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Description: The natural logarithm of
positive  raised to a
power. Property 4 of
[Cohen] p. 301-302, restricted to natural
logarithms and
nonnegative-integer powers  . (Contributed by Steve Rodriguez,
25-Nov-2007.) |
| Ref | Expression |
|---|---|
| logexptOLD |
    
  
logOLD    logOLD |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | efexpt 7314 |
. . . . 5
logOLD   logOLD logOLD | |
| 2 | logcltOLD 8703 |
. . . . . 6
logOLD  | |
| 3 | 2 | recnd 5287 |
. . . . 5
logOLD |
| 4 | 1, 3 | sylan 448 |
. . . 4
logOLD logOLD |
| 5 | eflogtOLD 8704 |
. . . . . 6
logOLD | |
| 6 | 5 | opreq1d 3960 |
. . . . 5
logOLD |
| 7 | 6 | adantr 389 |
. . . 4
logOLD |
| 8 | 4, 7 | eqtrd 1499 |
. . 3
logOLD |
| 9 | 8 | fveq2d 3713 |
. 2
logOLD logOLD logOLD |
| 10 | axmulrcl 5246 |
. . . . 5
logOLD
logOLD | |
| 11 | 10 | ancoms 436 |
. . . 4
logOLD
logOLD |
| 12 | nn0ret 6055 |
. . . 4
| |
| 13 | 11, 2, 12 | syl2an 454 |
. . 3
logOLD |
| 14 | logeftOLD 8705 |
. . 3
logOLD logOLD logOLD logOLD | |
| 15 | 13, 14 | syl 10 |
. 2
logOLD logOLD logOLD |
| 16 | 9, 15 | eqtr3d 1501 |
1
logOLD logOLD |
| Colors of variables: wff set class |
| Syntax hints: wi 3 wa 223 wceq 953 wcel 955 cfv 3172 (class class class)co 3948
cc 5204
cr 5205
cmul 5211
cn0 5269
crp 5272
cexp 6500
ce 7235
logOLDclogOLD 8699 |
| This theorem was proved from axioms: ax-1 4 ax-2 5 ax-3 6 ax-mp 7 ax-7 959 ax-gen 960 ax-8 961 ax-9 962 ax-10 963 ax-11 964 ax-12 965 ax-13 966 ax-14 967 ax-17 968 ax-4 970 ax-5o 972 ax-6o 975 ax-9o 1119 ax-10o 1136 ax-16 1206 ax-11o 1213 ax-ext 1452 ax-rep 2683 ax-sep 2693 ax-nul 2700 ax-pow 2732 ax-pr 2769 ax-un 2857 ax-inf2 4597 |
| This theorem depends on definitions: df-bi 147 df-or 224 df-an 225 df-3or 774 df-3an 775 df-ex 978 df-sb 1168 df-eu 1375 df-mo 1376 df-clab 1457 df-cleq 1462 df-clel 1465 df-ne 1579 df-nel 1580 df-ral 1641 df-rex 1642 df-reu 1643 df-rab 1644 df-v 1803 df-sbc 1932 df-csb 1992 df-dif 2039 df-un 2040 df-in 2041 df-ss 2043 df-pss 2045 df-nul 2271 df-if 2352 df-pw 2392 df-sn 2402 df-pr 2403 df-tp 2405 df-op 2406 df-uni 2494 df-int 2524 df-iun 2558 df-br 2610 df-opab 2657 df-tr 2671 df-eprel 2821 df-id 2824 df-po 2831 df-so 2841 df-fr 2907 df-we 2924 df-ord 2941 df-on 2942 df-lim 2943 df-suc 2944 df-om 3122 df-xp 3174 df-rel 3175 df-cnv 3176 df-co 3177 df-dm 3178 df-rn 3179 df-res 3180 df-ima 3181 df-fun 3182 df-fn 3183 df-f 3184 df-f1 3185 df-fo 3186 df-f1o 3187 df-fv 3188 df-rdg 3917 df-opr 3950 df-oprab 3951 df-1st 4063 df-2nd 4064 df-1o 4117 df-oadd 4119 df-omul 4120 df-er 4245 df-ec 4247 df-qs 4250 df-en 4351 df-dom 4352 df-sdom 4353 df-sup 4548 df-ni 4972 df-pli 4973 df-mi 4974 df-lti 4975 df-plpq 5007 df-mpq 5008 df-enq 5009 df-nq 5010 df-plq 5011 df-mq 5012 df-rq 5013 df-ltq 5014 df-1q 5015 df-np 5058 df-1p 5059 df-plp 5060 df-mp 5061 df-ltp 5062 df-plpr 5136 df-mpr 5137 df-enr 5138 df-nr 5139 df-plr 5140 df-mr 5141 df-ltr 5142 df-0r 5143 df-1r 5144 df-m1r 5145 df-c 5212 df-0 5213 df-1 5214 df-i 5215 df-r 5216 df-plus 5217 df-mul 5218 df-lt 5219 df-sub 5328 df-neg 5330 df-pnf 5459 df-mnf 5460 df-xr 5461 df-ltxr 5462 df-le 5463 df-div 5672 df-n 5873 df-2 5917 df-3 5918 df-4 5919 df-n0 6047 df-z 6083 df-fl 6172 df-q 6194 df-rp 6219 df-seq1 6245 df-shft 6278 df-ioo 6298 df-icc 6301 df-uz 6350 df-fz 6400 df-seqz 6465 df-seq0 6466 df-exp 6501 df-sqr 6600 df-re 6682 df-im 6683 df-cj 6684 df-abs 6685 df-fac 6869 df-bc 6894 df-clim 6913 df-sum 6918 df-cncf 7198 df-ef 7240 df-logOLD 8700 |