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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 6891 | . . 3 ⊢ (𝑥 = 𝐵 → (log‘𝑥) = (log‘𝐵)) | |
2 | 1 | oveq2d 7424 | . 2 ⊢ (𝑥 = 𝐵 → ((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵))) |
3 | fveq2 6891 | . . 3 ⊢ (𝑦 = 𝑋 → (log‘𝑦) = (log‘𝑋)) | |
4 | 3 | oveq1d 7423 | . 2 ⊢ (𝑦 = 𝑋 → ((log‘𝑦) / (log‘𝐵)) = ((log‘𝑋) / (log‘𝐵))) |
5 | df-logb 26267 | . 2 ⊢ logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥))) | |
6 | ovex 7441 | . 2 ⊢ ((log‘𝑋) / (log‘𝐵)) ∈ V | |
7 | 2, 4, 5, 6 | ovmpo 7567 | 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 3945 {csn 4628 {cpr 4630 ‘cfv 6543 (class class class)co 7408 ℂcc 11107 0cc0 11109 1c1 11110 / cdiv 11870 logclog 26062 logb clogb 26266 |
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 5299 ax-nul 5306 ax-pr 5427 |
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 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-logb 26267 |
This theorem is referenced by: logbcl 26269 logbid1 26270 logb1 26271 elogb 26272 logbchbase 26273 relogbval 26274 relogbcl 26275 relogbreexp 26277 relogbmul 26279 nnlogbexp 26283 relogbcxp 26287 cxplogb 26288 logbgt0b 26295 dvrelog2b 40926 dvrelogpow2b 40928 rege1logbrege0 47234 logb2aval 47799 |
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