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Mirrors > Home > MPE Home > Th. List > logcnlem5 | Structured version Visualization version GIF version |
Description: Lemma for logcn 26704. (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 4146 | . . 3 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ | |
3 | 1, 2 | eqsstri 4030 | . 2 ⊢ 𝐷 ⊆ ℂ |
4 | ax-resscn 11210 | . 2 ⊢ ℝ ⊆ ℂ | |
5 | eqid 2735 | . . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥))) = (𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥))) | |
6 | 1 | ellogdm 26696 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ+))) |
7 | 6 | simplbi 497 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ) |
8 | 1 | logdmn0 26697 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0) |
9 | 7, 8 | logcld 26627 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 → (log‘𝑥) ∈ ℂ) |
10 | 9 | imcld 15231 | . . . 4 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ∈ ℝ) |
11 | 5, 10 | fmpti 7132 | . . 3 ⊢ (𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥))):𝐷⟶ℝ |
12 | eqid 2735 | . . . 4 ⊢ if(𝑦 ∈ ℝ+, 𝑦, (abs‘(ℑ‘𝑦))) = if(𝑦 ∈ ℝ+, 𝑦, (abs‘(ℑ‘𝑦))) | |
13 | eqid 2735 | . . . 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 26700 | . . 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 767 | . . . . . 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 771 | . . . . . 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 769 | . . . . . 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 773 | . . . . . 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 26702 | . . . . 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 6912 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (ℑ‘(log‘𝑥)) = (ℑ‘(log‘𝑦))) | |
24 | fvex 6920 | . . . . . . . . 9 ⊢ (ℑ‘(log‘𝑦)) ∈ V | |
25 | 23, 5, 24 | fvmpt 7016 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐷 → ((𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥)))‘𝑦) = (ℑ‘(log‘𝑦))) |
26 | 25 | ad2antrr 726 | . . . . . . 7 ⊢ (((𝑦 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) ∧ 𝑧 ∈ ℝ+) → ((𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥)))‘𝑦) = (ℑ‘(log‘𝑦))) |
27 | 2fveq3 6912 | . . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (ℑ‘(log‘𝑥)) = (ℑ‘(log‘𝑤))) | |
28 | fvex 6920 | . . . . . . . . 9 ⊢ (ℑ‘(log‘𝑤)) ∈ V | |
29 | 27, 5, 28 | fvmpt 7016 | . . . . . . . 8 ⊢ (𝑤 ∈ 𝐷 → ((𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥)))‘𝑤) = (ℑ‘(log‘𝑤))) |
30 | 29 | ad2antlr 727 | . . . . . . 7 ⊢ (((𝑦 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) ∧ 𝑧 ∈ ℝ+) → ((𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥)))‘𝑤) = (ℑ‘(log‘𝑤))) |
31 | 26, 30 | oveq12d 7449 | . . . . . 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 6911 | . . . . 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 24936 | . 2 ⊢ ((𝐷 ⊆ ℂ ∧ ℝ ⊆ ℂ) → (𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥))) ∈ (𝐷–cn→ℝ)) |
36 | 3, 4, 35 | mp2an 692 | 1 ⊢ (𝑥 ∈ 𝐷 ↦ (ℑ‘(log‘𝑥))) ∈ (𝐷–cn→ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∖ cdif 3960 ⊆ wss 3963 ifcif 4531 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 -∞cmnf 11291 < clt 11293 ≤ cle 11294 − cmin 11490 / cdiv 11918 ℝ+crp 13032 (,]cioc 13385 ℑcim 15134 abscabs 15270 –cn→ccncf 24916 logclog 26611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-tan 16104 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613 |
This theorem is referenced by: logcn 26704 |
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