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Mirrors > Home > MPE Home > Th. List > psercnlem2 | Structured version Visualization version GIF version |
Description: Lemma for psercn 25021. (Contributed by Mario Carneiro, 18-Mar-2015.) |
Ref | Expression |
---|---|
pserf.g | ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) |
pserf.f | ⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) |
pserf.a | ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) |
pserf.r | ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) |
psercn.s | ⊢ 𝑆 = (◡abs “ (0[,)𝑅)) |
psercnlem2.i | ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑀 ∈ ℝ+ ∧ (abs‘𝑎) < 𝑀 ∧ 𝑀 < 𝑅)) |
Ref | Expression |
---|---|
psercnlem2 | ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ∧ (0(ball‘(abs ∘ − ))𝑀) ⊆ (◡abs “ (0[,]𝑀)) ∧ (◡abs “ (0[,]𝑀)) ⊆ 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psercn.s | . . . . . . 7 ⊢ 𝑆 = (◡abs “ (0[,)𝑅)) | |
2 | cnvimass 5916 | . . . . . . . 8 ⊢ (◡abs “ (0[,)𝑅)) ⊆ dom abs | |
3 | absf 14689 | . . . . . . . . 9 ⊢ abs:ℂ⟶ℝ | |
4 | 3 | fdmi 6498 | . . . . . . . 8 ⊢ dom abs = ℂ |
5 | 2, 4 | sseqtri 3951 | . . . . . . 7 ⊢ (◡abs “ (0[,)𝑅)) ⊆ ℂ |
6 | 1, 5 | eqsstri 3949 | . . . . . 6 ⊢ 𝑆 ⊆ ℂ |
7 | 6 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
8 | 7 | sselda 3915 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ℂ) |
9 | 8 | abscld 14788 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) ∈ ℝ) |
10 | 8 | absge0d 14796 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 ≤ (abs‘𝑎)) |
11 | psercnlem2.i | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑀 ∈ ℝ+ ∧ (abs‘𝑎) < 𝑀 ∧ 𝑀 < 𝑅)) | |
12 | 11 | simp2d 1140 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < 𝑀) |
13 | 0re 10632 | . . . . . 6 ⊢ 0 ∈ ℝ | |
14 | 11 | simp1d 1139 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ+) |
15 | 14 | rpxrd 12420 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ*) |
16 | elico2 12789 | . . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝑀 ∈ ℝ*) → ((abs‘𝑎) ∈ (0[,)𝑀) ↔ ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑀))) | |
17 | 13, 15, 16 | sylancr 590 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) ∈ (0[,)𝑀) ↔ ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑀))) |
18 | 9, 10, 12, 17 | mpbir3and 1339 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) ∈ (0[,)𝑀)) |
19 | ffn 6487 | . . . . 5 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ) | |
20 | elpreima 6805 | . . . . 5 ⊢ (abs Fn ℂ → (𝑎 ∈ (◡abs “ (0[,)𝑀)) ↔ (𝑎 ∈ ℂ ∧ (abs‘𝑎) ∈ (0[,)𝑀)))) | |
21 | 3, 19, 20 | mp2b 10 | . . . 4 ⊢ (𝑎 ∈ (◡abs “ (0[,)𝑀)) ↔ (𝑎 ∈ ℂ ∧ (abs‘𝑎) ∈ (0[,)𝑀))) |
22 | 8, 18, 21 | sylanbrc 586 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ (◡abs “ (0[,)𝑀))) |
23 | eqid 2798 | . . . . 5 ⊢ (abs ∘ − ) = (abs ∘ − ) | |
24 | 23 | cnbl0 23379 | . . . 4 ⊢ (𝑀 ∈ ℝ* → (◡abs “ (0[,)𝑀)) = (0(ball‘(abs ∘ − ))𝑀)) |
25 | 15, 24 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (◡abs “ (0[,)𝑀)) = (0(ball‘(abs ∘ − ))𝑀)) |
26 | 22, 25 | eleqtrd 2892 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀)) |
27 | icossicc 12814 | . . . 4 ⊢ (0[,)𝑀) ⊆ (0[,]𝑀) | |
28 | imass2 5932 | . . . 4 ⊢ ((0[,)𝑀) ⊆ (0[,]𝑀) → (◡abs “ (0[,)𝑀)) ⊆ (◡abs “ (0[,]𝑀))) | |
29 | 27, 28 | mp1i 13 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (◡abs “ (0[,)𝑀)) ⊆ (◡abs “ (0[,]𝑀))) |
30 | 25, 29 | eqsstrrd 3954 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ (◡abs “ (0[,]𝑀))) |
31 | iccssxr 12808 | . . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ* | |
32 | pserf.g | . . . . . . . 8 ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) | |
33 | pserf.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) | |
34 | pserf.r | . . . . . . . 8 ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) | |
35 | 32, 33, 34 | radcnvcl 25012 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (0[,]+∞)) |
36 | 35 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ (0[,]+∞)) |
37 | 31, 36 | sseldi 3913 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ ℝ*) |
38 | 11 | simp3d 1141 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 < 𝑅) |
39 | df-ico 12732 | . . . . . 6 ⊢ [,) = (𝑢 ∈ ℝ*, 𝑣 ∈ ℝ* ↦ {𝑤 ∈ ℝ* ∣ (𝑢 ≤ 𝑤 ∧ 𝑤 < 𝑣)}) | |
40 | df-icc 12733 | . . . . . 6 ⊢ [,] = (𝑢 ∈ ℝ*, 𝑣 ∈ ℝ* ↦ {𝑤 ∈ ℝ* ∣ (𝑢 ≤ 𝑤 ∧ 𝑤 ≤ 𝑣)}) | |
41 | xrlelttr 12537 | . . . . . 6 ⊢ ((𝑧 ∈ ℝ* ∧ 𝑀 ∈ ℝ* ∧ 𝑅 ∈ ℝ*) → ((𝑧 ≤ 𝑀 ∧ 𝑀 < 𝑅) → 𝑧 < 𝑅)) | |
42 | 39, 40, 41 | ixxss2 12745 | . . . . 5 ⊢ ((𝑅 ∈ ℝ* ∧ 𝑀 < 𝑅) → (0[,]𝑀) ⊆ (0[,)𝑅)) |
43 | 37, 38, 42 | syl2anc 587 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (0[,]𝑀) ⊆ (0[,)𝑅)) |
44 | imass2 5932 | . . . 4 ⊢ ((0[,]𝑀) ⊆ (0[,)𝑅) → (◡abs “ (0[,]𝑀)) ⊆ (◡abs “ (0[,)𝑅))) | |
45 | 43, 44 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (◡abs “ (0[,]𝑀)) ⊆ (◡abs “ (0[,)𝑅))) |
46 | 45, 1 | sseqtrrdi 3966 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (◡abs “ (0[,]𝑀)) ⊆ 𝑆) |
47 | 26, 30, 46 | 3jca 1125 | 1 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ∧ (0(ball‘(abs ∘ − ))𝑀) ⊆ (◡abs “ (0[,]𝑀)) ∧ (◡abs “ (0[,]𝑀)) ⊆ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 {crab 3110 ⊆ wss 3881 class class class wbr 5030 ↦ cmpt 5110 ◡ccnv 5518 dom cdm 5519 “ cima 5522 ∘ ccom 5523 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 supcsup 8888 ℂcc 10524 ℝcr 10525 0cc0 10526 + caddc 10529 · cmul 10531 +∞cpnf 10661 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 − cmin 10859 ℕ0cn0 11885 ℝ+crp 12377 [,)cico 12728 [,]cicc 12729 seqcseq 13364 ↑cexp 13425 abscabs 14585 ⇝ cli 14833 Σcsu 15034 ballcbl 20078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-xadd 12496 df-ico 12732 df-icc 12733 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 |
This theorem is referenced by: psercn 25021 pserdvlem2 25023 pserdv 25024 |
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