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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 6890 | . . 3 ⊢ (𝑥 = 𝐵 → (log‘𝑥) = (log‘𝐵)) | |
2 | 1 | oveq2d 7427 | . 2 ⊢ (𝑥 = 𝐵 → ((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵))) |
3 | fveq2 6890 | . . 3 ⊢ (𝑦 = 𝑋 → (log‘𝑦) = (log‘𝑋)) | |
4 | 3 | oveq1d 7426 | . 2 ⊢ (𝑦 = 𝑋 → ((log‘𝑦) / (log‘𝐵)) = ((log‘𝑋) / (log‘𝐵))) |
5 | df-logb 26506 | . 2 ⊢ logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥))) | |
6 | ovex 7444 | . 2 ⊢ ((log‘𝑋) / (log‘𝐵)) ∈ V | |
7 | 2, 4, 5, 6 | ovmpo 7570 | 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 394 = wceq 1539 ∈ wcel 2104 ∖ cdif 3944 {csn 4627 {cpr 4629 ‘cfv 6542 (class class class)co 7411 ℂcc 11110 0cc0 11112 1c1 11113 / cdiv 11875 logclog 26299 logb clogb 26505 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-logb 26506 |
This theorem is referenced by: logbcl 26508 logbid1 26509 logb1 26510 elogb 26511 logbchbase 26512 relogbval 26513 relogbcl 26514 relogbreexp 26516 relogbmul 26518 nnlogbexp 26522 relogbcxp 26526 cxplogb 26527 logbgt0b 26534 dvrelog2b 41237 dvrelogpow2b 41239 rege1logbrege0 47331 logb2aval 47896 |
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