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Theorem abelthlem7a 26482
Description: Lemma for abelth 26486. (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 3962 . 2 (𝜑𝑋𝑆)
3 oveq2 7440 . . . . 5 (𝑧 = 𝑋 → (1 − 𝑧) = (1 − 𝑋))
43fveq2d 6909 . . . 4 (𝑧 = 𝑋 → (abs‘(1 − 𝑧)) = (abs‘(1 − 𝑋)))
5 fveq2 6905 . . . . . 6 (𝑧 = 𝑋 → (abs‘𝑧) = (abs‘𝑋))
65oveq2d 7448 . . . . 5 (𝑧 = 𝑋 → (1 − (abs‘𝑧)) = (1 − (abs‘𝑋)))
76oveq2d 7448 . . . 4 (𝑧 = 𝑋 → (𝑀 · (1 − (abs‘𝑧))) = (𝑀 · (1 − (abs‘𝑋))))
84, 7breq12d 5155 . . 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 3694 . 2 (𝑋𝑆 ↔ (𝑋 ∈ ℂ ∧ (abs‘(1 − 𝑋)) ≤ (𝑀 · (1 − (abs‘𝑋)))))
112, 10sylib 218 1 (𝜑 → (𝑋 ∈ ℂ ∧ (abs‘(1 − 𝑋)) ≤ (𝑀 · (1 − (abs‘𝑋)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  {crab 3435  cdif 3947  {csn 4625   class class class wbr 5142  cmpt 5224  dom cdm 5684  wf 6556  cfv 6560  (class class class)co 7432  cc 11154  cr 11155  0cc0 11156  1c1 11157   + caddc 11159   · cmul 11161  cle 11297  cmin 11493  0cn0 12528  seqcseq 14043  cexp 14103  abscabs 15274  cli 15521  Σcsu 15723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435
This theorem is referenced by:  abelthlem7  26483
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