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Mirrors > Home > MPE Home > Th. List > logdiv2 | Structured version Visualization version GIF version |
Description: Generalization of relogdiv 26623 to a complex left argument. (Contributed by Mario Carneiro, 8-Jul-2017.) |
Ref | Expression |
---|---|
logdiv2 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℝ+) → (log‘(𝐴 / 𝐵)) = ((log‘𝐴) − (log‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | logcl 26598 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ) | |
2 | 1 | 3adant3 1129 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℝ+) → (log‘𝐴) ∈ ℂ) |
3 | relogcl 26605 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ+ → (log‘𝐵) ∈ ℝ) | |
4 | 3 | 3ad2ant3 1132 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℝ+) → (log‘𝐵) ∈ ℝ) |
5 | 4 | recnd 11294 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℝ+) → (log‘𝐵) ∈ ℂ) |
6 | efsub 16104 | . . . . 5 ⊢ (((log‘𝐴) ∈ ℂ ∧ (log‘𝐵) ∈ ℂ) → (exp‘((log‘𝐴) − (log‘𝐵))) = ((exp‘(log‘𝐴)) / (exp‘(log‘𝐵)))) | |
7 | 2, 5, 6 | syl2anc 582 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℝ+) → (exp‘((log‘𝐴) − (log‘𝐵))) = ((exp‘(log‘𝐴)) / (exp‘(log‘𝐵)))) |
8 | eflog 26606 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = 𝐴) | |
9 | 8 | 3adant3 1129 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℝ+) → (exp‘(log‘𝐴)) = 𝐴) |
10 | reeflog 26610 | . . . . . 6 ⊢ (𝐵 ∈ ℝ+ → (exp‘(log‘𝐵)) = 𝐵) | |
11 | 10 | 3ad2ant3 1132 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℝ+) → (exp‘(log‘𝐵)) = 𝐵) |
12 | 9, 11 | oveq12d 7444 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℝ+) → ((exp‘(log‘𝐴)) / (exp‘(log‘𝐵))) = (𝐴 / 𝐵)) |
13 | 7, 12 | eqtrd 2766 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℝ+) → (exp‘((log‘𝐴) − (log‘𝐵))) = (𝐴 / 𝐵)) |
14 | 13 | fveq2d 6907 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℝ+) → (log‘(exp‘((log‘𝐴) − (log‘𝐵)))) = (log‘(𝐴 / 𝐵))) |
15 | 2, 5 | negsubd 11629 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℝ+) → ((log‘𝐴) + -(log‘𝐵)) = ((log‘𝐴) − (log‘𝐵))) |
16 | logrncl 26597 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ran log) | |
17 | 16 | 3adant3 1129 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℝ+) → (log‘𝐴) ∈ ran log) |
18 | 4 | renegcld 11693 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℝ+) → -(log‘𝐵) ∈ ℝ) |
19 | logrnaddcl 26604 | . . . . 5 ⊢ (((log‘𝐴) ∈ ran log ∧ -(log‘𝐵) ∈ ℝ) → ((log‘𝐴) + -(log‘𝐵)) ∈ ran log) | |
20 | 17, 18, 19 | syl2anc 582 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℝ+) → ((log‘𝐴) + -(log‘𝐵)) ∈ ran log) |
21 | 15, 20 | eqeltrrd 2827 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℝ+) → ((log‘𝐴) − (log‘𝐵)) ∈ ran log) |
22 | logef 26611 | . . 3 ⊢ (((log‘𝐴) − (log‘𝐵)) ∈ ran log → (log‘(exp‘((log‘𝐴) − (log‘𝐵)))) = ((log‘𝐴) − (log‘𝐵))) | |
23 | 21, 22 | syl 17 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℝ+) → (log‘(exp‘((log‘𝐴) − (log‘𝐵)))) = ((log‘𝐴) − (log‘𝐵))) |
24 | 14, 23 | eqtr3d 2768 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℝ+) → (log‘(𝐴 / 𝐵)) = ((log‘𝐴) − (log‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ran crn 5685 ‘cfv 6556 (class class class)co 7426 ℂcc 11158 ℝcr 11159 0cc0 11160 + caddc 11163 − cmin 11496 -cneg 11497 / cdiv 11923 ℝ+crp 13030 expce 16065 logclog 26584 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-inf2 9686 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 ax-pre-sup 11238 ax-addf 11239 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4916 df-int 4957 df-iun 5005 df-iin 5006 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-se 5640 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-isom 6565 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8005 df-2nd 8006 df-supp 8177 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-2o 8499 df-er 8736 df-map 8859 df-pm 8860 df-ixp 8929 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-fsupp 9408 df-fi 9456 df-sup 9487 df-inf 9488 df-oi 9555 df-card 9984 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-div 11924 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12613 df-dec 12732 df-uz 12877 df-q 12987 df-rp 13031 df-xneg 13148 df-xadd 13149 df-xmul 13150 df-ioo 13384 df-ioc 13385 df-ico 13386 df-icc 13387 df-fz 13541 df-fzo 13684 df-fl 13814 df-mod 13892 df-seq 14024 df-exp 14084 df-fac 14293 df-bc 14322 df-hash 14350 df-shft 15074 df-cj 15106 df-re 15107 df-im 15108 df-sqrt 15242 df-abs 15243 df-limsup 15475 df-clim 15492 df-rlim 15493 df-sum 15693 df-ef 16071 df-sin 16073 df-cos 16074 df-pi 16076 df-struct 17151 df-sets 17168 df-slot 17186 df-ndx 17198 df-base 17216 df-ress 17245 df-plusg 17281 df-mulr 17282 df-starv 17283 df-sca 17284 df-vsca 17285 df-ip 17286 df-tset 17287 df-ple 17288 df-ds 17290 df-unif 17291 df-hom 17292 df-cco 17293 df-rest 17439 df-topn 17440 df-0g 17458 df-gsum 17459 df-topgen 17460 df-pt 17461 df-prds 17464 df-xrs 17519 df-qtop 17524 df-imas 17525 df-xps 17527 df-mre 17601 df-mrc 17602 df-acs 17604 df-mgm 18635 df-sgrp 18714 df-mnd 18730 df-submnd 18776 df-mulg 19064 df-cntz 19313 df-cmn 19782 df-psmet 21337 df-xmet 21338 df-met 21339 df-bl 21340 df-mopn 21341 df-fbas 21342 df-fg 21343 df-cnfld 21346 df-top 22890 df-topon 22907 df-topsp 22929 df-bases 22943 df-cld 23017 df-ntr 23018 df-cls 23019 df-nei 23096 df-lp 23134 df-perf 23135 df-cn 23225 df-cnp 23226 df-haus 23313 df-tx 23560 df-hmeo 23753 df-fil 23844 df-fm 23936 df-flim 23937 df-flf 23938 df-xms 24320 df-ms 24321 df-tms 24322 df-cncf 24892 df-limc 25889 df-dv 25890 df-log 26586 |
This theorem is referenced by: lgamcvg2 27086 |
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