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Mirrors > Home > MPE Home > Th. List > logblog | Structured version Visualization version GIF version |
Description: The general logarithm to the base being Euler's constant regarded as function is the natural logarithm. (Contributed by AV, 12-Jun-2020.) |
Ref | Expression |
---|---|
logblog | ⊢ (curry logb ‘e) = log |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | loge 25938 | . . . . . 6 ⊢ (log‘e) = 1 | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → (log‘e) = 1) |
3 | 2 | oveq2d 7370 | . . . 4 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → ((log‘𝑦) / (log‘e)) = ((log‘𝑦) / 1)) |
4 | eldifsn 4746 | . . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ {0}) ↔ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0)) | |
5 | logcl 25920 | . . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (log‘𝑦) ∈ ℂ) | |
6 | 4, 5 | sylbi 216 | . . . . 5 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → (log‘𝑦) ∈ ℂ) |
7 | 6 | div1d 11920 | . . . 4 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → ((log‘𝑦) / 1) = (log‘𝑦)) |
8 | 3, 7 | eqtrd 2776 | . . 3 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → ((log‘𝑦) / (log‘e)) = (log‘𝑦)) |
9 | 8 | mpteq2ia 5207 | . 2 ⊢ (𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘e))) = (𝑦 ∈ (ℂ ∖ {0}) ↦ (log‘𝑦)) |
10 | ere 15968 | . . . 4 ⊢ e ∈ ℝ | |
11 | 10 | recni 11166 | . . 3 ⊢ e ∈ ℂ |
12 | ene0 16088 | . . 3 ⊢ e ≠ 0 | |
13 | ene1 16089 | . . 3 ⊢ e ≠ 1 | |
14 | logbmpt 26134 | . . 3 ⊢ ((e ∈ ℂ ∧ e ≠ 0 ∧ e ≠ 1) → (curry logb ‘e) = (𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘e)))) | |
15 | 11, 12, 13, 14 | mp3an 1461 | . 2 ⊢ (curry logb ‘e) = (𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘e))) |
16 | logf1o 25916 | . . . 4 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log | |
17 | f1ofn 6783 | . . . 4 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → log Fn (ℂ ∖ {0})) | |
18 | 16, 17 | ax-mp 5 | . . 3 ⊢ log Fn (ℂ ∖ {0}) |
19 | dffn5 6899 | . . 3 ⊢ (log Fn (ℂ ∖ {0}) ↔ log = (𝑦 ∈ (ℂ ∖ {0}) ↦ (log‘𝑦))) | |
20 | 18, 19 | mpbi 229 | . 2 ⊢ log = (𝑦 ∈ (ℂ ∖ {0}) ↦ (log‘𝑦)) |
21 | 9, 15, 20 | 3eqtr4i 2774 | 1 ⊢ (curry logb ‘e) = log |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2942 ∖ cdif 3906 {csn 4585 ↦ cmpt 5187 ran crn 5633 Fn wfn 6489 –1-1-onto→wf1o 6493 ‘cfv 6494 (class class class)co 7354 curry ccur 8193 ℂcc 11046 0cc0 11048 1c1 11049 / cdiv 11809 eceu 15942 logclog 25906 logb clogb 26110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-inf2 9574 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 ax-pre-sup 11126 ax-addf 11127 ax-mulf 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-se 5588 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-of 7614 df-om 7800 df-1st 7918 df-2nd 7919 df-supp 8090 df-cur 8195 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-1o 8409 df-2o 8410 df-er 8645 df-map 8764 df-pm 8765 df-ixp 8833 df-en 8881 df-dom 8882 df-sdom 8883 df-fin 8884 df-fsupp 9303 df-fi 9344 df-sup 9375 df-inf 9376 df-oi 9443 df-card 9872 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-div 11810 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-5 12216 df-6 12217 df-7 12218 df-8 12219 df-9 12220 df-n0 12411 df-z 12497 df-dec 12616 df-uz 12761 df-q 12871 df-rp 12913 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-ioo 13265 df-ioc 13266 df-ico 13267 df-icc 13268 df-fz 13422 df-fzo 13565 df-fl 13694 df-mod 13772 df-seq 13904 df-exp 13965 df-fac 14171 df-bc 14200 df-hash 14228 df-shft 14949 df-cj 14981 df-re 14982 df-im 14983 df-sqrt 15117 df-abs 15118 df-limsup 15350 df-clim 15367 df-rlim 15368 df-sum 15568 df-ef 15947 df-e 15948 df-sin 15949 df-cos 15950 df-pi 15952 df-struct 17016 df-sets 17033 df-slot 17051 df-ndx 17063 df-base 17081 df-ress 17110 df-plusg 17143 df-mulr 17144 df-starv 17145 df-sca 17146 df-vsca 17147 df-ip 17148 df-tset 17149 df-ple 17150 df-ds 17152 df-unif 17153 df-hom 17154 df-cco 17155 df-rest 17301 df-topn 17302 df-0g 17320 df-gsum 17321 df-topgen 17322 df-pt 17323 df-prds 17326 df-xrs 17381 df-qtop 17386 df-imas 17387 df-xps 17389 df-mre 17463 df-mrc 17464 df-acs 17466 df-mgm 18494 df-sgrp 18543 df-mnd 18554 df-submnd 18599 df-mulg 18869 df-cntz 19093 df-cmn 19560 df-psmet 20784 df-xmet 20785 df-met 20786 df-bl 20787 df-mopn 20788 df-fbas 20789 df-fg 20790 df-cnfld 20793 df-top 22239 df-topon 22256 df-topsp 22278 df-bases 22292 df-cld 22366 df-ntr 22367 df-cls 22368 df-nei 22445 df-lp 22483 df-perf 22484 df-cn 22574 df-cnp 22575 df-haus 22662 df-tx 22909 df-hmeo 23102 df-fil 23193 df-fm 23285 df-flim 23286 df-flf 23287 df-xms 23669 df-ms 23670 df-tms 23671 df-cncf 24237 df-limc 25226 df-dv 25227 df-log 25908 df-logb 26111 |
This theorem is referenced by: (None) |
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