Theorem abelthlem7a 26757

Description: Lemma for abelth 26761. (Contributed by Mario Carneiro, 8-May-2015.)

Hypotheses

Ref	Expression
abelth.1	⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)
abelth.2	⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )
abelth.3	⊢ (𝜑 → 𝑀 ∈ ℝ)
abelth.4	⊢ (𝜑 → 0 ≤ 𝑀)
abelth.5	⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}
abelth.6	⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛)))
abelth.7	⊢ (𝜑 → seq0( + , 𝐴) ⇝ 0)
abelthlem6.1	⊢ (𝜑 → 𝑋 ∈ (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

Step	Hyp	Ref	Expression
1		abelthlem6.1	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑆 ∖ {1}))
2	1	eldifad 3911	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
3		oveq2 7426	. . . . 5 ⊢ (𝑧 = 𝑋 → (1 − 𝑧) = (1 − 𝑋))
4	3	fveq2d 6887	. . . 4 ⊢ (𝑧 = 𝑋 → (abs‘(1 − 𝑧)) = (abs‘(1 − 𝑋)))
5		fveq2 6883	. . . . . 6 ⊢ (𝑧 = 𝑋 → (abs‘𝑧) = (abs‘𝑋))
6	5	oveq2d 7434	. . . . 5 ⊢ (𝑧 = 𝑋 → (1 − (abs‘𝑧)) = (1 − (abs‘𝑋)))
7	6	oveq2d 7434	. . . 4 ⊢ (𝑧 = 𝑋 → (𝑀 · (1 − (abs‘𝑧))) = (𝑀 · (1 − (abs‘𝑋))))
8	4, 7	breq12d 5116	. . 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)))}
10	8, 9	elrab2 3649	. 2 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ ℂ ∧ (abs‘(1 − 𝑋)) ≤ (𝑀 · (1 − (abs‘𝑋)))))
11	2, 10	sylib 221	1 ⊢ (𝜑 → (𝑋 ∈ ℂ ∧ (abs‘(1 − 𝑋)) ≤ (𝑀 · (1 − (abs‘𝑋)))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3413   ∖ cdif 3896  {csn 4584   class class class wbr 5103   ↦ cmpt 5186  dom cdm 5651  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418  ℂcc 11191  ℝcr 11192  0cc0 11193  1c1 11194   + caddc 11196   · cmul 11198   ≤ cle 11337   − cmin 11534  ℕ₀cn0 12599  seqcseq 14137  ↑cexp 14197  abscabs 15394   ⇝ cli 15644  Σcsu 15846

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6493  df-fv 6545  df-ov 7421

This theorem is used by:  abelthlem7  26758