Theorem abelthlem7a 25949

Description: Lemma for abelth 25953. (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 3961	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
3		oveq2 7417	. . . . 5 ⊢ (𝑧 = 𝑋 → (1 − 𝑧) = (1 − 𝑋))
4	3	fveq2d 6896	. . . 4 ⊢ (𝑧 = 𝑋 → (abs‘(1 − 𝑧)) = (abs‘(1 − 𝑋)))
5		fveq2 6892	. . . . . 6 ⊢ (𝑧 = 𝑋 → (abs‘𝑧) = (abs‘𝑋))
6	5	oveq2d 7425	. . . . 5 ⊢ (𝑧 = 𝑋 → (1 − (abs‘𝑧)) = (1 − (abs‘𝑋)))
7	6	oveq2d 7425	. . . 4 ⊢ (𝑧 = 𝑋 → (𝑀 · (1 − (abs‘𝑧))) = (𝑀 · (1 − (abs‘𝑋))))
8	4, 7	breq12d 5162	. . 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 3687	. 2 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ ℂ ∧ (abs‘(1 − 𝑋)) ≤ (𝑀 · (1 − (abs‘𝑋)))))
11	2, 10	sylib 217	1 ⊢ (𝜑 → (𝑋 ∈ ℂ ∧ (abs‘(1 − 𝑋)) ≤ (𝑀 · (1 − (abs‘𝑋)))))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {crab 3433   ∖ cdif 3946  {csn 4629   class class class wbr 5149   ↦ cmpt 5232  dom cdm 5677  ⟶wf 6540  ‘cfv 6544  (class class class)co 7409  ℂcc 11108  ℝcr 11109  0cc0 11110  1c1 11111   + caddc 11113   · cmul 11115   ≤ cle 11249   − cmin 11444  ℕ₀cn0 12472  seqcseq 13966  ↑cexp 14027  abscabs 15181   ⇝ cli 15428  Σcsu 15632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412

This theorem is referenced by:  abelthlem7  25950