Theorem abelthlem7a 26369

Description: Lemma for abelth 26373. (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 3909	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
3		oveq2 7349	. . . . 5 ⊢ (𝑧 = 𝑋 → (1 − 𝑧) = (1 − 𝑋))
4	3	fveq2d 6821	. . . 4 ⊢ (𝑧 = 𝑋 → (abs‘(1 − 𝑧)) = (abs‘(1 − 𝑋)))
5		fveq2 6817	. . . . . 6 ⊢ (𝑧 = 𝑋 → (abs‘𝑧) = (abs‘𝑋))
6	5	oveq2d 7357	. . . . 5 ⊢ (𝑧 = 𝑋 → (1 − (abs‘𝑧)) = (1 − (abs‘𝑋)))
7	6	oveq2d 7357	. . . 4 ⊢ (𝑧 = 𝑋 → (𝑀 · (1 − (abs‘𝑧))) = (𝑀 · (1 − (abs‘𝑋))))
8	4, 7	breq12d 5099	. . 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 3645	. 2 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ ℂ ∧ (abs‘(1 − 𝑋)) ≤ (𝑀 · (1 − (abs‘𝑋)))))
11	2, 10	sylib 218	1 ⊢ (𝜑 → (𝑋 ∈ ℂ ∧ (abs‘(1 − 𝑋)) ≤ (𝑀 · (1 − (abs‘𝑋)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {crab 3395   ∖ cdif 3894  {csn 4571   class class class wbr 5086   ↦ cmpt 5167  dom cdm 5611  ⟶wf 6472  ‘cfv 6476  (class class class)co 7341  ℂcc 10999  ℝcr 11000  0cc0 11001  1c1 11002   + caddc 11004   · cmul 11006   ≤ cle 11142   − cmin 11339  ℕ₀cn0 12376  seqcseq 13903  ↑cexp 13963  abscabs 15136   ⇝ cli 15386  Σcsu 15588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-iota 6432  df-fv 6484  df-ov 7344

This theorem is referenced by:  abelthlem7  26370