Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > abelthlem7a | Structured version Visualization version GIF version |
Description: Lemma for abelth 24956. (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 3945 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
3 | oveq2 7153 | . . . . 5 ⊢ (𝑧 = 𝑋 → (1 − 𝑧) = (1 − 𝑋)) | |
4 | 3 | fveq2d 6667 | . . . 4 ⊢ (𝑧 = 𝑋 → (abs‘(1 − 𝑧)) = (abs‘(1 − 𝑋))) |
5 | fveq2 6663 | . . . . . 6 ⊢ (𝑧 = 𝑋 → (abs‘𝑧) = (abs‘𝑋)) | |
6 | 5 | oveq2d 7161 | . . . . 5 ⊢ (𝑧 = 𝑋 → (1 − (abs‘𝑧)) = (1 − (abs‘𝑋))) |
7 | 6 | oveq2d 7161 | . . . 4 ⊢ (𝑧 = 𝑋 → (𝑀 · (1 − (abs‘𝑧))) = (𝑀 · (1 − (abs‘𝑋)))) |
8 | 4, 7 | breq12d 5070 | . . 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 3680 | . 2 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ ℂ ∧ (abs‘(1 − 𝑋)) ≤ (𝑀 · (1 − (abs‘𝑋))))) |
11 | 2, 10 | sylib 219 | 1 ⊢ (𝜑 → (𝑋 ∈ ℂ ∧ (abs‘(1 − 𝑋)) ≤ (𝑀 · (1 − (abs‘𝑋))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {crab 3139 ∖ cdif 3930 {csn 4557 class class class wbr 5057 ↦ cmpt 5137 dom cdm 5548 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 ℝcr 10524 0cc0 10525 1c1 10526 + caddc 10528 · cmul 10530 ≤ cle 10664 − cmin 10858 ℕ0cn0 11885 seqcseq 13357 ↑cexp 13417 abscabs 14581 ⇝ cli 14829 Σcsu 15030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 |
This theorem is referenced by: abelthlem7 24953 |
Copyright terms: Public domain | W3C validator |