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Mirrors > Home > MPE Home > Th. List > abelthlem7a | Structured version Visualization version GIF version |
Description: Lemma for abelth 26500. (Contributed by Mario Carneiro, 8-May-2015.) |
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})) |
Ref | Expression |
---|---|
abelthlem7a | ⊢ (𝜑 → (𝑋 ∈ ℂ ∧ (abs‘(1 − 𝑋)) ≤ (𝑀 · (1 − (abs‘𝑋))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abelthlem6.1 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑆 ∖ {1})) | |
2 | 1 | eldifad 3975 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
3 | oveq2 7439 | . . . . 5 ⊢ (𝑧 = 𝑋 → (1 − 𝑧) = (1 − 𝑋)) | |
4 | 3 | fveq2d 6911 | . . . 4 ⊢ (𝑧 = 𝑋 → (abs‘(1 − 𝑧)) = (abs‘(1 − 𝑋))) |
5 | fveq2 6907 | . . . . . 6 ⊢ (𝑧 = 𝑋 → (abs‘𝑧) = (abs‘𝑋)) | |
6 | 5 | oveq2d 7447 | . . . . 5 ⊢ (𝑧 = 𝑋 → (1 − (abs‘𝑧)) = (1 − (abs‘𝑋))) |
7 | 6 | oveq2d 7447 | . . . 4 ⊢ (𝑧 = 𝑋 → (𝑀 · (1 − (abs‘𝑧))) = (𝑀 · (1 − (abs‘𝑋)))) |
8 | 4, 7 | breq12d 5161 | . . 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 3698 | . 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))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {crab 3433 ∖ cdif 3960 {csn 4631 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 ≤ cle 11294 − cmin 11490 ℕ0cn0 12524 seqcseq 14039 ↑cexp 14099 abscabs 15270 ⇝ cli 15517 Σcsu 15719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 |
This theorem is referenced by: abelthlem7 26497 |
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