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Mirrors > Home > MPE Home > Th. List > pserdvlem1 | Structured version Visualization version GIF version |
Description: Lemma for pserdv 25020. (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 5952 | . . . . . . . . . 10 ⊢ (◡abs “ (0[,)𝑅)) ⊆ dom abs | |
3 | absf 14700 | . . . . . . . . . . 11 ⊢ abs:ℂ⟶ℝ | |
4 | 3 | fdmi 6527 | . . . . . . . . . 10 ⊢ dom abs = ℂ |
5 | 2, 4 | sseqtri 4006 | . . . . . . . . 9 ⊢ (◡abs “ (0[,)𝑅)) ⊆ ℂ |
6 | 1, 5 | eqsstri 4004 | . . . . . . . 8 ⊢ 𝑆 ⊆ ℂ |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
8 | 7 | sselda 3970 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ℂ) |
9 | 8 | abscld 14799 | . . . . 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 25016 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑀 ∈ ℝ+ ∧ (abs‘𝑎) < 𝑀 ∧ 𝑀 < 𝑅)) |
16 | 15 | simp1d 1138 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ+) |
17 | 16 | rpred 12434 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ) |
18 | 9, 17 | readdcld 10673 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) + 𝑀) ∈ ℝ) |
19 | 0red 10647 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 ∈ ℝ) | |
20 | 8 | absge0d 14807 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 ≤ (abs‘𝑎)) |
21 | 9, 16 | ltaddrpd 12467 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < ((abs‘𝑎) + 𝑀)) |
22 | 19, 9, 18, 20, 21 | lelttrd 10801 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 < ((abs‘𝑎) + 𝑀)) |
23 | 18, 22 | elrpd 12431 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) + 𝑀) ∈ ℝ+) |
24 | 23 | rphalfcld 12446 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((abs‘𝑎) + 𝑀) / 2) ∈ ℝ+) |
25 | 15 | simp2d 1139 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < 𝑀) |
26 | avglt1 11878 | . . . 4 ⊢ (((abs‘𝑎) ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((abs‘𝑎) < 𝑀 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑀) / 2))) | |
27 | 9, 17, 26 | syl2anc 586 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) < 𝑀 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑀) / 2))) |
28 | 25, 27 | mpbid 234 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < (((abs‘𝑎) + 𝑀) / 2)) |
29 | 18 | rehalfcld 11887 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((abs‘𝑎) + 𝑀) / 2) ∈ ℝ) |
30 | 29 | rexrd 10694 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((abs‘𝑎) + 𝑀) / 2) ∈ ℝ*) |
31 | 17 | rexrd 10694 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ*) |
32 | iccssxr 12822 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
33 | 10, 12, 13 | radcnvcl 25008 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ (0[,]+∞)) |
34 | 32, 33 | sseldi 3968 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ*) |
35 | 34 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ ℝ*) |
36 | avglt2 11879 | . . . . 5 ⊢ (((abs‘𝑎) ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((abs‘𝑎) < 𝑀 ↔ (((abs‘𝑎) + 𝑀) / 2) < 𝑀)) | |
37 | 9, 17, 36 | syl2anc 586 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) < 𝑀 ↔ (((abs‘𝑎) + 𝑀) / 2) < 𝑀)) |
38 | 25, 37 | mpbid 234 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((abs‘𝑎) + 𝑀) / 2) < 𝑀) |
39 | 15 | simp3d 1140 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 < 𝑅) |
40 | 30, 31, 35, 38, 39 | xrlttrd 12555 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((abs‘𝑎) + 𝑀) / 2) < 𝑅) |
41 | 24, 28, 40 | 3jca 1124 | 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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 {crab 3145 ⊆ wss 3939 ifcif 4470 class class class wbr 5069 ↦ cmpt 5149 ◡ccnv 5557 dom cdm 5558 “ cima 5561 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 supcsup 8907 ℂcc 10538 ℝcr 10539 0cc0 10540 1c1 10541 + caddc 10543 · cmul 10545 +∞cpnf 10675 ℝ*cxr 10677 < clt 10678 / cdiv 11300 2c2 11695 ℕ0cn0 11900 ℝ+crp 12392 [,)cico 12743 [,]cicc 12744 seqcseq 13372 ↑cexp 13432 abscabs 14596 ⇝ cli 14844 Σcsu 15045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-ico 12747 df-icc 12748 df-fz 12896 df-seq 13373 df-exp 13433 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-clim 14848 |
This theorem is referenced by: pserdvlem2 25019 pserdv 25020 |
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