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Mirrors > Home > MPE Home > Th. List > abelthlem7a | Structured version Visualization version GIF version |
Description: Lemma for abelth 26467. (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 3960 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
3 | oveq2 7423 | . . . . 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 7431 | . . . . 5 ⊢ (𝑧 = 𝑋 → (1 − (abs‘𝑧)) = (1 − (abs‘𝑋))) |
7 | 6 | oveq2d 7431 | . . . 4 ⊢ (𝑧 = 𝑋 → (𝑀 · (1 − (abs‘𝑧))) = (𝑀 · (1 − (abs‘𝑋)))) |
8 | 4, 7 | breq12d 5158 | . . 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 3685 | . 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))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 {crab 3420 ∖ cdif 3945 {csn 4625 class class class wbr 5145 ↦ cmpt 5228 dom cdm 5674 ⟶wf 6541 ‘cfv 6545 (class class class)co 7415 ℂcc 11146 ℝcr 11147 0cc0 11148 1c1 11149 + caddc 11151 · cmul 11153 ≤ cle 11289 − cmin 11484 ℕ0cn0 12517 seqcseq 14014 ↑cexp 14074 abscabs 15233 ⇝ cli 15480 Σcsu 15684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rab 3421 df-v 3466 df-dif 3951 df-un 3953 df-ss 3965 df-nul 4325 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4908 df-br 5146 df-iota 6497 df-fv 6553 df-ov 7418 |
This theorem is referenced by: abelthlem7 26464 |
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