Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > abelthlem7a | Structured version Visualization version GIF version |
Description: Lemma for abelth 25029. (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 3948 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
3 | oveq2 7164 | . . . . 5 ⊢ (𝑧 = 𝑋 → (1 − 𝑧) = (1 − 𝑋)) | |
4 | 3 | fveq2d 6674 | . . . 4 ⊢ (𝑧 = 𝑋 → (abs‘(1 − 𝑧)) = (abs‘(1 − 𝑋))) |
5 | fveq2 6670 | . . . . . 6 ⊢ (𝑧 = 𝑋 → (abs‘𝑧) = (abs‘𝑋)) | |
6 | 5 | oveq2d 7172 | . . . . 5 ⊢ (𝑧 = 𝑋 → (1 − (abs‘𝑧)) = (1 − (abs‘𝑋))) |
7 | 6 | oveq2d 7172 | . . . 4 ⊢ (𝑧 = 𝑋 → (𝑀 · (1 − (abs‘𝑧))) = (𝑀 · (1 − (abs‘𝑋)))) |
8 | 4, 7 | breq12d 5079 | . . 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 3683 | . 2 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ ℂ ∧ (abs‘(1 − 𝑋)) ≤ (𝑀 · (1 − (abs‘𝑋))))) |
11 | 2, 10 | sylib 220 | 1 ⊢ (𝜑 → (𝑋 ∈ ℂ ∧ (abs‘(1 − 𝑋)) ≤ (𝑀 · (1 − (abs‘𝑋))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3142 ∖ cdif 3933 {csn 4567 class class class wbr 5066 ↦ cmpt 5146 dom cdm 5555 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 ≤ cle 10676 − cmin 10870 ℕ0cn0 11898 seqcseq 13370 ↑cexp 13430 abscabs 14593 ⇝ cli 14841 Σcsu 15042 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 |
This theorem is referenced by: abelthlem7 25026 |
Copyright terms: Public domain | W3C validator |