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Mirrors > Home > MPE Home > Th. List > abelthlem1 | Structured version Visualization version GIF version |
Description: Lemma for abelth 24408. (Contributed by Mario Carneiro, 1-Apr-2015.) |
Ref | Expression |
---|---|
abelth.1 | ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) |
abelth.2 | ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ ) |
Ref | Expression |
---|---|
abelthlem1 | ⊢ (𝜑 → 1 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑧 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑧↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abs1 14238 | . 2 ⊢ (abs‘1) = 1 | |
2 | eqid 2771 | . . 3 ⊢ (𝑧 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑧↑𝑛)))) = (𝑧 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑧↑𝑛)))) | |
3 | abelth.1 | . . 3 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) | |
4 | eqid 2771 | . . 3 ⊢ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑧 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑧↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) = sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑧 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑧↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) | |
5 | 1cnd 10256 | . . 3 ⊢ (𝜑 → 1 ∈ ℂ) | |
6 | 3 | feqmptd 6389 | . . . . . . 7 ⊢ (𝜑 → 𝐴 = (𝑛 ∈ ℕ0 ↦ (𝐴‘𝑛))) |
7 | 3 | ffvelrnda 6500 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝐴‘𝑛) ∈ ℂ) |
8 | 7 | mulid1d 10257 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝐴‘𝑛) · 1) = (𝐴‘𝑛)) |
9 | 8 | mpteq2dva 4878 | . . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · 1)) = (𝑛 ∈ ℕ0 ↦ (𝐴‘𝑛))) |
10 | 6, 9 | eqtr4d 2808 | . . . . . 6 ⊢ (𝜑 → 𝐴 = (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · 1))) |
11 | ax-1cn 10194 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
12 | oveq1 6798 | . . . . . . . . . . 11 ⊢ (𝑧 = 1 → (𝑧↑𝑛) = (1↑𝑛)) | |
13 | nn0z 11600 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℤ) | |
14 | 1exp 13089 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℤ → (1↑𝑛) = 1) | |
15 | 13, 14 | syl 17 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ0 → (1↑𝑛) = 1) |
16 | 12, 15 | sylan9eq 2825 | . . . . . . . . . 10 ⊢ ((𝑧 = 1 ∧ 𝑛 ∈ ℕ0) → (𝑧↑𝑛) = 1) |
17 | 16 | oveq2d 6807 | . . . . . . . . 9 ⊢ ((𝑧 = 1 ∧ 𝑛 ∈ ℕ0) → ((𝐴‘𝑛) · (𝑧↑𝑛)) = ((𝐴‘𝑛) · 1)) |
18 | 17 | mpteq2dva 4878 | . . . . . . . 8 ⊢ (𝑧 = 1 → (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑧↑𝑛))) = (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · 1))) |
19 | nn0ex 11498 | . . . . . . . . 9 ⊢ ℕ0 ∈ V | |
20 | 19 | mptex 6628 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · 1)) ∈ V |
21 | 18, 2, 20 | fvmpt 6422 | . . . . . . 7 ⊢ (1 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑧↑𝑛))))‘1) = (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · 1))) |
22 | 11, 21 | ax-mp 5 | . . . . . 6 ⊢ ((𝑧 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑧↑𝑛))))‘1) = (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · 1)) |
23 | 10, 22 | syl6eqr 2823 | . . . . 5 ⊢ (𝜑 → 𝐴 = ((𝑧 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑧↑𝑛))))‘1)) |
24 | 23 | seqeq3d 13009 | . . . 4 ⊢ (𝜑 → seq0( + , 𝐴) = seq0( + , ((𝑧 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑧↑𝑛))))‘1))) |
25 | abelth.2 | . . . 4 ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ ) | |
26 | 24, 25 | eqeltrrd 2851 | . . 3 ⊢ (𝜑 → seq0( + , ((𝑧 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑧↑𝑛))))‘1)) ∈ dom ⇝ ) |
27 | 2, 3, 4, 5, 26 | radcnvle 24387 | . 2 ⊢ (𝜑 → (abs‘1) ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑧 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑧↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )) |
28 | 1, 27 | syl5eqbrr 4822 | 1 ⊢ (𝜑 → 1 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑧 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑧↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 {crab 3065 class class class wbr 4786 ↦ cmpt 4863 dom cdm 5249 ⟶wf 6025 ‘cfv 6029 (class class class)co 6791 supcsup 8500 ℂcc 10134 ℝcr 10135 0cc0 10136 1c1 10137 + caddc 10139 · cmul 10141 ℝ*cxr 10273 < clt 10274 ≤ cle 10275 ℕ0cn0 11492 ℤcz 11577 seqcseq 13001 ↑cexp 13060 abscabs 14175 ⇝ cli 14416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-inf2 8700 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213 ax-pre-sup 10214 ax-addf 10215 ax-mulf 10216 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-isom 6038 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-om 7211 df-1st 7313 df-2nd 7314 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-1o 7711 df-oadd 7715 df-er 7894 df-pm 8010 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-sup 8502 df-inf 8503 df-oi 8569 df-card 8963 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-div 10885 df-nn 11221 df-2 11279 df-3 11280 df-n0 11493 df-z 11578 df-uz 11887 df-rp 12029 df-ico 12379 df-icc 12380 df-fz 12527 df-fzo 12667 df-fl 12794 df-seq 13002 df-exp 13061 df-hash 13315 df-cj 14040 df-re 14041 df-im 14042 df-sqrt 14176 df-abs 14177 df-limsup 14403 df-clim 14420 df-rlim 14421 df-sum 14618 |
This theorem is referenced by: abelthlem3 24400 abelth 24408 |
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