemma 2 for ply1mulgsum 42779. (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 42775 . 2 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑧 ∈ ℕ0𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))))
12 2nn0 11556 . . . . . . . 8 2 ∈ ℕ0
1312a1i 11 . . . . . . 7 (𝑧 ∈ ℕ0 → 2 ∈ ℕ0)
14 id 22 . . . . . . 7 (𝑧 ∈ ℕ0𝑧 ∈ ℕ0)
1513, 14nn0mulcld 11602 . . . . . 6 (𝑧 ∈ ℕ0 → (2 · 𝑧) ∈ ℕ0)
1615ad2antrr 717 . . . . 5 (((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) → (2 · 𝑧) ∈ ℕ0)
17 breq1 4811 . . . . . . . 8 (𝑠 = (2 · 𝑧) → (𝑠 < 𝑛 ↔ (2 · 𝑧) < 𝑛))
1817imbi1d 332 . . . . . . 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 3132 . . . . . 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 473 . . . . 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 11345 . . . . . . . . . . . . . . . . . . . . . . . . 25 2 ∈ ℝ
2221a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 ∈ ℕ0 → 2 ∈ ℝ)
23 nn0re 11547 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 ∈ ℕ0𝑧 ∈ ℝ)
2422, 23remulcld 10323 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ ℕ0 → (2 · 𝑧) ∈ ℝ)
2524ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (2 · 𝑧) ∈ ℝ)
26 nn0re 11547 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ ℕ0𝑛 ∈ ℝ)
2726adantl 473 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) → 𝑛 ∈ ℝ)
2827adantr 472 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑛 ∈ ℝ)
29 elfznn0 12639 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑙 ∈ (0...𝑛) → 𝑙 ∈ ℕ0)
30 nn0re 11547 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑙 ∈ ℕ0𝑙 ∈ ℝ)
3129, 30syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 ∈ (0...𝑛) → 𝑙 ∈ ℝ)
3231adantl 473 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑙 ∈ ℝ)
3325, 28, 32ltsub1d 10889 . . . . . . . . . . . . . . . . . . . . 21 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) < 𝑛 ↔ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)))
3423ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑧 ∈ ℝ)
3532, 34, 25lesub2d 10888 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑙𝑧 ↔ ((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙)))
3635adantr 472 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → (𝑙𝑧 ↔ ((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙)))
3724, 23resubcld 10711 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ ℕ0 → ((2 · 𝑧) − 𝑧) ∈ ℝ)
3837ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑧) ∈ ℝ)
3924adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) → (2 · 𝑧) ∈ ℝ)
40 resubcl 10598 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((2 · 𝑧) ∈ ℝ ∧ 𝑙 ∈ ℝ) → ((2 · 𝑧) − 𝑙) ∈ ℝ)
4139, 31, 40syl2an 589 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑙) ∈ ℝ)
42 resubcl 10598 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑛 ∈ ℝ ∧ 𝑙 ∈ ℝ) → (𝑛𝑙) ∈ ℝ)
4327, 31, 42syl2an 589 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑛𝑙) ∈ ℝ)
44 lelttr 10381 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1490 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → ((2 · 𝑧) − 𝑧) < (𝑛𝑙)))
46 nn0cn 11548 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 ∈ ℕ0𝑧 ∈ ℂ)
47 2txmxeqx 11418 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 ∈ ℂ → ((2 · 𝑧) − 𝑧) = 𝑧)
4846, 47syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ ℕ0 → ((2 · 𝑧) − 𝑧) = 𝑧)
4948ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑧) = 𝑧)
5049breq1d 4818 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (((2 · 𝑧) − 𝑧) < (𝑛𝑙) ↔ 𝑧 < (𝑛𝑙)))
5145, 50sylibd 230 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → 𝑧 < (𝑛𝑙)))
5251expcomd 406 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (((2 · 𝑧) − 𝑙) < (𝑛𝑙) → (((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) → 𝑧 < (𝑛𝑙))))
5352imp 395 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → (((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) → 𝑧 < (𝑛𝑙)))
5436, 53sylbid 231 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → (𝑙𝑧𝑧 < (𝑛𝑙)))
5554ex 401 . . . . . . . . . . . . . . . . . . . . 21 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (((2 · 𝑧) − 𝑙) < (𝑛𝑙) → (𝑙𝑧𝑧 < (𝑛𝑙))))
5633, 55sylbid 231 . . . . . . . . . . . . . . . . . . . 20 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) < 𝑛 → (𝑙𝑧𝑧 < (𝑛𝑙))))
5756ex 401 . . . . . . . . . . . . . . . . . . 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 401 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℕ0 → (𝑛 ∈ ℕ0 → ((2 · 𝑧) < 𝑛 → (𝑙 ∈ (0...𝑛) → (𝑙𝑧𝑧 < (𝑛𝑙))))))
6059ad2antrr 717 . . . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . . . 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 396 . . . . . . . . . . . . . 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 12653 . . . . . . . . . . . . . . . . . . 19 (𝑙 ∈ (0...𝑛) → (𝑛𝑙) ∈ (0...𝑛))
64 elfznn0 12639 . . . . . . . . . . . . . . . . . . 19 ((𝑛𝑙) ∈ (0...𝑛) → (𝑛𝑙) ∈ ℕ0)
65 breq2 4812 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = (𝑛𝑙) → (𝑧 < 𝑥𝑧 < (𝑛𝑙)))
66 fveqeq2 6383 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = (𝑛𝑙) → ((𝐴𝑥) = (0g𝑅) ↔ (𝐴‘(𝑛𝑙)) = (0g𝑅)))
67 fveqeq2 6383 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = (𝑛𝑙) → ((𝐶𝑥) = (0g𝑅) ↔ (𝐶‘(𝑛𝑙)) = (0g𝑅)))
6866, 67anbi12d 624 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = (𝑛𝑙) → (((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)) ↔ ((𝐴‘(𝑛𝑙)) = (0g𝑅) ∧ (𝐶‘(𝑛𝑙)) = (0g𝑅))))
6965, 68imbi12d 335 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = (𝑛𝑙) → ((𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) ↔ (𝑧 < (𝑛𝑙) → ((𝐴‘(𝑛𝑙)) = (0g𝑅) ∧ (𝐶‘(𝑛𝑙)) = (0g𝑅)))))
7069rspcva 3458 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛𝑙) ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) → (𝑧 < (𝑛𝑙) → ((𝐴‘(𝑛𝑙)) = (0g𝑅) ∧ (𝐶‘(𝑛𝑙)) = (0g𝑅))))
71 simpr 477 . . . . . . . . . . . . . . . . . . . . 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 401 . . . . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . 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 395 . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . 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 6857 . . . . . . . . . . . 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 1275 . . . . . . . . . . . . . . 15 ((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
8281ad2antrr 717 . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . 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 1277 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝐾𝐵)
8584adantr 472 . . . . . . . . . . . . . . . 16 (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → 𝐾𝐵)
8685, 29anim12i 606 . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . 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 2764 . . . . . . . . . . . . . . 15 (Base‘𝑅) = (Base‘𝑅)
893, 2, 1, 88coe1fvalcl 19854 . . . . . . . . . . . . . 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 2764 . . . . . . . . . . . . . 14 (0g𝑅) = (0g𝑅)
9288, 8, 91ringrz 18854 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ (𝐴𝑙) ∈ (Base‘𝑅)) → ((𝐴𝑙) (0g𝑅)) = (0g𝑅))
9383, 90, 92syl2anc 579 . . . . . . . . . . . 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 2798 . . . . . . . . . . 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 10370 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧 ∈ ℝ ∧ 𝑙 ∈ ℝ) → (𝑧 < 𝑙 ↔ ¬ 𝑙𝑧))
9623, 30, 95syl2an 589 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 ∈ ℕ0𝑙 ∈ ℕ0) → (𝑧 < 𝑙 ↔ ¬ 𝑙𝑧))
9796bicomd 214 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ ℕ0𝑙 ∈ ℕ0) → (¬ 𝑙𝑧𝑧 < 𝑙))
9897expcom 402 . . . . . . . . . . . . . . . . . 18 (𝑙 ∈ ℕ0 → (𝑧 ∈ ℕ0 → (¬ 𝑙𝑧𝑧 < 𝑙)))
9998, 29syl11 33 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℕ0 → (𝑙 ∈ (0...𝑛) → (¬ 𝑙𝑧𝑧 < 𝑙)))
10099ad4antr 724 . . . . . . . . . . . . . . . 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 395 . . . . . . . . . . . . . . 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 4812 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑙 → (𝑧 < 𝑥𝑧 < 𝑙))
103 fveqeq2 6383 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑙 → ((𝐴𝑥) = (0g𝑅) ↔ (𝐴𝑙) = (0g𝑅)))
104 fveqeq2 6383 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑙 → ((𝐶𝑥) = (0g𝑅) ↔ (𝐶𝑙) = (0g𝑅)))
105103, 104anbi12d 624 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑙 → (((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)) ↔ ((𝐴𝑙) = (0g𝑅) ∧ (𝐶𝑙) = (0g𝑅))))
106102, 105imbi12d 335 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑙 → ((𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) ↔ (𝑧 < 𝑙 → ((𝐴𝑙) = (0g𝑅) ∧ (𝐶𝑙) = (0g𝑅)))))
107106rspcva 3458 . . . . . . . . . . . . . . . . . . . 20 ((𝑙 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) → (𝑧 < 𝑙 → ((𝐴𝑙) = (0g𝑅) ∧ (𝐶𝑙) = (0g𝑅))))
108 simpl 474 . . . . . . . . . . . . . . . . . . . 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 401 . . . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . 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 395 . . . . . . . . . . . . . . 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 231 . . . . . . . . . . . . . 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 396 . . . . . . . . . . . . 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 6856 . . . . . . . . . . . 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 473 . . . . . . . . . . . . 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 1279 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝐿𝐵)
119118adantr 472 . . . . . . . . . . . . . . . 16 (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → 𝐿𝐵)
120 fznn0sub 12579 . . . . . . . . . . . . . . . 16 (𝑙 ∈ (0...𝑛) → (𝑛𝑙) ∈ ℕ0)
121119, 120anim12i 606 . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . 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 19854 . . . . . . . . . . . . . 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 18853 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ (𝐶‘(𝑛𝑙)) ∈ (Base‘𝑅)) → ((0g𝑅) (𝐶‘(𝑛𝑙))) = (0g𝑅))
126117, 124, 125syl2anc 579 . . . . . . . . . . . 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 2798 . . . . . . . . . . 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 846 . . . . . . . . . 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 4902 . . . . . . . . 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 6857 . . . . . . . 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 18822 . . . . . . . . . . . 12 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
1321313ad2ant1 1163 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → 𝑅 ∈ Mnd)
133 ovex 6873 . . . . . . . . . . 11 (0...𝑛) ∈ V
134132, 133jctir 516 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → (𝑅 ∈ Mnd ∧ (0...𝑛) ∈ V))
135134ad3antlr 722 . . . . . . . . 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 17641 . . . . . . . . 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 2798 . . . . . . 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 401 . . . . . 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 3112 . . . . 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 3467 . . . 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 401 . . 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 3174 . 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3054  wrex 3055  Vcvv 3349   class class class wbr 4808  cmpt 4887  cfv 6067  (class class class)co 6841  cc 10186  cr 10187  0cc0 10188   · cmul 10193   < clt 10327  cle 10328  cmin 10519  2c2 11326  0cn0 11537  ...cfz 12532  Basecbs 16131  .rcmulr 16216   ·𝑠 cvsca 16219  0gc0g 16367   Σg cgsu 16368  Mndcmnd 17561  .gcmg 17808  mulGrpcmgp 18755  Ringcrg 18813  var1cv1 19818  Poly1cpl1 19819  coe1cco1 19820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146  ax-cnex 10244  ax-resscn 10245  ax-1cn 10246  ax-icn 10247  ax-addcl 10248  ax-addrcl 10249  ax-mulcl 10250  ax-mulrcl 10251  ax-mulcom 10252  ax-addass 10253  ax-mulass 10254  ax-distr 10255  ax-i2m1 10256  ax-1ne0 10257  ax-1rid 10258  ax-rnegex 10259  ax-rrecex 10260  ax-cnre 10261  ax-pre-lttri 10262  ax-pre-lttrn 10263  ax-pre-ltadd 10264  ax-pre-mulgt0 10265
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-int 4633  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-riota 6802  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-of 7094  df-om 7263  df-1st 7365  df-2nd 7366  df-supp 7497  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-1o 7763  df-oadd 7767  df-er 7946  df-map 8061  df-en 8160  df-dom 8161  df-sdom 8162  df-fin 8163  df-fsupp 8482  df-pnf 10329  df-mnf 10330  df-xr 10331  df-ltxr 10332  df-le 10333  df-sub 10521  df-neg 10522  df-nn 11274  df-2 11334  df-3 11335  df-4 11336  df-5 11337  df-6 11338  df-7 11339  df-8 11340  df-9 11341  df-n0 11538  df-z 11624  df-dec 11740  df-uz 11886  df-fz 12533  df-seq 13008  df-struct 16133  df-ndx 16134  df-slot 16135  df-base 16137  df-sets 16138  df-ress 16139  df-plusg 16228  df-mulr 16229  df-sca 16231  df-vsca 16232  df-tset 16234  df-ple 16235  df-0g 16369  df-gsum 16370  df-mgm 17509  df-sgrp 17551  df-mnd 17562  df-grp 17693  df-minusg 17694  df-mgp 18756  df-ring 18815  df-psr 19629  df-mpl 19631  df-opsr 19633  df-psr1 19822  df-ply1 19824  df-coe1 19825
This theorem is referenced by:  ply1mulgsumlem3  42777  ply1mulgsumlem4  42778
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