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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 6645 | . . 3 ⊢ (𝑥 = 𝐵 → (log‘𝑥) = (log‘𝐵)) | |
2 | 1 | oveq2d 7151 | . 2 ⊢ (𝑥 = 𝐵 → ((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵))) |
3 | fveq2 6645 | . . 3 ⊢ (𝑦 = 𝑋 → (log‘𝑦) = (log‘𝑋)) | |
4 | 3 | oveq1d 7150 | . 2 ⊢ (𝑦 = 𝑋 → ((log‘𝑦) / (log‘𝐵)) = ((log‘𝑋) / (log‘𝐵))) |
5 | df-logb 25351 | . 2 ⊢ logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥))) | |
6 | ovex 7168 | . 2 ⊢ ((log‘𝑋) / (log‘𝐵)) ∈ V | |
7 | 2, 4, 5, 6 | ovmpo 7289 | 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 399 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 {csn 4525 {cpr 4527 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 0cc0 10526 1c1 10527 / cdiv 11286 logclog 25146 logb clogb 25350 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-logb 25351 |
This theorem is referenced by: logbcl 25353 logbid1 25354 logb1 25355 elogb 25356 logbchbase 25357 relogbval 25358 relogbcl 25359 relogbreexp 25361 relogbmul 25363 nnlogbexp 25367 relogbcxp 25371 cxplogb 25372 logbgt0b 25379 rege1logbrege0 44972 logb2aval 45290 |
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