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| Mirrors > Home > MPE Home > Th. List > pserdvlem1 | Structured version Visualization version GIF version | ||
| Description: Lemma for pserdv 26473. (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 6100 | . . . . . . . . . 10 ⊢ (◡abs “ (0[,)𝑅)) ⊆ dom abs | |
| 3 | absf 15376 | . . . . . . . . . . 11 ⊢ abs:ℂ⟶ℝ | |
| 4 | 3 | fdmi 6747 | . . . . . . . . . 10 ⊢ dom abs = ℂ |
| 5 | 2, 4 | sseqtri 4032 | . . . . . . . . 9 ⊢ (◡abs “ (0[,)𝑅)) ⊆ ℂ |
| 6 | 1, 5 | eqsstri 4030 | . . . . . . . 8 ⊢ 𝑆 ⊆ ℂ |
| 7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
| 8 | 7 | sselda 3983 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ℂ) |
| 9 | 8 | abscld 15475 | . . . . 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 26469 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑀 ∈ ℝ+ ∧ (abs‘𝑎) < 𝑀 ∧ 𝑀 < 𝑅)) |
| 16 | 15 | simp1d 1143 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ+) |
| 17 | 16 | rpred 13077 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ) |
| 18 | 9, 17 | readdcld 11290 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) + 𝑀) ∈ ℝ) |
| 19 | 0red 11264 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 ∈ ℝ) | |
| 20 | 8 | absge0d 15483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 ≤ (abs‘𝑎)) |
| 21 | 9, 16 | ltaddrpd 13110 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < ((abs‘𝑎) + 𝑀)) |
| 22 | 19, 9, 18, 20, 21 | lelttrd 11419 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 < ((abs‘𝑎) + 𝑀)) |
| 23 | 18, 22 | elrpd 13074 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) + 𝑀) ∈ ℝ+) |
| 24 | 23 | rphalfcld 13089 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((abs‘𝑎) + 𝑀) / 2) ∈ ℝ+) |
| 25 | 15 | simp2d 1144 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < 𝑀) |
| 26 | avglt1 12504 | . . . 4 ⊢ (((abs‘𝑎) ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((abs‘𝑎) < 𝑀 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑀) / 2))) | |
| 27 | 9, 17, 26 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) < 𝑀 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑀) / 2))) |
| 28 | 25, 27 | mpbid 232 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < (((abs‘𝑎) + 𝑀) / 2)) |
| 29 | 18 | rehalfcld 12513 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((abs‘𝑎) + 𝑀) / 2) ∈ ℝ) |
| 30 | 29 | rexrd 11311 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((abs‘𝑎) + 𝑀) / 2) ∈ ℝ*) |
| 31 | 17 | rexrd 11311 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ*) |
| 32 | iccssxr 13470 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
| 33 | 10, 12, 13 | radcnvcl 26460 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ (0[,]+∞)) |
| 34 | 32, 33 | sselid 3981 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ*) |
| 35 | 34 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ ℝ*) |
| 36 | avglt2 12505 | . . . . 5 ⊢ (((abs‘𝑎) ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((abs‘𝑎) < 𝑀 ↔ (((abs‘𝑎) + 𝑀) / 2) < 𝑀)) | |
| 37 | 9, 17, 36 | syl2anc 584 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) < 𝑀 ↔ (((abs‘𝑎) + 𝑀) / 2) < 𝑀)) |
| 38 | 25, 37 | mpbid 232 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((abs‘𝑎) + 𝑀) / 2) < 𝑀) |
| 39 | 15 | simp3d 1145 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 < 𝑅) |
| 40 | 30, 31, 35, 38, 39 | xrlttrd 13201 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((abs‘𝑎) + 𝑀) / 2) < 𝑅) |
| 41 | 24, 28, 40 | 3jca 1129 | 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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 {crab 3436 ⊆ wss 3951 ifcif 4525 class class class wbr 5143 ↦ cmpt 5225 ◡ccnv 5684 dom cdm 5685 “ cima 5688 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 supcsup 9480 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 +∞cpnf 11292 ℝ*cxr 11294 < clt 11295 / cdiv 11920 2c2 12321 ℕ0cn0 12526 ℝ+crp 13034 [,)cico 13389 [,]cicc 13390 seqcseq 14042 ↑cexp 14102 abscabs 15273 ⇝ cli 15520 Σcsu 15722 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-ico 13393 df-icc 13394 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 |
| This theorem is referenced by: pserdvlem2 26472 pserdv 26473 |
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