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Mirrors > Home > MPE Home > Th. List > pserdvlem1 | Structured version Visualization version GIF version |
Description: Lemma for pserdv 24303. (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 5595 | . . . . . . . . . 10 ⊢ (◡abs “ (0[,)𝑅)) ⊆ dom abs | |
3 | absf 14197 | . . . . . . . . . . 11 ⊢ abs:ℂ⟶ℝ | |
4 | 3 | fdmi 6165 | . . . . . . . . . 10 ⊢ dom abs = ℂ |
5 | 2, 4 | sseqtri 3743 | . . . . . . . . 9 ⊢ (◡abs “ (0[,)𝑅)) ⊆ ℂ |
6 | 1, 5 | eqsstri 3741 | . . . . . . . 8 ⊢ 𝑆 ⊆ ℂ |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
8 | 7 | sselda 3709 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ℂ) |
9 | 8 | abscld 14295 | . . . . 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 24299 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑀 ∈ ℝ+ ∧ (abs‘𝑎) < 𝑀 ∧ 𝑀 < 𝑅)) |
16 | 15 | simp1d 1134 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ+) |
17 | 16 | rpred 11986 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ) |
18 | 9, 17 | readdcld 10182 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) + 𝑀) ∈ ℝ) |
19 | 0red 10154 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 ∈ ℝ) | |
20 | 8 | absge0d 14303 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 ≤ (abs‘𝑎)) |
21 | 9, 16 | ltaddrpd 12019 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < ((abs‘𝑎) + 𝑀)) |
22 | 19, 9, 18, 20, 21 | lelttrd 10308 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 < ((abs‘𝑎) + 𝑀)) |
23 | 18, 22 | elrpd 11983 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) + 𝑀) ∈ ℝ+) |
24 | 23 | rphalfcld 11998 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((abs‘𝑎) + 𝑀) / 2) ∈ ℝ+) |
25 | 15 | simp2d 1135 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < 𝑀) |
26 | avglt1 11383 | . . . 4 ⊢ (((abs‘𝑎) ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((abs‘𝑎) < 𝑀 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑀) / 2))) | |
27 | 9, 17, 26 | syl2anc 696 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) < 𝑀 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑀) / 2))) |
28 | 25, 27 | mpbid 222 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < (((abs‘𝑎) + 𝑀) / 2)) |
29 | 18 | rehalfcld 11392 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((abs‘𝑎) + 𝑀) / 2) ∈ ℝ) |
30 | 29 | rexrd 10202 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((abs‘𝑎) + 𝑀) / 2) ∈ ℝ*) |
31 | 17 | rexrd 10202 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ*) |
32 | iccssxr 12370 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
33 | 10, 12, 13 | radcnvcl 24291 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ (0[,]+∞)) |
34 | 32, 33 | sseldi 3707 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ*) |
35 | 34 | adantr 472 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ ℝ*) |
36 | avglt2 11384 | . . . . 5 ⊢ (((abs‘𝑎) ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((abs‘𝑎) < 𝑀 ↔ (((abs‘𝑎) + 𝑀) / 2) < 𝑀)) | |
37 | 9, 17, 36 | syl2anc 696 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) < 𝑀 ↔ (((abs‘𝑎) + 𝑀) / 2) < 𝑀)) |
38 | 25, 37 | mpbid 222 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((abs‘𝑎) + 𝑀) / 2) < 𝑀) |
39 | 15 | simp3d 1136 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 < 𝑅) |
40 | 30, 31, 35, 38, 39 | xrlttrd 12104 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((abs‘𝑎) + 𝑀) / 2) < 𝑅) |
41 | 24, 28, 40 | 3jca 1379 | 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 196 ∧ wa 383 ∧ w3a 1072 = wceq 1596 ∈ wcel 2103 {crab 3018 ⊆ wss 3680 ifcif 4194 class class class wbr 4760 ↦ cmpt 4837 ◡ccnv 5217 dom cdm 5218 “ cima 5221 ⟶wf 5997 ‘cfv 6001 (class class class)co 6765 supcsup 8462 ℂcc 10047 ℝcr 10048 0cc0 10049 1c1 10050 + caddc 10052 · cmul 10054 +∞cpnf 10184 ℝ*cxr 10186 < clt 10187 / cdiv 10797 2c2 11183 ℕ0cn0 11405 ℝ+crp 11946 [,)cico 12291 [,]cicc 12292 seqcseq 12916 ↑cexp 12975 abscabs 14094 ⇝ cli 14335 Σcsu 14536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1835 ax-4 1850 ax-5 1952 ax-6 2018 ax-7 2054 ax-8 2105 ax-9 2112 ax-10 2132 ax-11 2147 ax-12 2160 ax-13 2355 ax-ext 2704 ax-rep 4879 ax-sep 4889 ax-nul 4897 ax-pow 4948 ax-pr 5011 ax-un 7066 ax-inf2 8651 ax-cnex 10105 ax-resscn 10106 ax-1cn 10107 ax-icn 10108 ax-addcl 10109 ax-addrcl 10110 ax-mulcl 10111 ax-mulrcl 10112 ax-mulcom 10113 ax-addass 10114 ax-mulass 10115 ax-distr 10116 ax-i2m1 10117 ax-1ne0 10118 ax-1rid 10119 ax-rnegex 10120 ax-rrecex 10121 ax-cnre 10122 ax-pre-lttri 10123 ax-pre-lttrn 10124 ax-pre-ltadd 10125 ax-pre-mulgt0 10126 ax-pre-sup 10127 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1599 df-ex 1818 df-nf 1823 df-sb 2011 df-eu 2575 df-mo 2576 df-clab 2711 df-cleq 2717 df-clel 2720 df-nfc 2855 df-ne 2897 df-nel 3000 df-ral 3019 df-rex 3020 df-reu 3021 df-rmo 3022 df-rab 3023 df-v 3306 df-sbc 3542 df-csb 3640 df-dif 3683 df-un 3685 df-in 3687 df-ss 3694 df-pss 3696 df-nul 4024 df-if 4195 df-pw 4268 df-sn 4286 df-pr 4288 df-tp 4290 df-op 4292 df-uni 4545 df-iun 4630 df-br 4761 df-opab 4821 df-mpt 4838 df-tr 4861 df-id 5128 df-eprel 5133 df-po 5139 df-so 5140 df-fr 5177 df-we 5179 df-xp 5224 df-rel 5225 df-cnv 5226 df-co 5227 df-dm 5228 df-rn 5229 df-res 5230 df-ima 5231 df-pred 5793 df-ord 5839 df-on 5840 df-lim 5841 df-suc 5842 df-iota 5964 df-fun 6003 df-fn 6004 df-f 6005 df-f1 6006 df-fo 6007 df-f1o 6008 df-fv 6009 df-riota 6726 df-ov 6768 df-oprab 6769 df-mpt2 6770 df-om 7183 df-1st 7285 df-2nd 7286 df-wrecs 7527 df-recs 7588 df-rdg 7626 df-1o 7680 df-er 7862 df-en 8073 df-dom 8074 df-sdom 8075 df-fin 8076 df-sup 8464 df-pnf 10189 df-mnf 10190 df-xr 10191 df-ltxr 10192 df-le 10193 df-sub 10381 df-neg 10382 df-div 10798 df-nn 11134 df-2 11192 df-3 11193 df-n0 11406 df-z 11491 df-uz 11801 df-rp 11947 df-ico 12295 df-icc 12296 df-fz 12441 df-seq 12917 df-exp 12976 df-cj 13959 df-re 13960 df-im 13961 df-sqrt 14095 df-abs 14096 df-clim 14339 |
This theorem is referenced by: pserdvlem2 24302 pserdv 24303 |
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