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Mirrors > Home > MPE Home > Th. List > logbval | Structured version Visualization version GIF 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 David A. Wheeler, 16-Jul-2017.) |
Ref | Expression |
---|---|
logbval | ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6879 | . . 3 ⊢ (𝑥 = 𝐵 → (log‘𝑥) = (log‘𝐵)) | |
2 | 1 | oveq2d 7410 | . 2 ⊢ (𝑥 = 𝐵 → ((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵))) |
3 | fveq2 6879 | . . 3 ⊢ (𝑦 = 𝑋 → (log‘𝑦) = (log‘𝑋)) | |
4 | 3 | oveq1d 7409 | . 2 ⊢ (𝑦 = 𝑋 → ((log‘𝑦) / (log‘𝐵)) = ((log‘𝑋) / (log‘𝐵))) |
5 | df-logb 26199 | . 2 ⊢ logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥))) | |
6 | ovex 7427 | . 2 ⊢ ((log‘𝑋) / (log‘𝐵)) ∈ V | |
7 | 2, 4, 5, 6 | ovmpo 7552 | 1 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∖ cdif 3942 {csn 4623 {cpr 4625 ‘cfv 6533 (class class class)co 7394 ℂcc 11092 0cc0 11094 1c1 11095 / cdiv 11855 logclog 25994 logb clogb 26198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pr 5421 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3775 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-iota 6485 df-fun 6535 df-fv 6541 df-ov 7397 df-oprab 7398 df-mpo 7399 df-logb 26199 |
This theorem is referenced by: logbcl 26201 logbid1 26202 logb1 26203 elogb 26204 logbchbase 26205 relogbval 26206 relogbcl 26207 relogbreexp 26209 relogbmul 26211 nnlogbexp 26215 relogbcxp 26219 cxplogb 26220 logbgt0b 26227 dvrelog2b 40800 dvrelogpow2b 40802 rege1logbrege0 46956 logb2aval 47521 |
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