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Mirrors > Home > MPE Home > Th. List > logneg | Structured version Visualization version GIF version |
Description: The natural logarithm of a negative real number. (Contributed by Mario Carneiro, 13-May-2014.) (Revised by Mario Carneiro, 3-Apr-2015.) |
Ref | Expression |
---|---|
logneg | ⊢ (𝐴 ∈ ℝ+ → (log‘-𝐴) = ((log‘𝐴) + (i · π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relogcl 26529 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ) | |
2 | 1 | recnd 11280 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℂ) |
3 | ax-icn 11205 | . . . . . 6 ⊢ i ∈ ℂ | |
4 | picn 26414 | . . . . . 6 ⊢ π ∈ ℂ | |
5 | 3, 4 | mulcli 11259 | . . . . 5 ⊢ (i · π) ∈ ℂ |
6 | efadd 16078 | . . . . 5 ⊢ (((log‘𝐴) ∈ ℂ ∧ (i · π) ∈ ℂ) → (exp‘((log‘𝐴) + (i · π))) = ((exp‘(log‘𝐴)) · (exp‘(i · π)))) | |
7 | 2, 5, 6 | sylancl 584 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (exp‘((log‘𝐴) + (i · π))) = ((exp‘(log‘𝐴)) · (exp‘(i · π)))) |
8 | efipi 26428 | . . . . . 6 ⊢ (exp‘(i · π)) = -1 | |
9 | 8 | oveq2i 7437 | . . . . 5 ⊢ ((exp‘(log‘𝐴)) · (exp‘(i · π))) = ((exp‘(log‘𝐴)) · -1) |
10 | reeflog 26534 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → (exp‘(log‘𝐴)) = 𝐴) | |
11 | 10 | oveq1d 7441 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → ((exp‘(log‘𝐴)) · -1) = (𝐴 · -1)) |
12 | 9, 11 | eqtrid 2780 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → ((exp‘(log‘𝐴)) · (exp‘(i · π))) = (𝐴 · -1)) |
13 | rpcn 13024 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) | |
14 | neg1cn 12364 | . . . . . 6 ⊢ -1 ∈ ℂ | |
15 | mulcom 11232 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ -1 ∈ ℂ) → (𝐴 · -1) = (-1 · 𝐴)) | |
16 | 13, 14, 15 | sylancl 584 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (𝐴 · -1) = (-1 · 𝐴)) |
17 | 13 | mulm1d 11704 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (-1 · 𝐴) = -𝐴) |
18 | 16, 17 | eqtrd 2768 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (𝐴 · -1) = -𝐴) |
19 | 7, 12, 18 | 3eqtrd 2772 | . . 3 ⊢ (𝐴 ∈ ℝ+ → (exp‘((log‘𝐴) + (i · π))) = -𝐴) |
20 | 19 | fveq2d 6906 | . 2 ⊢ (𝐴 ∈ ℝ+ → (log‘(exp‘((log‘𝐴) + (i · π)))) = (log‘-𝐴)) |
21 | addcl 11228 | . . . . 5 ⊢ (((log‘𝐴) ∈ ℂ ∧ (i · π) ∈ ℂ) → ((log‘𝐴) + (i · π)) ∈ ℂ) | |
22 | 2, 5, 21 | sylancl 584 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → ((log‘𝐴) + (i · π)) ∈ ℂ) |
23 | pipos 26415 | . . . . . . 7 ⊢ 0 < π | |
24 | pire 26413 | . . . . . . . 8 ⊢ π ∈ ℝ | |
25 | lt0neg2 11759 | . . . . . . . 8 ⊢ (π ∈ ℝ → (0 < π ↔ -π < 0)) | |
26 | 24, 25 | ax-mp 5 | . . . . . . 7 ⊢ (0 < π ↔ -π < 0) |
27 | 23, 26 | mpbi 229 | . . . . . 6 ⊢ -π < 0 |
28 | 24 | renegcli 11559 | . . . . . . 7 ⊢ -π ∈ ℝ |
29 | 0re 11254 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
30 | 28, 29, 24 | lttri 11378 | . . . . . 6 ⊢ ((-π < 0 ∧ 0 < π) → -π < π) |
31 | 27, 23, 30 | mp2an 690 | . . . . 5 ⊢ -π < π |
32 | crim 15102 | . . . . . 6 ⊢ (((log‘𝐴) ∈ ℝ ∧ π ∈ ℝ) → (ℑ‘((log‘𝐴) + (i · π))) = π) | |
33 | 1, 24, 32 | sylancl 584 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (ℑ‘((log‘𝐴) + (i · π))) = π) |
34 | 31, 33 | breqtrrid 5190 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → -π < (ℑ‘((log‘𝐴) + (i · π)))) |
35 | 24 | leidi 11786 | . . . . 5 ⊢ π ≤ π |
36 | 33, 35 | eqbrtrdi 5191 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (ℑ‘((log‘𝐴) + (i · π))) ≤ π) |
37 | ellogrn 26513 | . . . 4 ⊢ (((log‘𝐴) + (i · π)) ∈ ran log ↔ (((log‘𝐴) + (i · π)) ∈ ℂ ∧ -π < (ℑ‘((log‘𝐴) + (i · π))) ∧ (ℑ‘((log‘𝐴) + (i · π))) ≤ π)) | |
38 | 22, 34, 36, 37 | syl3anbrc 1340 | . . 3 ⊢ (𝐴 ∈ ℝ+ → ((log‘𝐴) + (i · π)) ∈ ran log) |
39 | logef 26535 | . . 3 ⊢ (((log‘𝐴) + (i · π)) ∈ ran log → (log‘(exp‘((log‘𝐴) + (i · π)))) = ((log‘𝐴) + (i · π))) | |
40 | 38, 39 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ+ → (log‘(exp‘((log‘𝐴) + (i · π)))) = ((log‘𝐴) + (i · π))) |
41 | 20, 40 | eqtr3d 2770 | 1 ⊢ (𝐴 ∈ ℝ+ → (log‘-𝐴) = ((log‘𝐴) + (i · π))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 class class class wbr 5152 ran crn 5683 ‘cfv 6553 (class class class)co 7426 ℂcc 11144 ℝcr 11145 0cc0 11146 1c1 11147 ici 11148 + caddc 11149 · cmul 11151 < clt 11286 ≤ cle 11287 -cneg 11483 ℝ+crp 13014 ℑcim 15085 expce 16045 πcpi 16050 logclog 26508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ioc 13369 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-mod 13875 df-seq 14007 df-exp 14067 df-fac 14273 df-bc 14302 df-hash 14330 df-shft 15054 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-limsup 15455 df-clim 15472 df-rlim 15473 df-sum 15673 df-ef 16051 df-sin 16053 df-cos 16054 df-pi 16056 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-pt 17433 df-prds 17436 df-xrs 17491 df-qtop 17496 df-imas 17497 df-xps 17499 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-mulg 19031 df-cntz 19275 df-cmn 19744 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cld 22943 df-ntr 22944 df-cls 22945 df-nei 23022 df-lp 23060 df-perf 23061 df-cn 23151 df-cnp 23152 df-haus 23239 df-tx 23486 df-hmeo 23679 df-fil 23770 df-fm 23862 df-flim 23863 df-flf 23864 df-xms 24246 df-ms 24247 df-tms 24248 df-cncf 24818 df-limc 25815 df-dv 25816 df-log 26510 |
This theorem is referenced by: logm1 26543 lognegb 26544 cxpsqrt 26657 |
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