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Mirrors > Home > MPE Home > Th. List > logrnaddcl | Structured version Visualization version GIF version |
Description: The range of the natural logarithm is closed under addition with reals. (Contributed by Mario Carneiro, 3-Apr-2015.) |
Ref | Expression |
---|---|
logrnaddcl | ⊢ ((𝐴 ∈ ran log ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ran log) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | logrncn 25934 | . . 3 ⊢ (𝐴 ∈ ran log → 𝐴 ∈ ℂ) | |
2 | recn 11146 | . . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
3 | addcl 11138 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) | |
4 | 1, 2, 3 | syl2an 597 | . 2 ⊢ ((𝐴 ∈ ran log ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℂ) |
5 | ellogrn 25931 | . . . . 5 ⊢ (𝐴 ∈ ran log ↔ (𝐴 ∈ ℂ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)) | |
6 | 5 | simp2bi 1147 | . . . 4 ⊢ (𝐴 ∈ ran log → -π < (ℑ‘𝐴)) |
7 | 6 | adantr 482 | . . 3 ⊢ ((𝐴 ∈ ran log ∧ 𝐵 ∈ ℝ) → -π < (ℑ‘𝐴)) |
8 | imadd 15025 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 + 𝐵)) = ((ℑ‘𝐴) + (ℑ‘𝐵))) | |
9 | 1, 2, 8 | syl2an 597 | . . . 4 ⊢ ((𝐴 ∈ ran log ∧ 𝐵 ∈ ℝ) → (ℑ‘(𝐴 + 𝐵)) = ((ℑ‘𝐴) + (ℑ‘𝐵))) |
10 | reim0 15009 | . . . . . 6 ⊢ (𝐵 ∈ ℝ → (ℑ‘𝐵) = 0) | |
11 | 10 | adantl 483 | . . . . 5 ⊢ ((𝐴 ∈ ran log ∧ 𝐵 ∈ ℝ) → (ℑ‘𝐵) = 0) |
12 | 11 | oveq2d 7374 | . . . 4 ⊢ ((𝐴 ∈ ran log ∧ 𝐵 ∈ ℝ) → ((ℑ‘𝐴) + (ℑ‘𝐵)) = ((ℑ‘𝐴) + 0)) |
13 | 1 | adantr 482 | . . . . . . 7 ⊢ ((𝐴 ∈ ran log ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℂ) |
14 | 13 | imcld 15086 | . . . . . 6 ⊢ ((𝐴 ∈ ran log ∧ 𝐵 ∈ ℝ) → (ℑ‘𝐴) ∈ ℝ) |
15 | 14 | recnd 11188 | . . . . 5 ⊢ ((𝐴 ∈ ran log ∧ 𝐵 ∈ ℝ) → (ℑ‘𝐴) ∈ ℂ) |
16 | 15 | addid1d 11360 | . . . 4 ⊢ ((𝐴 ∈ ran log ∧ 𝐵 ∈ ℝ) → ((ℑ‘𝐴) + 0) = (ℑ‘𝐴)) |
17 | 9, 12, 16 | 3eqtrd 2777 | . . 3 ⊢ ((𝐴 ∈ ran log ∧ 𝐵 ∈ ℝ) → (ℑ‘(𝐴 + 𝐵)) = (ℑ‘𝐴)) |
18 | 7, 17 | breqtrrd 5134 | . 2 ⊢ ((𝐴 ∈ ran log ∧ 𝐵 ∈ ℝ) → -π < (ℑ‘(𝐴 + 𝐵))) |
19 | 5 | simp3bi 1148 | . . . 4 ⊢ (𝐴 ∈ ran log → (ℑ‘𝐴) ≤ π) |
20 | 19 | adantr 482 | . . 3 ⊢ ((𝐴 ∈ ran log ∧ 𝐵 ∈ ℝ) → (ℑ‘𝐴) ≤ π) |
21 | 17, 20 | eqbrtrd 5128 | . 2 ⊢ ((𝐴 ∈ ran log ∧ 𝐵 ∈ ℝ) → (ℑ‘(𝐴 + 𝐵)) ≤ π) |
22 | ellogrn 25931 | . 2 ⊢ ((𝐴 + 𝐵) ∈ ran log ↔ ((𝐴 + 𝐵) ∈ ℂ ∧ -π < (ℑ‘(𝐴 + 𝐵)) ∧ (ℑ‘(𝐴 + 𝐵)) ≤ π)) | |
23 | 4, 18, 21, 22 | syl3anbrc 1344 | 1 ⊢ ((𝐴 ∈ ran log ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ran log) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5106 ran crn 5635 ‘cfv 6497 (class class class)co 7358 ℂcc 11054 ℝcr 11055 0cc0 11056 + caddc 11059 < clt 11194 ≤ cle 11195 -cneg 11391 ℑcim 14989 πcpi 15954 logclog 25926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 ax-addf 11135 ax-mulf 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-er 8651 df-map 8770 df-pm 8771 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-fi 9352 df-sup 9383 df-inf 9384 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-q 12879 df-rp 12921 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13274 df-ioc 13275 df-ico 13276 df-icc 13277 df-fz 13431 df-fzo 13574 df-fl 13703 df-mod 13781 df-seq 13913 df-exp 13974 df-fac 14180 df-bc 14209 df-hash 14237 df-shft 14958 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-limsup 15359 df-clim 15376 df-rlim 15377 df-sum 15577 df-ef 15955 df-sin 15957 df-cos 15958 df-pi 15960 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-starv 17153 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-unif 17161 df-hom 17162 df-cco 17163 df-rest 17309 df-topn 17310 df-0g 17328 df-gsum 17329 df-topgen 17330 df-pt 17331 df-prds 17334 df-xrs 17389 df-qtop 17394 df-imas 17395 df-xps 17397 df-mre 17471 df-mrc 17472 df-acs 17474 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-submnd 18607 df-mulg 18878 df-cntz 19102 df-cmn 19569 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-fbas 20809 df-fg 20810 df-cnfld 20813 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cld 22386 df-ntr 22387 df-cls 22388 df-nei 22465 df-lp 22503 df-perf 22504 df-cn 22594 df-cnp 22595 df-haus 22682 df-tx 22929 df-hmeo 23122 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-xms 23689 df-ms 23690 df-tms 23691 df-cncf 24257 df-limc 25246 df-dv 25247 df-log 25928 |
This theorem is referenced by: logmul2 25987 logdiv2 25988 atanlogaddlem 26279 |
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