Theorem abelthlem7a 26488

Description: Lemma for abelth 26492. (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 3914	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
3		oveq2 7399	. . . . 5 ⊢ (𝑧 = 𝑋 → (1 − 𝑧) = (1 − 𝑋))
4	3	fveq2d 6866	. . . 4 ⊢ (𝑧 = 𝑋 → (abs‘(1 − 𝑧)) = (abs‘(1 − 𝑋)))
5		fveq2 6862	. . . . . 6 ⊢ (𝑧 = 𝑋 → (abs‘𝑧) = (abs‘𝑋))
6	5	oveq2d 7407	. . . . 5 ⊢ (𝑧 = 𝑋 → (1 − (abs‘𝑧)) = (1 − (abs‘𝑋)))
7	6	oveq2d 7407	. . . 4 ⊢ (𝑧 = 𝑋 → (𝑀 · (1 − (abs‘𝑧))) = (𝑀 · (1 − (abs‘𝑋))))
8	4, 7	breq12d 5110	. . 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 3652	. 2 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ ℂ ∧ (abs‘(1 − 𝑋)) ≤ (𝑀 · (1 − (abs‘𝑋)))))
11	2, 10	sylib 220	1 ⊢ (𝜑 → (𝑋 ∈ ℂ ∧ (abs‘(1 − 𝑋)) ≤ (𝑀 · (1 − (abs‘𝑋)))))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  {crab 3413   ∖ cdif 3899  {csn 4579   class class class wbr 5097   ↦ cmpt 5178  dom cdm 5643  ⟶wf 6512  ‘cfv 6516  (class class class)co 7391  ℂcc 11065  ℝcr 11066  0cc0 11067  1c1 11068   + caddc 11070   · cmul 11072   ≤ cle 11211   − cmin 11408  ℕ₀cn0 12475  seqcseq 14008  ↑cexp 14068  abscabs 15252   ⇝ cli 15502  Σcsu 15704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-iota 6472  df-fv 6524  df-ov 7394

This theorem is referenced by:  abelthlem7  26489