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Mirrors > Home > MPE Home > Th. List > logcnlem5 | Structured version Visualization version GIF version |
Description: Lemma for logcn 26496. (Contributed by Mario Carneiro, 18-Feb-2015.) |
Ref | Expression |
---|---|
logcn.d | ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) |
Ref | Expression |
---|---|
logcnlem5 | ⊢ (𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥))) ∈ (𝐷–cn→ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | logcn.d | . . 3 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
2 | difss 4131 | . . 3 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ | |
3 | 1, 2 | eqsstri 4016 | . 2 ⊢ 𝐷 ⊆ ℂ |
4 | ax-resscn 11173 | . 2 ⊢ ℝ ⊆ ℂ | |
5 | eqid 2731 | . . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥))) = (𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥))) | |
6 | 1 | ellogdm 26488 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ+))) |
7 | 6 | simplbi 497 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ) |
8 | 1 | logdmn0 26489 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0) |
9 | 7, 8 | logcld 26420 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 → (log‘𝑥) ∈ ℂ) |
10 | 9 | imcld 15149 | . . . 4 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ∈ ℝ) |
11 | 5, 10 | fmpti 7113 | . . 3 ⊢ (𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥))):𝐷⟶ℝ |
12 | eqid 2731 | . . . 4 ⊢ if(𝑦 ∈ ℝ+, 𝑦, (abs‘(ℑ‘𝑦))) = if(𝑦 ∈ ℝ+, 𝑦, (abs‘(ℑ‘𝑦))) | |
13 | eqid 2731 | . . . 4 ⊢ ((abs‘𝑦) · (𝑧 / (1 + 𝑧))) = ((abs‘𝑦) · (𝑧 / (1 + 𝑧))) | |
14 | simpl 482 | . . . 4 ⊢ ((𝑦 ∈ 𝐷 ∧ 𝑧 ∈ ℝ+) → 𝑦 ∈ 𝐷) | |
15 | simpr 484 | . . . 4 ⊢ ((𝑦 ∈ 𝐷 ∧ 𝑧 ∈ ℝ+) → 𝑧 ∈ ℝ+) | |
16 | 1, 12, 13, 14, 15 | logcnlem2 26492 | . . 3 ⊢ ((𝑦 ∈ 𝐷 ∧ 𝑧 ∈ ℝ+) → if(if(𝑦 ∈ ℝ+, 𝑦, (abs‘(ℑ‘𝑦))) ≤ ((abs‘𝑦) · (𝑧 / (1 + 𝑧))), if(𝑦 ∈ ℝ+, 𝑦, (abs‘(ℑ‘𝑦))), ((abs‘𝑦) · (𝑧 / (1 + 𝑧)))) ∈ ℝ+) |
17 | simpll 764 | . . . . . 6 ⊢ (((𝑦 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) ∧ (𝑧 ∈ ℝ+ ∧ (abs‘(𝑦 − 𝑤)) < if(if(𝑦 ∈ ℝ+, 𝑦, (abs‘(ℑ‘𝑦))) ≤ ((abs‘𝑦) · (𝑧 / (1 + 𝑧))), if(𝑦 ∈ ℝ+, 𝑦, (abs‘(ℑ‘𝑦))), ((abs‘𝑦) · (𝑧 / (1 + 𝑧)))))) → 𝑦 ∈ 𝐷) | |
18 | simprl 768 | . . . . . 6 ⊢ (((𝑦 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) ∧ (𝑧 ∈ ℝ+ ∧ (abs‘(𝑦 − 𝑤)) < if(if(𝑦 ∈ ℝ+, 𝑦, (abs‘(ℑ‘𝑦))) ≤ ((abs‘𝑦) · (𝑧 / (1 + 𝑧))), if(𝑦 ∈ ℝ+, 𝑦, (abs‘(ℑ‘𝑦))), ((abs‘𝑦) · (𝑧 / (1 + 𝑧)))))) → 𝑧 ∈ ℝ+) | |
19 | simplr 766 | . . . . . 6 ⊢ (((𝑦 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) ∧ (𝑧 ∈ ℝ+ ∧ (abs‘(𝑦 − 𝑤)) < if(if(𝑦 ∈ ℝ+, 𝑦, (abs‘(ℑ‘𝑦))) ≤ ((abs‘𝑦) · (𝑧 / (1 + 𝑧))), if(𝑦 ∈ ℝ+, 𝑦, (abs‘(ℑ‘𝑦))), ((abs‘𝑦) · (𝑧 / (1 + 𝑧)))))) → 𝑤 ∈ 𝐷) | |
20 | simprr 770 | . . . . . 6 ⊢ (((𝑦 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) ∧ (𝑧 ∈ ℝ+ ∧ (abs‘(𝑦 − 𝑤)) < if(if(𝑦 ∈ ℝ+, 𝑦, (abs‘(ℑ‘𝑦))) ≤ ((abs‘𝑦) · (𝑧 / (1 + 𝑧))), if(𝑦 ∈ ℝ+, 𝑦, (abs‘(ℑ‘𝑦))), ((abs‘𝑦) · (𝑧 / (1 + 𝑧)))))) → (abs‘(𝑦 − 𝑤)) < if(if(𝑦 ∈ ℝ+, 𝑦, (abs‘(ℑ‘𝑦))) ≤ ((abs‘𝑦) · (𝑧 / (1 + 𝑧))), if(𝑦 ∈ ℝ+, 𝑦, (abs‘(ℑ‘𝑦))), ((abs‘𝑦) · (𝑧 / (1 + 𝑧))))) | |
21 | 1, 12, 13, 17, 18, 19, 20 | logcnlem4 26494 | . . . . 5 ⊢ (((𝑦 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) ∧ (𝑧 ∈ ℝ+ ∧ (abs‘(𝑦 − 𝑤)) < if(if(𝑦 ∈ ℝ+, 𝑦, (abs‘(ℑ‘𝑦))) ≤ ((abs‘𝑦) · (𝑧 / (1 + 𝑧))), if(𝑦 ∈ ℝ+, 𝑦, (abs‘(ℑ‘𝑦))), ((abs‘𝑦) · (𝑧 / (1 + 𝑧)))))) → (abs‘((ℑ‘(log‘𝑦)) − (ℑ‘(log‘𝑤)))) < 𝑧) |
22 | 21 | expr 456 | . . . 4 ⊢ (((𝑦 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) ∧ 𝑧 ∈ ℝ+) → ((abs‘(𝑦 − 𝑤)) < if(if(𝑦 ∈ ℝ+, 𝑦, (abs‘(ℑ‘𝑦))) ≤ ((abs‘𝑦) · (𝑧 / (1 + 𝑧))), if(𝑦 ∈ ℝ+, 𝑦, (abs‘(ℑ‘𝑦))), ((abs‘𝑦) · (𝑧 / (1 + 𝑧)))) → (abs‘((ℑ‘(log‘𝑦)) − (ℑ‘(log‘𝑤)))) < 𝑧)) |
23 | 2fveq3 6896 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (ℑ‘(log‘𝑥)) = (ℑ‘(log‘𝑦))) | |
24 | fvex 6904 | . . . . . . . . 9 ⊢ (ℑ‘(log‘𝑦)) ∈ V | |
25 | 23, 5, 24 | fvmpt 6998 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐷 → ((𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥)))‘𝑦) = (ℑ‘(log‘𝑦))) |
26 | 25 | ad2antrr 723 | . . . . . . 7 ⊢ (((𝑦 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) ∧ 𝑧 ∈ ℝ+) → ((𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥)))‘𝑦) = (ℑ‘(log‘𝑦))) |
27 | 2fveq3 6896 | . . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (ℑ‘(log‘𝑥)) = (ℑ‘(log‘𝑤))) | |
28 | fvex 6904 | . . . . . . . . 9 ⊢ (ℑ‘(log‘𝑤)) ∈ V | |
29 | 27, 5, 28 | fvmpt 6998 | . . . . . . . 8 ⊢ (𝑤 ∈ 𝐷 → ((𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥)))‘𝑤) = (ℑ‘(log‘𝑤))) |
30 | 29 | ad2antlr 724 | . . . . . . 7 ⊢ (((𝑦 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) ∧ 𝑧 ∈ ℝ+) → ((𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥)))‘𝑤) = (ℑ‘(log‘𝑤))) |
31 | 26, 30 | oveq12d 7430 | . . . . . 6 ⊢ (((𝑦 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) ∧ 𝑧 ∈ ℝ+) → (((𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥)))‘𝑦) − ((𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥)))‘𝑤)) = ((ℑ‘(log‘𝑦)) − (ℑ‘(log‘𝑤)))) |
32 | 31 | fveq2d 6895 | . . . . 5 ⊢ (((𝑦 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) ∧ 𝑧 ∈ ℝ+) → (abs‘(((𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥)))‘𝑦) − ((𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥)))‘𝑤))) = (abs‘((ℑ‘(log‘𝑦)) − (ℑ‘(log‘𝑤))))) |
33 | 32 | breq1d 5158 | . . . 4 ⊢ (((𝑦 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) ∧ 𝑧 ∈ ℝ+) → ((abs‘(((𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥)))‘𝑦) − ((𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥)))‘𝑤))) < 𝑧 ↔ (abs‘((ℑ‘(log‘𝑦)) − (ℑ‘(log‘𝑤)))) < 𝑧)) |
34 | 22, 33 | sylibrd 259 | . . 3 ⊢ (((𝑦 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) ∧ 𝑧 ∈ ℝ+) → ((abs‘(𝑦 − 𝑤)) < if(if(𝑦 ∈ ℝ+, 𝑦, (abs‘(ℑ‘𝑦))) ≤ ((abs‘𝑦) · (𝑧 / (1 + 𝑧))), if(𝑦 ∈ ℝ+, 𝑦, (abs‘(ℑ‘𝑦))), ((abs‘𝑦) · (𝑧 / (1 + 𝑧)))) → (abs‘(((𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥)))‘𝑦) − ((𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥)))‘𝑤))) < 𝑧)) |
35 | 11, 16, 34 | elcncf1ii 24737 | . 2 ⊢ ((𝐷 ⊆ ℂ ∧ ℝ ⊆ ℂ) → (𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥))) ∈ (𝐷–cn→ℝ)) |
36 | 3, 4, 35 | mp2an 689 | 1 ⊢ (𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥))) ∈ (𝐷–cn→ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∖ cdif 3945 ⊆ wss 3948 ifcif 4528 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 ℝcr 11115 0cc0 11116 1c1 11117 + caddc 11119 · cmul 11121 -∞cmnf 11253 < clt 11255 ≤ cle 11256 − cmin 11451 / cdiv 11878 ℝ+crp 12981 (,]cioc 13332 ℑcim 15052 abscabs 15188 –cn→ccncf 24717 logclog 26404 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-er 8709 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-fi 9412 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ioo 13335 df-ioc 13336 df-ico 13337 df-icc 13338 df-fz 13492 df-fzo 13635 df-fl 13764 df-mod 13842 df-seq 13974 df-exp 14035 df-fac 14241 df-bc 14270 df-hash 14298 df-shft 15021 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-limsup 15422 df-clim 15439 df-rlim 15440 df-sum 15640 df-ef 16018 df-sin 16020 df-cos 16021 df-tan 16022 df-pi 16023 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-mulg 18994 df-cntz 19229 df-cmn 19698 df-psmet 21226 df-xmet 21227 df-met 21228 df-bl 21229 df-mopn 21230 df-fbas 21231 df-fg 21232 df-cnfld 21235 df-top 22717 df-topon 22734 df-topsp 22756 df-bases 22770 df-cld 22844 df-ntr 22845 df-cls 22846 df-nei 22923 df-lp 22961 df-perf 22962 df-cn 23052 df-cnp 23053 df-haus 23140 df-tx 23387 df-hmeo 23580 df-fil 23671 df-fm 23763 df-flim 23764 df-flf 23765 df-xms 24147 df-ms 24148 df-tms 24149 df-cncf 24719 df-limc 25716 df-dv 25717 df-log 26406 |
This theorem is referenced by: logcn 26496 |
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