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Mirrors > Home > MPE Home > Th. List > dfrelog | Structured version Visualization version GIF version |
Description: The natural logarithm function on the positive reals in terms of the real exponential function. (Contributed by Paul Chapman, 21-Apr-2008.) |
Ref | Expression |
---|---|
dfrelog | ⊢ (log ↾ ℝ+) = ◡(exp ↾ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 5682 | . . . 4 ⊢ ((exp ↾ ran log) “ ℝ) = ran ((exp ↾ ran log) ↾ ℝ) | |
2 | relogrn 26450 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ran log) | |
3 | 2 | ssriv 3981 | . . . . . 6 ⊢ ℝ ⊆ ran log |
4 | resabs1 6005 | . . . . . 6 ⊢ (ℝ ⊆ ran log → ((exp ↾ ran log) ↾ ℝ) = (exp ↾ ℝ)) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ ((exp ↾ ran log) ↾ ℝ) = (exp ↾ ℝ) |
6 | 5 | rneqi 5930 | . . . 4 ⊢ ran ((exp ↾ ran log) ↾ ℝ) = ran (exp ↾ ℝ) |
7 | reeff1o 26339 | . . . . . 6 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ+ | |
8 | dff1o2 6832 | . . . . . 6 ⊢ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ ↔ ((exp ↾ ℝ) Fn ℝ ∧ Fun ◡(exp ↾ ℝ) ∧ ran (exp ↾ ℝ) = ℝ+)) | |
9 | 7, 8 | mpbi 229 | . . . . 5 ⊢ ((exp ↾ ℝ) Fn ℝ ∧ Fun ◡(exp ↾ ℝ) ∧ ran (exp ↾ ℝ) = ℝ+) |
10 | 9 | simp3i 1138 | . . . 4 ⊢ ran (exp ↾ ℝ) = ℝ+ |
11 | 1, 6, 10 | 3eqtri 2758 | . . 3 ⊢ ((exp ↾ ran log) “ ℝ) = ℝ+ |
12 | 11 | reseq2i 5972 | . 2 ⊢ (◡(exp ↾ ran log) ↾ ((exp ↾ ran log) “ ℝ)) = (◡(exp ↾ ran log) ↾ ℝ+) |
13 | 5 | cnveqi 5868 | . . 3 ⊢ ◡((exp ↾ ran log) ↾ ℝ) = ◡(exp ↾ ℝ) |
14 | logf1o 26453 | . . . . . 6 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log | |
15 | f1ofun 6829 | . . . . . 6 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → Fun log) | |
16 | 14, 15 | ax-mp 5 | . . . . 5 ⊢ Fun log |
17 | dflog2 26449 | . . . . . 6 ⊢ log = ◡(exp ↾ ran log) | |
18 | 17 | funeqi 6563 | . . . . 5 ⊢ (Fun log ↔ Fun ◡(exp ↾ ran log)) |
19 | 16, 18 | mpbi 229 | . . . 4 ⊢ Fun ◡(exp ↾ ran log) |
20 | funcnvres 6620 | . . . 4 ⊢ (Fun ◡(exp ↾ ran log) → ◡((exp ↾ ran log) ↾ ℝ) = (◡(exp ↾ ran log) ↾ ((exp ↾ ran log) “ ℝ))) | |
21 | 19, 20 | ax-mp 5 | . . 3 ⊢ ◡((exp ↾ ran log) ↾ ℝ) = (◡(exp ↾ ran log) ↾ ((exp ↾ ran log) “ ℝ)) |
22 | 13, 21 | eqtr3i 2756 | . 2 ⊢ ◡(exp ↾ ℝ) = (◡(exp ↾ ran log) ↾ ((exp ↾ ran log) “ ℝ)) |
23 | 17 | reseq1i 5971 | . 2 ⊢ (log ↾ ℝ+) = (◡(exp ↾ ran log) ↾ ℝ+) |
24 | 12, 22, 23 | 3eqtr4ri 2765 | 1 ⊢ (log ↾ ℝ+) = ◡(exp ↾ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1084 = wceq 1533 ∖ cdif 3940 ⊆ wss 3943 {csn 4623 ◡ccnv 5668 ran crn 5670 ↾ cres 5671 “ cima 5672 Fun wfun 6531 Fn wfn 6532 –1-1-onto→wf1o 6536 ℂcc 11110 ℝcr 11111 0cc0 11112 ℝ+crp 12980 expce 16011 logclog 26443 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ioo 13334 df-ioc 13335 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15421 df-clim 15438 df-rlim 15439 df-sum 15639 df-ef 16017 df-sin 16019 df-cos 16020 df-pi 16022 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-mulg 18996 df-cntz 19233 df-cmn 19702 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cld 22878 df-ntr 22879 df-cls 22880 df-nei 22957 df-lp 22995 df-perf 22996 df-cn 23086 df-cnp 23087 df-haus 23174 df-tx 23421 df-hmeo 23614 df-fil 23705 df-fm 23797 df-flim 23798 df-flf 23799 df-xms 24181 df-ms 24182 df-tms 24183 df-cncf 24753 df-limc 25750 df-dv 25751 df-log 26445 |
This theorem is referenced by: relogiso 26487 reloggim 26488 dvrelog 26526 |
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