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Mirrors > Home > ILE Home > Th. List > rplogbval | Unicode version |
Description: Define the value of the logb function, the logarithm generalized to an arbitrary base, when used as infix. Most Metamath statements select variables in order of their use, but to make the order clearer we use "B" for base and "X" for the argument of the logarithm function here. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by Jim Kingdon, 3-Jul-2024.) |
Ref | Expression |
---|---|
rplogbval | # logb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpcn 9589 | . . . 4 | |
2 | 1 | 3ad2ant1 1007 | . . 3 # |
3 | rpne0 9596 | . . . 4 | |
4 | 3 | 3ad2ant1 1007 | . . 3 # |
5 | simp2 987 | . . . 4 # # | |
6 | 1cnd 7906 | . . . . 5 # | |
7 | apne 8512 | . . . . 5 # | |
8 | 2, 6, 7 | syl2anc 409 | . . . 4 # # |
9 | 5, 8 | mpd 13 | . . 3 # |
10 | eldifpr 3597 | . . 3 | |
11 | 2, 4, 9, 10 | syl3anbrc 1170 | . 2 # |
12 | rpcn 9589 | . . . 4 | |
13 | 12 | 3ad2ant3 1009 | . . 3 # |
14 | rpne0 9596 | . . . 4 | |
15 | 14 | 3ad2ant3 1009 | . . 3 # |
16 | eldifsn 3697 | . . 3 | |
17 | 13, 15, 16 | sylanbrc 414 | . 2 # |
18 | simp3 988 | . . . 4 # | |
19 | 18 | relogcld 13350 | . . 3 # |
20 | simp1 986 | . . . 4 # | |
21 | 20 | relogcld 13350 | . . 3 # |
22 | 20, 5 | logrpap0d 13346 | . . 3 # # |
23 | 19, 21, 22 | redivclapd 8722 | . 2 # |
24 | fveq2 5480 | . . . 4 | |
25 | 24 | oveq2d 5852 | . . 3 |
26 | fveq2 5480 | . . . 4 | |
27 | 26 | oveq1d 5851 | . . 3 |
28 | df-logb 13409 | . . 3 logb | |
29 | 25, 27, 28 | ovmpog 5967 | . 2 logb |
30 | 11, 17, 23, 29 | syl3anc 1227 | 1 # logb |
Colors of variables: wff set class |
Syntax hints: wi 4 w3a 967 wceq 1342 wcel 2135 wne 2334 cdif 3108 csn 3570 cpr 3571 class class class wbr 3976 cfv 5182 (class class class)co 5836 cc 7742 cr 7743 cc0 7744 c1 7745 # cap 8470 cdiv 8559 crp 9580 clog 13324 logb clogb 13408 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 ax-pre-suploc 7865 ax-addf 7866 ax-mulf 7867 |
This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-disj 3954 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-isom 5191 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-of 6044 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-frec 6350 df-1o 6375 df-oadd 6379 df-er 6492 df-map 6607 df-pm 6608 df-en 6698 df-dom 6699 df-fin 6700 df-sup 6940 df-inf 6941 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-xneg 9699 df-xadd 9700 df-ioo 9819 df-ico 9821 df-icc 9822 df-fz 9936 df-fzo 10068 df-seqfrec 10371 df-exp 10445 df-fac 10628 df-bc 10650 df-ihash 10678 df-shft 10743 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-clim 11206 df-sumdc 11281 df-ef 11575 df-e 11576 df-rest 12500 df-topgen 12519 df-psmet 12534 df-xmet 12535 df-met 12536 df-bl 12537 df-mopn 12538 df-top 12543 df-topon 12556 df-bases 12588 df-ntr 12643 df-cn 12735 df-cnp 12736 df-tx 12800 df-cncf 13105 df-limced 13172 df-dvap 13173 df-relog 13326 df-logb 13409 |
This theorem is referenced by: rplogbcl 13411 rplogbid1 13412 rplogb1 13413 rpelogb 13414 rplogbchbase 13415 relogbval 13416 rplogbreexp 13418 rprelogbmul 13420 rpcxplogb 13429 logbgt0b 13431 |
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