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Mirrors > Home > MPE Home > Th. List > pserdvlem1 | Structured version Visualization version GIF version |
Description: Lemma for pserdv 25321. (Contributed by Mario Carneiro, 7-May-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)) |
psercn.m | ⊢ 𝑀 = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) |
Ref | Expression |
---|---|
pserdvlem1 | ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((((abs‘𝑎) + 𝑀) / 2) ∈ ℝ+ ∧ (abs‘𝑎) < (((abs‘𝑎) + 𝑀) / 2) ∧ (((abs‘𝑎) + 𝑀) / 2) < 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psercn.s | . . . . . . . . 9 ⊢ 𝑆 = (◡abs “ (0[,)𝑅)) | |
2 | cnvimass 5949 | . . . . . . . . . 10 ⊢ (◡abs “ (0[,)𝑅)) ⊆ dom abs | |
3 | absf 14901 | . . . . . . . . . . 11 ⊢ abs:ℂ⟶ℝ | |
4 | 3 | fdmi 6557 | . . . . . . . . . 10 ⊢ dom abs = ℂ |
5 | 2, 4 | sseqtri 3937 | . . . . . . . . 9 ⊢ (◡abs “ (0[,)𝑅)) ⊆ ℂ |
6 | 1, 5 | eqsstri 3935 | . . . . . . . 8 ⊢ 𝑆 ⊆ ℂ |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
8 | 7 | sselda 3901 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ℂ) |
9 | 8 | abscld 15000 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) ∈ ℝ) |
10 | pserf.g | . . . . . . . 8 ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) | |
11 | pserf.f | . . . . . . . 8 ⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗)) | |
12 | pserf.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) | |
13 | pserf.r | . . . . . . . 8 ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) | |
14 | psercn.m | . . . . . . . 8 ⊢ 𝑀 = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) | |
15 | 10, 11, 12, 13, 1, 14 | psercnlem1 25317 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑀 ∈ ℝ+ ∧ (abs‘𝑎) < 𝑀 ∧ 𝑀 < 𝑅)) |
16 | 15 | simp1d 1144 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ+) |
17 | 16 | rpred 12628 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ) |
18 | 9, 17 | readdcld 10862 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) + 𝑀) ∈ ℝ) |
19 | 0red 10836 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 ∈ ℝ) | |
20 | 8 | absge0d 15008 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 ≤ (abs‘𝑎)) |
21 | 9, 16 | ltaddrpd 12661 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < ((abs‘𝑎) + 𝑀)) |
22 | 19, 9, 18, 20, 21 | lelttrd 10990 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 < ((abs‘𝑎) + 𝑀)) |
23 | 18, 22 | elrpd 12625 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) + 𝑀) ∈ ℝ+) |
24 | 23 | rphalfcld 12640 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((abs‘𝑎) + 𝑀) / 2) ∈ ℝ+) |
25 | 15 | simp2d 1145 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < 𝑀) |
26 | avglt1 12068 | . . . 4 ⊢ (((abs‘𝑎) ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((abs‘𝑎) < 𝑀 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑀) / 2))) | |
27 | 9, 17, 26 | syl2anc 587 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) < 𝑀 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑀) / 2))) |
28 | 25, 27 | mpbid 235 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < (((abs‘𝑎) + 𝑀) / 2)) |
29 | 18 | rehalfcld 12077 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((abs‘𝑎) + 𝑀) / 2) ∈ ℝ) |
30 | 29 | rexrd 10883 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((abs‘𝑎) + 𝑀) / 2) ∈ ℝ*) |
31 | 17 | rexrd 10883 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ*) |
32 | iccssxr 13018 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
33 | 10, 12, 13 | radcnvcl 25309 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ (0[,]+∞)) |
34 | 32, 33 | sseldi 3899 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ*) |
35 | 34 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ ℝ*) |
36 | avglt2 12069 | . . . . 5 ⊢ (((abs‘𝑎) ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((abs‘𝑎) < 𝑀 ↔ (((abs‘𝑎) + 𝑀) / 2) < 𝑀)) | |
37 | 9, 17, 36 | syl2anc 587 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) < 𝑀 ↔ (((abs‘𝑎) + 𝑀) / 2) < 𝑀)) |
38 | 25, 37 | mpbid 235 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((abs‘𝑎) + 𝑀) / 2) < 𝑀) |
39 | 15 | simp3d 1146 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 < 𝑅) |
40 | 30, 31, 35, 38, 39 | xrlttrd 12749 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((abs‘𝑎) + 𝑀) / 2) < 𝑅) |
41 | 24, 28, 40 | 3jca 1130 | 1 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((((abs‘𝑎) + 𝑀) / 2) ∈ ℝ+ ∧ (abs‘𝑎) < (((abs‘𝑎) + 𝑀) / 2) ∧ (((abs‘𝑎) + 𝑀) / 2) < 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 {crab 3065 ⊆ wss 3866 ifcif 4439 class class class wbr 5053 ↦ cmpt 5135 ◡ccnv 5550 dom cdm 5551 “ cima 5554 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 supcsup 9056 ℂcc 10727 ℝcr 10728 0cc0 10729 1c1 10730 + caddc 10732 · cmul 10734 +∞cpnf 10864 ℝ*cxr 10866 < clt 10867 / cdiv 11489 2c2 11885 ℕ0cn0 12090 ℝ+crp 12586 [,)cico 12937 [,]cicc 12938 seqcseq 13574 ↑cexp 13635 abscabs 14797 ⇝ cli 15045 Σcsu 15249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-ico 12941 df-icc 12942 df-fz 13096 df-seq 13575 df-exp 13636 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 |
This theorem is referenced by: pserdvlem2 25320 pserdv 25321 |
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