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Mirrors > Home > MPE Home > Th. List > Mathboxes > relogbmulbexp | Structured version Visualization version GIF version |
Description: The logarithm of the product of a positive real number and the base to the power of a real number is the logarithm of the positive real number plus the real number. (Contributed by AV, 29-May-2020.) |
Ref | Expression |
---|---|
relogbmulbexp | ⊢ ((𝐵 ∈ (ℝ+ ∖ {1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ)) → (𝐵 logb (𝐴 · (𝐵↑𝑐𝐶))) = ((𝐵 logb 𝐴) + 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpcn 12685 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℂ) | |
2 | 1 | adantr 480 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ+ ∧ 𝐵 ≠ 1) → 𝐵 ∈ ℂ) |
3 | rpne0 12691 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ≠ 0) | |
4 | 3 | adantr 480 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ+ ∧ 𝐵 ≠ 1) → 𝐵 ≠ 0) |
5 | simpr 484 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ+ ∧ 𝐵 ≠ 1) → 𝐵 ≠ 1) | |
6 | 2, 4, 5 | 3jca 1126 | . . . . 5 ⊢ ((𝐵 ∈ ℝ+ ∧ 𝐵 ≠ 1) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1)) |
7 | eldifsn 4722 | . . . . 5 ⊢ (𝐵 ∈ (ℝ+ ∖ {1}) ↔ (𝐵 ∈ ℝ+ ∧ 𝐵 ≠ 1)) | |
8 | eldifpr 4595 | . . . . 5 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1)) | |
9 | 6, 7, 8 | 3imtr4i 291 | . . . 4 ⊢ (𝐵 ∈ (ℝ+ ∖ {1}) → 𝐵 ∈ (ℂ ∖ {0, 1})) |
10 | 9 | adantr 480 | . . 3 ⊢ ((𝐵 ∈ (ℝ+ ∖ {1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ)) → 𝐵 ∈ (ℂ ∖ {0, 1})) |
11 | simprl 767 | . . 3 ⊢ ((𝐵 ∈ (ℝ+ ∖ {1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ)) → 𝐴 ∈ ℝ+) | |
12 | eldifi 4062 | . . . 4 ⊢ (𝐵 ∈ (ℝ+ ∖ {1}) → 𝐵 ∈ ℝ+) | |
13 | 12 | adantr 480 | . . 3 ⊢ ((𝐵 ∈ (ℝ+ ∖ {1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ)) → 𝐵 ∈ ℝ+) |
14 | simpr 484 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℝ) | |
15 | 14 | adantl 481 | . . 3 ⊢ ((𝐵 ∈ (ℝ+ ∖ {1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ)) → 𝐶 ∈ ℝ) |
16 | relogbmulexp 25871 | . . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ∧ 𝐶 ∈ ℝ)) → (𝐵 logb (𝐴 · (𝐵↑𝑐𝐶))) = ((𝐵 logb 𝐴) + (𝐶 · (𝐵 logb 𝐵)))) | |
17 | 10, 11, 13, 15, 16 | syl13anc 1370 | . 2 ⊢ ((𝐵 ∈ (ℝ+ ∖ {1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ)) → (𝐵 logb (𝐴 · (𝐵↑𝑐𝐶))) = ((𝐵 logb 𝐴) + (𝐶 · (𝐵 logb 𝐵)))) |
18 | 7, 6 | sylbi 216 | . . . . . . 7 ⊢ (𝐵 ∈ (ℝ+ ∖ {1}) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1)) |
19 | logbid1 25861 | . . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (𝐵 logb 𝐵) = 1) | |
20 | 18, 19 | syl 17 | . . . . . 6 ⊢ (𝐵 ∈ (ℝ+ ∖ {1}) → (𝐵 logb 𝐵) = 1) |
21 | 20 | adantr 480 | . . . . 5 ⊢ ((𝐵 ∈ (ℝ+ ∖ {1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ)) → (𝐵 logb 𝐵) = 1) |
22 | 21 | oveq2d 7276 | . . . 4 ⊢ ((𝐵 ∈ (ℝ+ ∖ {1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ)) → (𝐶 · (𝐵 logb 𝐵)) = (𝐶 · 1)) |
23 | ax-1rid 10888 | . . . . . 6 ⊢ (𝐶 ∈ ℝ → (𝐶 · 1) = 𝐶) | |
24 | 23 | adantl 481 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ) → (𝐶 · 1) = 𝐶) |
25 | 24 | adantl 481 | . . . 4 ⊢ ((𝐵 ∈ (ℝ+ ∖ {1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ)) → (𝐶 · 1) = 𝐶) |
26 | 22, 25 | eqtrd 2777 | . . 3 ⊢ ((𝐵 ∈ (ℝ+ ∖ {1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ)) → (𝐶 · (𝐵 logb 𝐵)) = 𝐶) |
27 | 26 | oveq2d 7276 | . 2 ⊢ ((𝐵 ∈ (ℝ+ ∖ {1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ)) → ((𝐵 logb 𝐴) + (𝐶 · (𝐵 logb 𝐵))) = ((𝐵 logb 𝐴) + 𝐶)) |
28 | 17, 27 | eqtrd 2777 | 1 ⊢ ((𝐵 ∈ (ℝ+ ∖ {1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ)) → (𝐵 logb (𝐴 · (𝐵↑𝑐𝐶))) = ((𝐵 logb 𝐴) + 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2107 ≠ wne 2941 ∖ cdif 3885 {csn 4563 {cpr 4565 (class class class)co 7260 ℂcc 10816 ℝcr 10817 0cc0 10818 1c1 10819 + caddc 10821 · cmul 10823 ℝ+crp 12675 ↑𝑐ccxp 25654 logb clogb 25857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5210 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7571 ax-inf2 9345 ax-cnex 10874 ax-resscn 10875 ax-1cn 10876 ax-icn 10877 ax-addcl 10878 ax-addrcl 10879 ax-mulcl 10880 ax-mulrcl 10881 ax-mulcom 10882 ax-addass 10883 ax-mulass 10884 ax-distr 10885 ax-i2m1 10886 ax-1ne0 10887 ax-1rid 10888 ax-rnegex 10889 ax-rrecex 10890 ax-cnre 10891 ax-pre-lttri 10892 ax-pre-lttrn 10893 ax-pre-ltadd 10894 ax-pre-mulgt0 10895 ax-pre-sup 10896 ax-addf 10897 ax-mulf 10898 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3429 df-sbc 3717 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-iin 4929 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-se 5541 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6259 df-on 6260 df-lim 6261 df-suc 6262 df-iota 6381 df-fun 6425 df-fn 6426 df-f 6427 df-f1 6428 df-fo 6429 df-f1o 6430 df-fv 6431 df-isom 6432 df-riota 7217 df-ov 7263 df-oprab 7264 df-mpo 7265 df-of 7516 df-om 7693 df-1st 7809 df-2nd 7810 df-supp 7954 df-frecs 8073 df-wrecs 8104 df-recs 8178 df-rdg 8217 df-1o 8272 df-2o 8273 df-er 8461 df-map 8580 df-pm 8581 df-ixp 8649 df-en 8697 df-dom 8698 df-sdom 8699 df-fin 8700 df-fsupp 9075 df-fi 9116 df-sup 9147 df-inf 9148 df-oi 9215 df-card 9644 df-pnf 10958 df-mnf 10959 df-xr 10960 df-ltxr 10961 df-le 10962 df-sub 11153 df-neg 11154 df-div 11579 df-nn 11920 df-2 11982 df-3 11983 df-4 11984 df-5 11985 df-6 11986 df-7 11987 df-8 11988 df-9 11989 df-n0 12180 df-z 12266 df-dec 12383 df-uz 12528 df-q 12634 df-rp 12676 df-xneg 12793 df-xadd 12794 df-xmul 12795 df-ioo 13028 df-ioc 13029 df-ico 13030 df-icc 13031 df-fz 13185 df-fzo 13328 df-fl 13456 df-mod 13534 df-seq 13666 df-exp 13727 df-fac 13932 df-bc 13961 df-hash 13989 df-shft 14722 df-cj 14754 df-re 14755 df-im 14756 df-sqrt 14890 df-abs 14891 df-limsup 15124 df-clim 15141 df-rlim 15142 df-sum 15342 df-ef 15721 df-sin 15723 df-cos 15724 df-pi 15726 df-struct 16792 df-sets 16809 df-slot 16827 df-ndx 16839 df-base 16857 df-ress 16886 df-plusg 16919 df-mulr 16920 df-starv 16921 df-sca 16922 df-vsca 16923 df-ip 16924 df-tset 16925 df-ple 16926 df-ds 16928 df-unif 16929 df-hom 16930 df-cco 16931 df-rest 17077 df-topn 17078 df-0g 17096 df-gsum 17097 df-topgen 17098 df-pt 17099 df-prds 17102 df-xrs 17157 df-qtop 17162 df-imas 17163 df-xps 17165 df-mre 17239 df-mrc 17240 df-acs 17242 df-mgm 18270 df-sgrp 18319 df-mnd 18330 df-submnd 18375 df-mulg 18645 df-cntz 18867 df-cmn 19332 df-psmet 20533 df-xmet 20534 df-met 20535 df-bl 20536 df-mopn 20537 df-fbas 20538 df-fg 20539 df-cnfld 20542 df-top 21987 df-topon 22004 df-topsp 22026 df-bases 22040 df-cld 22114 df-ntr 22115 df-cls 22116 df-nei 22193 df-lp 22231 df-perf 22232 df-cn 22322 df-cnp 22323 df-haus 22410 df-tx 22657 df-hmeo 22850 df-fil 22941 df-fm 23033 df-flim 23034 df-flf 23035 df-xms 23417 df-ms 23418 df-tms 23419 df-cncf 23985 df-limc 24973 df-dv 24974 df-log 25655 df-cxp 25656 df-logb 25858 |
This theorem is referenced by: (None) |
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