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Theorem abelthlem7a 26463
Description: Lemma for abelth 26467. (Contributed by Mario Carneiro, 8-May-2015.)
Hypotheses
Ref Expression
abelth.1 (𝜑𝐴:ℕ0⟶ℂ)
abelth.2 (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )
abelth.3 (𝜑𝑀 ∈ ℝ)
abelth.4 (𝜑 → 0 ≤ 𝑀)
abelth.5 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}
abelth.6 𝐹 = (𝑥𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛)))
abelth.7 (𝜑 → seq0( + , 𝐴) ⇝ 0)
abelthlem6.1 (𝜑𝑋 ∈ (𝑆 ∖ {1}))
Assertion
Ref Expression
abelthlem7a (𝜑 → (𝑋 ∈ ℂ ∧ (abs‘(1 − 𝑋)) ≤ (𝑀 · (1 − (abs‘𝑋)))))
Distinct variable groups:   𝑥,𝑛,𝑧,𝑀   𝑛,𝑋,𝑥,𝑧   𝐴,𝑛,𝑥,𝑧   𝜑,𝑛,𝑥   𝑆,𝑛,𝑥
Allowed substitution hints:   𝜑(𝑧)   𝑆(𝑧)   𝐹(𝑥,𝑧,𝑛)

Proof of Theorem abelthlem7a
StepHypRef Expression
1 abelthlem6.1 . . 3 (𝜑𝑋 ∈ (𝑆 ∖ {1}))
21eldifad 3960 . 2 (𝜑𝑋𝑆)
3 oveq2 7423 . . . . 5 (𝑧 = 𝑋 → (1 − 𝑧) = (1 − 𝑋))
43fveq2d 6896 . . . 4 (𝑧 = 𝑋 → (abs‘(1 − 𝑧)) = (abs‘(1 − 𝑋)))
5 fveq2 6892 . . . . . 6 (𝑧 = 𝑋 → (abs‘𝑧) = (abs‘𝑋))
65oveq2d 7431 . . . . 5 (𝑧 = 𝑋 → (1 − (abs‘𝑧)) = (1 − (abs‘𝑋)))
76oveq2d 7431 . . . 4 (𝑧 = 𝑋 → (𝑀 · (1 − (abs‘𝑧))) = (𝑀 · (1 − (abs‘𝑋))))
84, 7breq12d 5158 . . 3 (𝑧 = 𝑋 → ((abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧))) ↔ (abs‘(1 − 𝑋)) ≤ (𝑀 · (1 − (abs‘𝑋)))))
9 abelth.5 . . 3 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}
108, 9elrab2 3685 . 2 (𝑋𝑆 ↔ (𝑋 ∈ ℂ ∧ (abs‘(1 − 𝑋)) ≤ (𝑀 · (1 − (abs‘𝑋)))))
112, 10sylib 217 1 (𝜑 → (𝑋 ∈ ℂ ∧ (abs‘(1 − 𝑋)) ≤ (𝑀 · (1 − (abs‘𝑋)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  {crab 3420  cdif 3945  {csn 4625   class class class wbr 5145  cmpt 5228  dom cdm 5674  wf 6541  cfv 6545  (class class class)co 7415  cc 11146  cr 11147  0cc0 11148  1c1 11149   + caddc 11151   · cmul 11153  cle 11289  cmin 11484  0cn0 12517  seqcseq 14014  cexp 14074  abscabs 15233  cli 15480  Σcsu 15684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3421  df-v 3466  df-dif 3951  df-un 3953  df-ss 3965  df-nul 4325  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4908  df-br 5146  df-iota 6497  df-fv 6553  df-ov 7418
This theorem is referenced by:  abelthlem7  26464
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