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Theorem ply1mulgsumlem2 41969
Description: Lemma 2 for ply1mulgsum 41972. (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 41968 . 2 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑧 ∈ ℕ0𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))))
12 2nn0 11186 . . . . . . . 8 2 ∈ ℕ0
1312a1i 11 . . . . . . 7 (𝑧 ∈ ℕ0 → 2 ∈ ℕ0)
14 id 22 . . . . . . 7 (𝑧 ∈ ℕ0𝑧 ∈ ℕ0)
1513, 14nn0mulcld 11233 . . . . . 6 (𝑧 ∈ ℕ0 → (2 · 𝑧) ∈ ℕ0)
1615ad2antrr 758 . . . . 5 (((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) → (2 · 𝑧) ∈ ℕ0)
17 breq1 4586 . . . . . . . 8 (𝑠 = (2 · 𝑧) → (𝑠 < 𝑛 ↔ (2 · 𝑧) < 𝑛))
1817imbi1d 330 . . . . . . 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 2969 . . . . . 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 10967 . . . . . . . . . . . . . . . . . . . . . . . . 25 2 ∈ ℝ
2221a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 ∈ ℕ0 → 2 ∈ ℝ)
23 nn0re 11178 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 ∈ ℕ0𝑧 ∈ ℝ)
2422, 23remulcld 9949 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ ℕ0 → (2 · 𝑧) ∈ ℝ)
2524ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (2 · 𝑧) ∈ ℝ)
26 nn0re 11178 . . . . . . . . . . . . . . . . . . . . . . . 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 12302 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑙 ∈ (0...𝑛) → 𝑙 ∈ ℕ0)
30 nn0re 11178 . . . . . . . . . . . . . . . . . . . . . . . 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 10515 . . . . . . . . . . . . . . . . . . . . 21 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) < 𝑛 ↔ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)))
3423ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑧 ∈ ℝ)
3532, 34, 25lesub2d 10514 . . . . . . . . . . . . . . . . . . . . . . . 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 10337 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ ℕ0 → ((2 · 𝑧) − 𝑧) ∈ ℝ)
3837ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑧) ∈ ℝ)
3924adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) → (2 · 𝑧) ∈ ℝ)
40 resubcl 10224 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((2 · 𝑧) ∈ ℝ ∧ 𝑙 ∈ ℝ) → ((2 · 𝑧) − 𝑙) ∈ ℝ)
4139, 31, 40syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑙) ∈ ℝ)
42 resubcl 10224 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑛 ∈ ℝ ∧ 𝑙 ∈ ℝ) → (𝑛𝑙) ∈ ℝ)
4327, 31, 42syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑛𝑙) ∈ ℝ)
44 lelttr 10007 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1318 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → ((2 · 𝑧) − 𝑧) < (𝑛𝑙)))
46 nn0cn 11179 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 ∈ ℕ0𝑧 ∈ ℂ)
47 2txmxeqx 11026 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 ∈ ℂ → ((2 · 𝑧) − 𝑧) = 𝑧)
4846, 47syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ ℕ0 → ((2 · 𝑧) − 𝑧) = 𝑧)
4948ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) − 𝑧) = 𝑧)
5049breq1d 4593 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (((2 · 𝑧) − 𝑧) < (𝑛𝑙) ↔ 𝑧 < (𝑛𝑙)))
5145, 50sylibd 228 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → 𝑧 < (𝑛𝑙)))
5251expcomd 453 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (((2 · 𝑧) − 𝑙) < (𝑛𝑙) → (((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) → 𝑧 < (𝑛𝑙))))
5352imp 444 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → (((2 · 𝑧) − 𝑧) ≤ ((2 · 𝑧) − 𝑙) → 𝑧 < (𝑛𝑙)))
5436, 53sylbid 229 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ ((2 · 𝑧) − 𝑙) < (𝑛𝑙)) → (𝑙𝑧𝑧 < (𝑛𝑙)))
5554ex 449 . . . . . . . . . . . . . . . . . . . . 21 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (((2 · 𝑧) − 𝑙) < (𝑛𝑙) → (𝑙𝑧𝑧 < (𝑛𝑙))))
5633, 55sylbid 229 . . . . . . . . . . . . . . . . . . . 20 (((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((2 · 𝑧) < 𝑛 → (𝑙𝑧𝑧 < (𝑛𝑙))))
5756ex 449 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) → (𝑙 ∈ (0...𝑛) → ((2 · 𝑧) < 𝑛 → (𝑙𝑧𝑧 < (𝑛𝑙)))))
5857com23 84 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ ℕ0𝑛 ∈ ℕ0) → ((2 · 𝑧) < 𝑛 → (𝑙 ∈ (0...𝑛) → (𝑙𝑧𝑧 < (𝑛𝑙)))))
5958ex 449 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℕ0 → (𝑛 ∈ ℕ0 → ((2 · 𝑧) < 𝑛 → (𝑙 ∈ (0...𝑛) → (𝑙𝑧𝑧 < (𝑛𝑙))))))
6059ad2antrr 758 . . . . . . . . . . . . . . . 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 617 . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . 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 12315 . . . . . . . . . . . . . . . . . . 19 (𝑙 ∈ (0...𝑛) → (𝑛𝑙) ∈ (0...𝑛))
64 elfznn0 12302 . . . . . . . . . . . . . . . . . . 19 ((𝑛𝑙) ∈ (0...𝑛) → (𝑛𝑙) ∈ ℕ0)
65 breq2 4587 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = (𝑛𝑙) → (𝑧 < 𝑥𝑧 < (𝑛𝑙)))
66 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = (𝑛𝑙) → (𝐴𝑥) = (𝐴‘(𝑛𝑙)))
6766eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = (𝑛𝑙) → ((𝐴𝑥) = (0g𝑅) ↔ (𝐴‘(𝑛𝑙)) = (0g𝑅)))
68 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = (𝑛𝑙) → (𝐶𝑥) = (𝐶‘(𝑛𝑙)))
6968eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = (𝑛𝑙) → ((𝐶𝑥) = (0g𝑅) ↔ (𝐶‘(𝑛𝑙)) = (0g𝑅)))
7067, 69anbi12d 743 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = (𝑛𝑙) → (((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)) ↔ ((𝐴‘(𝑛𝑙)) = (0g𝑅) ∧ (𝐶‘(𝑛𝑙)) = (0g𝑅))))
7165, 70imbi12d 333 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = (𝑛𝑙) → ((𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) ↔ (𝑧 < (𝑛𝑙) → ((𝐴‘(𝑛𝑙)) = (0g𝑅) ∧ (𝐶‘(𝑛𝑙)) = (0g𝑅)))))
7271rspcva 3280 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛𝑙) ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) → (𝑧 < (𝑛𝑙) → ((𝐴‘(𝑛𝑙)) = (0g𝑅) ∧ (𝐶‘(𝑛𝑙)) = (0g𝑅))))
73 simpr 476 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴‘(𝑛𝑙)) = (0g𝑅) ∧ (𝐶‘(𝑛𝑙)) = (0g𝑅)) → (𝐶‘(𝑛𝑙)) = (0g𝑅))
7472, 73syl6 34 . . . . . . . . . . . . . . . . . . . 20 (((𝑛𝑙) ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) → (𝑧 < (𝑛𝑙) → (𝐶‘(𝑛𝑙)) = (0g𝑅)))
7574ex 449 . . . . . . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . 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 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)))
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 6565 . . . . . . . . . . . 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 1096 . . . . . . . . . . . . . . 15 ((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
8483ad2antrr 758 . . . . . . . . . . . . . 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 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)
86 simplr2 1097 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝐾𝐵)
8786adantr 480 . . . . . . . . . . . . . . . 16 (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → 𝐾𝐵)
8887, 29anim12i 588 . . . . . . . . . . . . . . 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 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))
90 eqid 2610 . . . . . . . . . . . . . . 15 (Base‘𝑅) = (Base‘𝑅)
913, 2, 1, 90coe1fvalcl 19403 . . . . . . . . . . . . . 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 2610 . . . . . . . . . . . . . 14 (0g𝑅) = (0g𝑅)
9490, 8, 93ringrz 18411 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ (𝐴𝑙) ∈ (Base‘𝑅)) → ((𝐴𝑙) (0g𝑅)) = (0g𝑅))
9585, 92, 94syl2anc 691 . . . . . . . . . . . 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 2644 . . . . . . . . . . 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 9996 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧 ∈ ℝ ∧ 𝑙 ∈ ℝ) → (𝑧 < 𝑙 ↔ ¬ 𝑙𝑧))
9823, 30, 97syl2an 493 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧 ∈ ℕ0𝑙 ∈ ℕ0) → (𝑧 < 𝑙 ↔ ¬ 𝑙𝑧))
9998bicomd 212 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 ∈ ℕ0𝑙 ∈ ℕ0) → (¬ 𝑙𝑧𝑧 < 𝑙))
10099expcom 450 . . . . . . . . . . . . . . . . . . 19 (𝑙 ∈ ℕ0 → (𝑧 ∈ ℕ0 → (¬ 𝑙𝑧𝑧 < 𝑙)))
10129, 100syl 17 . . . . . . . . . . . . . . . . . 18 (𝑙 ∈ (0...𝑛) → (𝑧 ∈ ℕ0 → (¬ 𝑙𝑧𝑧 < 𝑙)))
102101com12 32 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℕ0 → (𝑙 ∈ (0...𝑛) → (¬ 𝑙𝑧𝑧 < 𝑙)))
103102ad4antr 764 . . . . . . . . . . . . . . . 16 (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → (𝑙 ∈ (0...𝑛) → (¬ 𝑙𝑧𝑧 < 𝑙)))
104103imp 444 . . . . . . . . . . . . . . 15 ((((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) ∧ 𝑙 ∈ (0...𝑛)) → (¬ 𝑙𝑧𝑧 < 𝑙))
105 breq2 4587 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑙 → (𝑧 < 𝑥𝑧 < 𝑙))
106 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑙 → (𝐴𝑥) = (𝐴𝑙))
107106eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑙 → ((𝐴𝑥) = (0g𝑅) ↔ (𝐴𝑙) = (0g𝑅)))
108 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑙 → (𝐶𝑥) = (𝐶𝑙))
109108eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑙 → ((𝐶𝑥) = (0g𝑅) ↔ (𝐶𝑙) = (0g𝑅)))
110107, 109anbi12d 743 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑙 → (((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)) ↔ ((𝐴𝑙) = (0g𝑅) ∧ (𝐶𝑙) = (0g𝑅))))
111105, 110imbi12d 333 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑙 → ((𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) ↔ (𝑧 < 𝑙 → ((𝐴𝑙) = (0g𝑅) ∧ (𝐶𝑙) = (0g𝑅)))))
112111rspcva 3280 . . . . . . . . . . . . . . . . . . . . 21 ((𝑙 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) → (𝑧 < 𝑙 → ((𝐴𝑙) = (0g𝑅) ∧ (𝐶𝑙) = (0g𝑅))))
113 simpl 472 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑙) = (0g𝑅) ∧ (𝐶𝑙) = (0g𝑅)) → (𝐴𝑙) = (0g𝑅))
114112, 113syl6 34 . . . . . . . . . . . . . . . . . . . 20 ((𝑙 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) → (𝑧 < 𝑙 → (𝐴𝑙) = (0g𝑅)))
115114ex 449 . . . . . . . . . . . . . . . . . . 19 (𝑙 ∈ ℕ0 → (∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) → (𝑧 < 𝑙 → (𝐴𝑙) = (0g𝑅))))
11629, 115syl 17 . . . . . . . . . . . . . . . . . 18 (𝑙 ∈ (0...𝑛) → (∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) → (𝑧 < 𝑙 → (𝐴𝑙) = (0g𝑅))))
117116com12 32 . . . . . . . . . . . . . . . . 17 (∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅))) → (𝑙 ∈ (0...𝑛) → (𝑧 < 𝑙 → (𝐴𝑙) = (0g𝑅))))
118117ad4antlr 765 . . . . . . . . . . . . . . . 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))))
119118imp 444 . . . . . . . . . . . . . . 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)))
120104, 119sylbid 229 . . . . . . . . . . . . . 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)))
121120impcom 445 . . . . . . . . . . . . 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))
122121oveq1d 6564 . . . . . . . . . . . 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))))
12384adantl 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)
124 simplr3 1098 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝐿𝐵)
125124adantr 480 . . . . . . . . . . . . . . . 16 (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → 𝐿𝐵)
126 fznn0sub 12244 . . . . . . . . . . . . . . . 16 (𝑙 ∈ (0...𝑛) → (𝑛𝑙) ∈ ℕ0)
127125, 126anim12i 588 . . . . . . . . . . . . . . 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))
128127adantl 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))
1294, 2, 1, 90coe1fvalcl 19403 . . . . . . . . . . . . . 14 ((𝐿𝐵 ∧ (𝑛𝑙) ∈ ℕ0) → (𝐶‘(𝑛𝑙)) ∈ (Base‘𝑅))
130128, 129syl 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))
13190, 8, 93ringlz 18410 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ (𝐶‘(𝑛𝑙)) ∈ (Base‘𝑅)) → ((0g𝑅) (𝐶‘(𝑛𝑙))) = (0g𝑅))
132123, 130, 131syl2anc 691 . . . . . . . . . . . 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))
133122, 132eqtrd 2644 . . . . . . . . . . 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))
13496, 133pm2.61ian 827 . . . . . . . . . 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))
135134mpteq2dva 4672 . . . . . . . . 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)))
136135oveq2d 6565 . . . . . . . 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))))
137 ringmnd 18379 . . . . . . . . . . . 12 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
1381373ad2ant1 1075 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → 𝑅 ∈ Mnd)
139 ovex 6577 . . . . . . . . . . 11 (0...𝑛) ∈ V
140138, 139jctir 559 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → (𝑅 ∈ Mnd ∧ (0...𝑛) ∈ V))
141140ad3antlr 763 . . . . . . . . 9 (((((𝑧 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑧 < 𝑥 → ((𝐴𝑥) = (0g𝑅) ∧ (𝐶𝑥) = (0g𝑅)))) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (2 · 𝑧) < 𝑛) → (𝑅 ∈ Mnd ∧ (0...𝑛) ∈ V))
14293gsumz 17197 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ (0...𝑛) ∈ V) → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ (0g𝑅))) = (0g𝑅))
143141, 142syl 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))
144136, 143eqtrd 2644 . . . . . . 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))
145144ex 449 . . . . . 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)))
146145ralrimiva 2949 . . . . 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)))
14716, 20, 146rspcedvd 3289 . . . 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)))
148147ex 449 . . 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))))
149148rexlimiva 3010 . 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))))
15011, 149mpcom 37 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 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  wrex 2897  Vcvv 3173   class class class wbr 4583  cmpt 4643  cfv 5804  (class class class)co 6549  cc 9813  cr 9814  0cc0 9815   · cmul 9820   < clt 9953  cle 9954  cmin 10145  2c2 10947  0cn0 11169  ...cfz 12197  Basecbs 15695  .rcmulr 15769   ·𝑠 cvsca 15772  0gc0g 15923   Σg cgsu 15924  Mndcmnd 17117  .gcmg 17363  mulGrpcmgp 18312  Ringcrg 18370  var1cv1 19367  Poly1cpl1 19368  coe1cco1 19369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-seq 12664  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-tset 15787  df-ple 15788  df-0g 15925  df-gsum 15926  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-mgp 18313  df-ring 18372  df-psr 19177  df-mpl 19179  df-opsr 19181  df-psr1 19371  df-ply1 19373  df-coe1 19374
This theorem is referenced by:  ply1mulgsumlem3  41970  ply1mulgsumlem4  41971
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