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Mirrors > Home > MPE Home > Th. List > abelthlem1 | Structured version Visualization version GIF version |
Description: Lemma for abelth 24744. (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 14516 | . 2 ⊢ (abs‘1) = 1 | |
2 | eqid 2772 | . . 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 2772 | . . 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 10432 | . . 3 ⊢ (𝜑 → 1 ∈ ℂ) | |
6 | 3 | feqmptd 6560 | . . . . . . 7 ⊢ (𝜑 → 𝐴 = (𝑛 ∈ ℕ0 ↦ (𝐴‘𝑛))) |
7 | 3 | ffvelrnda 6674 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝐴‘𝑛) ∈ ℂ) |
8 | 7 | mulid1d 10455 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝐴‘𝑛) · 1) = (𝐴‘𝑛)) |
9 | 8 | mpteq2dva 5018 | . . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · 1)) = (𝑛 ∈ ℕ0 ↦ (𝐴‘𝑛))) |
10 | 6, 9 | eqtr4d 2811 | . . . . . 6 ⊢ (𝜑 → 𝐴 = (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · 1))) |
11 | ax-1cn 10391 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
12 | oveq1 6981 | . . . . . . . . . . 11 ⊢ (𝑧 = 1 → (𝑧↑𝑛) = (1↑𝑛)) | |
13 | nn0z 11816 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℤ) | |
14 | 1exp 13271 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℤ → (1↑𝑛) = 1) | |
15 | 13, 14 | syl 17 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ0 → (1↑𝑛) = 1) |
16 | 12, 15 | sylan9eq 2828 | . . . . . . . . . 10 ⊢ ((𝑧 = 1 ∧ 𝑛 ∈ ℕ0) → (𝑧↑𝑛) = 1) |
17 | 16 | oveq2d 6990 | . . . . . . . . 9 ⊢ ((𝑧 = 1 ∧ 𝑛 ∈ ℕ0) → ((𝐴‘𝑛) · (𝑧↑𝑛)) = ((𝐴‘𝑛) · 1)) |
18 | 17 | mpteq2dva 5018 | . . . . . . . 8 ⊢ (𝑧 = 1 → (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑧↑𝑛))) = (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · 1))) |
19 | nn0ex 11712 | . . . . . . . . 9 ⊢ ℕ0 ∈ V | |
20 | 19 | mptex 6810 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · 1)) ∈ V |
21 | 18, 2, 20 | fvmpt 6593 | . . . . . . 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 2826 | . . . . 5 ⊢ (𝜑 → 𝐴 = ((𝑧 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑧↑𝑛))))‘1)) |
24 | 23 | seqeq3d 13190 | . . . 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 2861 | . . 3 ⊢ (𝜑 → seq0( + , ((𝑧 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑧↑𝑛))))‘1)) ∈ dom ⇝ ) |
27 | 2, 3, 4, 5, 26 | radcnvle 24723 | . 2 ⊢ (𝜑 → (abs‘1) ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑧 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑧↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )) |
28 | 1, 27 | syl5eqbrr 4961 | 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 387 = wceq 1507 ∈ wcel 2050 {crab 3086 class class class wbr 4925 ↦ cmpt 5004 dom cdm 5403 ⟶wf 6181 ‘cfv 6185 (class class class)co 6974 supcsup 8697 ℂcc 10331 ℝcr 10332 0cc0 10333 1c1 10334 + caddc 10336 · cmul 10338 ℝ*cxr 10471 < clt 10472 ≤ cle 10473 ℕ0cn0 11705 ℤcz 11791 seqcseq 13182 ↑cexp 13242 abscabs 14452 ⇝ cli 14700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-inf2 8896 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 ax-addf 10412 ax-mulf 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-pm 8207 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-sup 8699 df-inf 8700 df-oi 8767 df-card 9160 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-3 11502 df-n0 11706 df-z 11792 df-uz 12057 df-rp 12203 df-ico 12558 df-icc 12559 df-fz 12707 df-fzo 12848 df-fl 12975 df-seq 13183 df-exp 13243 df-hash 13504 df-cj 14317 df-re 14318 df-im 14319 df-sqrt 14453 df-abs 14454 df-limsup 14687 df-clim 14704 df-rlim 14705 df-sum 14902 |
This theorem is referenced by: abelthlem3 24736 abelth 24744 |
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