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Mirrors > Home > MPE Home > Th. List > relogbdiv | Structured version Visualization version GIF version |
Description: The logarithm of the quotient of two positive real numbers is the difference of logarithms. Property 3 of [Cohen4] p. 361. (Contributed by AV, 29-May-2020.) |
Ref | Expression |
---|---|
relogbdiv | ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+)) → (𝐵 logb (𝐴 / 𝐶)) = ((𝐵 logb 𝐴) − (𝐵 logb 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neg1rr 11755 | . . 3 ⊢ -1 ∈ ℝ | |
2 | relogbmulexp 25359 | . . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+ ∧ -1 ∈ ℝ)) → (𝐵 logb (𝐴 · (𝐶↑𝑐-1))) = ((𝐵 logb 𝐴) + (-1 · (𝐵 logb 𝐶)))) | |
3 | 1, 2 | mp3anr3 1456 | . 2 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+)) → (𝐵 logb (𝐴 · (𝐶↑𝑐-1))) = ((𝐵 logb 𝐴) + (-1 · (𝐵 logb 𝐶)))) |
4 | rpcn 12402 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) | |
5 | 4 | adantr 483 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+) → 𝐴 ∈ ℂ) |
6 | rpcn 12402 | . . . . . . 7 ⊢ (𝐶 ∈ ℝ+ → 𝐶 ∈ ℂ) | |
7 | 6 | adantl 484 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+) → 𝐶 ∈ ℂ) |
8 | rpne0 12408 | . . . . . . 7 ⊢ (𝐶 ∈ ℝ+ → 𝐶 ≠ 0) | |
9 | 8 | adantl 484 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+) → 𝐶 ≠ 0) |
10 | 5, 7, 9 | divrecd 11422 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+) → (𝐴 / 𝐶) = (𝐴 · (1 / 𝐶))) |
11 | 1cnd 10639 | . . . . . . . . 9 ⊢ (𝐶 ∈ ℝ+ → 1 ∈ ℂ) | |
12 | 6, 8, 11 | cxpnegd 25301 | . . . . . . . 8 ⊢ (𝐶 ∈ ℝ+ → (𝐶↑𝑐-1) = (1 / (𝐶↑𝑐1))) |
13 | 6 | cxp1d 25292 | . . . . . . . . 9 ⊢ (𝐶 ∈ ℝ+ → (𝐶↑𝑐1) = 𝐶) |
14 | 13 | oveq2d 7175 | . . . . . . . 8 ⊢ (𝐶 ∈ ℝ+ → (1 / (𝐶↑𝑐1)) = (1 / 𝐶)) |
15 | 12, 14 | eqtrd 2859 | . . . . . . 7 ⊢ (𝐶 ∈ ℝ+ → (𝐶↑𝑐-1) = (1 / 𝐶)) |
16 | 15 | adantl 484 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+) → (𝐶↑𝑐-1) = (1 / 𝐶)) |
17 | 16 | oveq2d 7175 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+) → (𝐴 · (𝐶↑𝑐-1)) = (𝐴 · (1 / 𝐶))) |
18 | 10, 17 | eqtr4d 2862 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+) → (𝐴 / 𝐶) = (𝐴 · (𝐶↑𝑐-1))) |
19 | 18 | adantl 484 | . . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+)) → (𝐴 / 𝐶) = (𝐴 · (𝐶↑𝑐-1))) |
20 | 19 | oveq2d 7175 | . 2 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+)) → (𝐵 logb (𝐴 / 𝐶)) = (𝐵 logb (𝐴 · (𝐶↑𝑐-1)))) |
21 | rpcndif0 12411 | . . . . . 6 ⊢ (𝐶 ∈ ℝ+ → 𝐶 ∈ (ℂ ∖ {0})) | |
22 | 21 | adantl 484 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+) → 𝐶 ∈ (ℂ ∖ {0})) |
23 | logbcl 25348 | . . . . 5 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝐶 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝐶) ∈ ℂ) | |
24 | 22, 23 | sylan2 594 | . . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+)) → (𝐵 logb 𝐶) ∈ ℂ) |
25 | mulm1 11084 | . . . . 5 ⊢ ((𝐵 logb 𝐶) ∈ ℂ → (-1 · (𝐵 logb 𝐶)) = -(𝐵 logb 𝐶)) | |
26 | 25 | oveq2d 7175 | . . . 4 ⊢ ((𝐵 logb 𝐶) ∈ ℂ → ((𝐵 logb 𝐴) + (-1 · (𝐵 logb 𝐶))) = ((𝐵 logb 𝐴) + -(𝐵 logb 𝐶))) |
27 | 24, 26 | syl 17 | . . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+)) → ((𝐵 logb 𝐴) + (-1 · (𝐵 logb 𝐶))) = ((𝐵 logb 𝐴) + -(𝐵 logb 𝐶))) |
28 | rpcndif0 12411 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ (ℂ ∖ {0})) | |
29 | 28 | adantr 483 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+) → 𝐴 ∈ (ℂ ∖ {0})) |
30 | logbcl 25348 | . . . . 5 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝐴) ∈ ℂ) | |
31 | 29, 30 | sylan2 594 | . . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+)) → (𝐵 logb 𝐴) ∈ ℂ) |
32 | 31, 24 | negsubd 11006 | . . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+)) → ((𝐵 logb 𝐴) + -(𝐵 logb 𝐶)) = ((𝐵 logb 𝐴) − (𝐵 logb 𝐶))) |
33 | 27, 32 | eqtr2d 2860 | . 2 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+)) → ((𝐵 logb 𝐴) − (𝐵 logb 𝐶)) = ((𝐵 logb 𝐴) + (-1 · (𝐵 logb 𝐶)))) |
34 | 3, 20, 33 | 3eqtr4d 2869 | 1 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+)) → (𝐵 logb (𝐴 / 𝐶)) = ((𝐵 logb 𝐴) − (𝐵 logb 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∖ cdif 3936 {csn 4570 {cpr 4572 (class class class)co 7159 ℂcc 10538 ℝcr 10539 0cc0 10540 1c1 10541 + caddc 10543 · cmul 10545 − cmin 10873 -cneg 10874 / cdiv 11300 ℝ+crp 12392 ↑𝑐ccxp 25142 logb clogb 25345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-fi 8878 df-sup 8909 df-inf 8910 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14429 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-limsup 14831 df-clim 14848 df-rlim 14849 df-sum 15046 df-ef 15424 df-sin 15426 df-cos 15427 df-pi 15429 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-hom 16592 df-cco 16593 df-rest 16699 df-topn 16700 df-0g 16718 df-gsum 16719 df-topgen 16720 df-pt 16721 df-prds 16724 df-xrs 16778 df-qtop 16783 df-imas 16784 df-xps 16786 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-mulg 18228 df-cntz 18450 df-cmn 18911 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-fbas 20545 df-fg 20546 df-cnfld 20549 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-cld 21630 df-ntr 21631 df-cls 21632 df-nei 21709 df-lp 21747 df-perf 21748 df-cn 21838 df-cnp 21839 df-haus 21926 df-tx 22173 df-hmeo 22366 df-fil 22457 df-fm 22549 df-flim 22550 df-flf 22551 df-xms 22933 df-ms 22934 df-tms 22935 df-cncf 23489 df-limc 24467 df-dv 24468 df-log 25143 df-cxp 25144 df-logb 25346 |
This theorem is referenced by: relogbdivb 44629 |
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