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Theorem ply1mulgsumlem2 41484
Description: Lemma 2 for ply1mulgsum 41487. (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 41483 . 2 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑧 ∈ ℕ0𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))))
12 2nn0 11260 . . . . . . . 8 2 ∈ ℕ0
1312a1i 11 . . . . . . 7 (𝑧 ∈ ℕ0 → 2 ∈ ℕ0)
14 id 22 . . . . . . 7 (𝑧 ∈ ℕ0𝑧 ∈ ℕ0)
1513, 14nn0mulcld 11307 . . . . . 6 (𝑧 ∈ ℕ0 → (2 · 𝑧) ∈ ℕ0)
1615ad2antrr 761 . . . . 5 (((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) → (2 · 𝑧) ∈ ℕ0)
17 breq1 4621 . . . . . . . 8 (𝑠 = (2 · 𝑧) → (𝑠 < 𝑛 ↔ (2 · 𝑧) < 𝑛))
1817imbi1d 331 . . . . . . 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 2981 . . . . . 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 11041 . . . . . . . . . . . . . . . . . . . . . . . . 25 2 ∈ ℝ
2221a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 ∈ ℕ0 → 2 ∈ ℝ)
23 nn0re 11252 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 ∈ ℕ0𝑧 ∈ ℝ)
2422, 23remulcld 10021 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ ℕ0 → (2 · 𝑧) ∈ ℝ)
2524ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (2 · 𝑧) ∈ ℝ)
26 nn0re 11252 . . . . . . . . . . . . . . . . . . . . . . . 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 12381 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑙 ∈ (0...𝑛) → 𝑙 ∈ ℕ0)
30 nn0re 11252 . . . . . . . . . . . . . . . . . . . . . . . 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 10587 . . . . . . . . . . . . . . . . . . . . 21 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) < 𝑛 ↔ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)))
3423ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑧 ∈ ℝ)
3532, 34, 25lesub2d 10586 . . . . . . . . . . . . . . . . . . . . . . . 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 10409 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ ℕ0 → ((2 · 𝑧) − 𝑧) ∈ ℝ)
3837ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑧) ∈ ℝ)
3924adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) → (2 · 𝑧) ∈ ℝ)
40 resubcl 10296 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((2 · 𝑧) ∈ ℝ ∧ 𝑙 ∈ ℝ) → ((2 · 𝑧) − 𝑙) ∈ ℝ)
4139, 31, 40syl2an 494 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑙) ∈ ℝ)
42 resubcl 10296 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑛 ∈ ℝ ∧ 𝑙 ∈ ℝ) → (𝑛𝑙) ∈ ℝ)
4327, 31, 42syl2an 494 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑛𝑙) ∈ ℝ)
44 lelttr 10079 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1323 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → ((2 · 𝑧) − 𝑧) < (𝑛𝑙)))
46 nn0cn 11253 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 ∈ ℕ0𝑧 ∈ ℂ)
47 2txmxeqx 11100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 ∈ ℂ → ((2 · 𝑧) − 𝑧) = 𝑧)
4846, 47syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ ℕ0 → ((2 · 𝑧) − 𝑧) = 𝑧)
4948ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑧) = 𝑧)
5049breq1d 4628 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (((2 · 𝑧) − 𝑧) < (𝑛𝑙) ↔ 𝑧 < (𝑛𝑙)))
5145, 50sylibd 229 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → 𝑧 < (𝑛𝑙)))
5251expcomd 454 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (((2 · 𝑧) − 𝑙) < (𝑛𝑙) → (((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) → 𝑧 < (𝑛𝑙))))
5352imp 445 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → (((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) → 𝑧 < (𝑛𝑙)))
5436, 53sylbid 230 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → (𝑙𝑧𝑧 < (𝑛𝑙)))
5554ex 450 . . . . . . . . . . . . . . . . . . . . 21 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (((2 · 𝑧) − 𝑙) < (𝑛𝑙) → (𝑙𝑧𝑧 < (𝑛𝑙))))
5633, 55sylbid 230 . . . . . . . . . . . . . . . . . . . 20 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) < 𝑛 → (𝑙𝑧𝑧 < (𝑛𝑙))))
5756ex 450 . . . . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℕ0 → (𝑛 ∈ ℕ0 → ((2 · 𝑧) < 𝑛 → (𝑙 ∈ (0...𝑛) → (𝑙𝑧𝑧 < (𝑛𝑙))))))
6059ad2antrr 761 . . . . . . . . . . . . . . . 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 618 . . . . . . . . . . . . . . 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 446 . . . . . . . . . . . . . 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 12394 . . . . . . . . . . . . . . . . . . 19 (𝑙 ∈ (0...𝑛) → (𝑛𝑙) ∈ (0...𝑛))
64 elfznn0 12381 . . . . . . . . . . . . . . . . . . 19 ((𝑛𝑙) ∈ (0...𝑛) → (𝑛𝑙) ∈ ℕ0)
65 breq2 4622 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = (𝑛𝑙) → (𝑧 < 𝑥𝑧 < (𝑛𝑙)))
66 fveq2 6153 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = (𝑛𝑙) → (𝐴𝑥) = (𝐴‘(𝑛𝑙)))
6766eqeq1d 2623 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = (𝑛𝑙) → ((𝐴𝑥) = (0g𝑅) ↔ (𝐴‘(𝑛𝑙)) = (0g𝑅)))
68 fveq2 6153 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = (𝑛𝑙) → (𝐶𝑥) = (𝐶‘(𝑛𝑙)))
6968eqeq1d 2623 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = (𝑛𝑙) → ((𝐶𝑥) = (0g𝑅) ↔ (𝐶‘(𝑛𝑙)) = (0g𝑅)))
7067, 69anbi12d 746 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = (𝑛𝑙) → (((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)) ↔ ((𝐴‘(𝑛𝑙)) = (0g𝑅) ∧ (𝐶‘(𝑛𝑙)) = (0g𝑅))))
7165, 70imbi12d 334 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = (𝑛𝑙) → ((𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) ↔ (𝑧 < (𝑛𝑙) → ((𝐴‘(𝑛𝑙)) = (0g𝑅) ∧ (𝐶‘(𝑛𝑙)) = (0g𝑅)))))
7271rspcva 3296 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛𝑙) ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) → (𝑧 < (𝑛𝑙) → ((𝐴‘(𝑛𝑙)) = (0g𝑅) ∧ (𝐶‘(𝑛𝑙)) = (0g𝑅))))
73 simpr 477 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴‘(𝑛𝑙)) = (0g𝑅) ∧ (𝐶‘(𝑛𝑙)) = (0g𝑅)) → (𝐶‘(𝑛𝑙)) = (0g𝑅))
7472, 73syl6 35 . . . . . . . . . . . . . . . . . . . 20 (((𝑛𝑙) ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) → (𝑧 < (𝑛𝑙) → (𝐶‘(𝑛𝑙)) = (0g𝑅)))
7574ex 450 . . . . . . . . . . . . . . . . . . 19 ((𝑛𝑙) ∈ ℕ0 → (∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) → (𝑧 < (𝑛𝑙) → (𝐶‘(𝑛𝑙)) = (0g𝑅))))
7663, 64, 753syl 18 . . . . . . . . . . . . . . . . . 18 (𝑙 ∈ (0...𝑛) → (∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) → (𝑧 < (𝑛𝑙) → (𝐶‘(𝑛𝑙)) = (0g𝑅))))
7776com12 32 . . . . . . . . . . . . . . . . 17 (∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) → (𝑙 ∈ (0...𝑛) → (𝑧 < (𝑛𝑙) → (𝐶‘(𝑛𝑙)) = (0g𝑅))))
7877ad4antlr 768 . . . . . . . . . . . . . . . 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))))
7978imp 445 . . . . . . . . . . . . . . 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)))
8079adantl 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)))
8162, 80mpd 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))
8281oveq2d 6626 . . . . . . . . . . . 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)))
83 simplr1 1101 . . . . . . . . . . . . . . 15 ((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
8483ad2antrr 761 . . . . . . . . . . . . . 14 ((((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛)) → 𝑅 ∈ Ring)
8584adantl 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)
86 simplr2 1102 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝐾𝐵)
8786adantr 481 . . . . . . . . . . . . . . . 16 (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → 𝐾𝐵)
8887, 29anim12i 589 . . . . . . . . . . . . . . 15 ((((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛)) → (𝐾𝐵𝑙 ∈ ℕ0))
8988adantl 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))
90 eqid 2621 . . . . . . . . . . . . . . 15 (Base‘𝑅) = (Base‘𝑅)
913, 2, 1, 90coe1fvalcl 19510 . . . . . . . . . . . . . 14 ((𝐾𝐵𝑙 ∈ ℕ0) → (𝐴𝑙) ∈ (Base‘𝑅))
9289, 91syl 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))
93 eqid 2621 . . . . . . . . . . . . . 14 (0g𝑅) = (0g𝑅)
9490, 8, 93ringrz 18516 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ (𝐴𝑙) ∈ (Base‘𝑅)) → ((𝐴𝑙) (0g𝑅)) = (0g𝑅))
9585, 92, 94syl2anc 692 . . . . . . . . . . . 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))
9682, 95eqtrd 2655 . . . . . . . . . . 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))
97 ltnle 10068 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧 ∈ ℝ ∧ 𝑙 ∈ ℝ) → (𝑧 < 𝑙 ↔ ¬ 𝑙𝑧))
9823, 30, 97syl2an 494 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 ∈ ℕ0𝑙 ∈ ℕ0) → (𝑧 < 𝑙 ↔ ¬ 𝑙𝑧))
9998bicomd 213 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ ℕ0𝑙 ∈ ℕ0) → (¬ 𝑙𝑧𝑧 < 𝑙))
10099expcom 451 . . . . . . . . . . . . . . . . . 18 (𝑙 ∈ ℕ0 → (𝑧 ∈ ℕ0 → (¬ 𝑙𝑧𝑧 < 𝑙)))
101100, 29syl11 33 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℕ0 → (𝑙 ∈ (0...𝑛) → (¬ 𝑙𝑧𝑧 < 𝑙)))
102101ad4antr 767 . . . . . . . . . . . . . . . 16 (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → (𝑙 ∈ (0...𝑛) → (¬ 𝑙𝑧𝑧 < 𝑙)))
103102imp 445 . . . . . . . . . . . . . . 15 ((((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛)) → (¬ 𝑙𝑧𝑧 < 𝑙))
104 breq2 4622 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑙 → (𝑧 < 𝑥𝑧 < 𝑙))
105 fveq2 6153 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑙 → (𝐴𝑥) = (𝐴𝑙))
106105eqeq1d 2623 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑙 → ((𝐴𝑥) = (0g𝑅) ↔ (𝐴𝑙) = (0g𝑅)))
107 fveq2 6153 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑙 → (𝐶𝑥) = (𝐶𝑙))
108107eqeq1d 2623 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑙 → ((𝐶𝑥) = (0g𝑅) ↔ (𝐶𝑙) = (0g𝑅)))
109106, 108anbi12d 746 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑙 → (((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)) ↔ ((𝐴𝑙) = (0g𝑅) ∧ (𝐶𝑙) = (0g𝑅))))
110104, 109imbi12d 334 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑙 → ((𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) ↔ (𝑧 < 𝑙 → ((𝐴𝑙) = (0g𝑅) ∧ (𝐶𝑙) = (0g𝑅)))))
111110rspcva 3296 . . . . . . . . . . . . . . . . . . . 20 ((𝑙 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) → (𝑧 < 𝑙 → ((𝐴𝑙) = (0g𝑅) ∧ (𝐶𝑙) = (0g𝑅))))
112 simpl 473 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑙) = (0g𝑅) ∧ (𝐶𝑙) = (0g𝑅)) → (𝐴𝑙) = (0g𝑅))
113111, 112syl6 35 . . . . . . . . . . . . . . . . . . 19 ((𝑙 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) → (𝑧 < 𝑙 → (𝐴𝑙) = (0g𝑅)))
114113ex 450 . . . . . . . . . . . . . . . . . 18 (𝑙 ∈ ℕ0 → (∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) → (𝑧 < 𝑙 → (𝐴𝑙) = (0g𝑅))))
115114, 29syl11 33 . . . . . . . . . . . . . . . . 17 (∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) → (𝑙 ∈ (0...𝑛) → (𝑧 < 𝑙 → (𝐴𝑙) = (0g𝑅))))
116115ad4antlr 768 . . . . . . . . . . . . . . . 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))))
117116imp 445 . . . . . . . . . . . . . . 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)))
118103, 117sylbid 230 . . . . . . . . . . . . . 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)))
119118impcom 446 . . . . . . . . . . . . 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))
120119oveq1d 6625 . . . . . . . . . . . 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))))
12184adantl 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)
122 simplr3 1103 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝐿𝐵)
123122adantr 481 . . . . . . . . . . . . . . . 16 (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → 𝐿𝐵)
124 fznn0sub 12322 . . . . . . . . . . . . . . . 16 (𝑙 ∈ (0...𝑛) → (𝑛𝑙) ∈ ℕ0)
125123, 124anim12i 589 . . . . . . . . . . . . . . 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))
126125adantl 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))
1274, 2, 1, 90coe1fvalcl 19510 . . . . . . . . . . . . . 14 ((𝐿𝐵 ∧ (𝑛𝑙) ∈ ℕ0) → (𝐶‘(𝑛𝑙)) ∈ (Base‘𝑅))
128126, 127syl 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))
12990, 8, 93ringlz 18515 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ (𝐶‘(𝑛𝑙)) ∈ (Base‘𝑅)) → ((0g𝑅) (𝐶‘(𝑛𝑙))) = (0g𝑅))
130121, 128, 129syl2anc 692 . . . . . . . . . . . 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))
131120, 130eqtrd 2655 . . . . . . . . . . 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))
13296, 131pm2.61ian 830 . . . . . . . . . 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))
133132mpteq2dva 4709 . . . . . . . . 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)))
134133oveq2d 6626 . . . . . . . 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))))
135 ringmnd 18484 . . . . . . . . . . . 12 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
1361353ad2ant1 1080 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → 𝑅 ∈ Mnd)
137 ovex 6638 . . . . . . . . . . 11 (0...𝑛) ∈ V
138136, 137jctir 560 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → (𝑅 ∈ Mnd ∧ (0...𝑛) ∈ V))
139138ad3antlr 766 . . . . . . . . 9 (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → (𝑅 ∈ Mnd ∧ (0...𝑛) ∈ V))
14093gsumz 17302 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ (0...𝑛) ∈ V) → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ (0g𝑅))) = (0g𝑅))
141139, 140syl 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))
142134, 141eqtrd 2655 . . . . . . 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))
143142ex 450 . . . . . 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)))
144143ralrimiva 2961 . . . . 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)))
14516, 20, 144rspcedvd 3305 . . . 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)))
146145ex 450 . . 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))))
147146rexlimiva 3022 . 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))))
14811, 147mpcom 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3189   class class class wbr 4618  cmpt 4678  cfv 5852  (class class class)co 6610  cc 9885  cr 9886  0cc0 9887   · cmul 9892   < clt 10025  cle 10026  cmin 10217  2c2 11021  0cn0 11243  ...cfz 12275  Basecbs 15788  .rcmulr 15870   ·𝑠 cvsca 15873  0gc0g 16028   Σg cgsu 16029  Mndcmnd 17222  .gcmg 17468  mulGrpcmgp 18417  Ringcrg 18475  var1cv1 19474  Poly1cpl1 19475  coe1cco1 19476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-fsupp 8227  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-2 11030  df-3 11031  df-4 11032  df-5 11033  df-6 11034  df-7 11035  df-8 11036  df-9 11037  df-n0 11244  df-z 11329  df-dec 11445  df-uz 11639  df-fz 12276  df-seq 12749  df-struct 15790  df-ndx 15791  df-slot 15792  df-base 15793  df-sets 15794  df-ress 15795  df-plusg 15882  df-mulr 15883  df-sca 15885  df-vsca 15886  df-tset 15888  df-ple 15889  df-0g 16030  df-gsum 16031  df-mgm 17170  df-sgrp 17212  df-mnd 17223  df-grp 17353  df-minusg 17354  df-mgp 18418  df-ring 18477  df-psr 19284  df-mpl 19286  df-opsr 19288  df-psr1 19478  df-ply1 19480  df-coe1 19481
This theorem is referenced by:  ply1mulgsumlem3  41485  ply1mulgsumlem4  41486
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