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| Description: Lemma for abelth 26486. (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 3962 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | 
| 3 | oveq2 7440 | . . . . 5 ⊢ (𝑧 = 𝑋 → (1 − 𝑧) = (1 − 𝑋)) | |
| 4 | 3 | fveq2d 6909 | . . . 4 ⊢ (𝑧 = 𝑋 → (abs‘(1 − 𝑧)) = (abs‘(1 − 𝑋))) | 
| 5 | fveq2 6905 | . . . . . 6 ⊢ (𝑧 = 𝑋 → (abs‘𝑧) = (abs‘𝑋)) | |
| 6 | 5 | oveq2d 7448 | . . . . 5 ⊢ (𝑧 = 𝑋 → (1 − (abs‘𝑧)) = (1 − (abs‘𝑋))) | 
| 7 | 6 | oveq2d 7448 | . . . 4 ⊢ (𝑧 = 𝑋 → (𝑀 · (1 − (abs‘𝑧))) = (𝑀 · (1 − (abs‘𝑋)))) | 
| 8 | 4, 7 | breq12d 5155 | . . 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 3694 | . 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 1539 ∈ wcel 2107 {crab 3435 ∖ cdif 3947 {csn 4625 class class class wbr 5142 ↦ cmpt 5224 dom cdm 5684 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 ℝcr 11155 0cc0 11156 1c1 11157 + caddc 11159 · cmul 11161 ≤ cle 11297 − cmin 11493 ℕ0cn0 12528 seqcseq 14043 ↑cexp 14103 abscabs 15274 ⇝ cli 15521 Σcsu 15723 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 | 
| This theorem is referenced by: abelthlem7 26483 | 
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