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Theorem logbmpt 26845
Description: The general logarithm to a fixed base regarded as mapping. (Contributed by AV, 11-Jun-2020.)
Assertion
Ref Expression
logbmpt ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (curry logb𝐵) = (𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝐵))))
Distinct variable group:   𝑦,𝐵

Proof of Theorem logbmpt
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-logb 26822 . . 3 logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))
2 ovexd 7465 . . . 4 (((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ (𝑥 ∈ (ℂ ∖ {0, 1}) ∧ 𝑦 ∈ (ℂ ∖ {0}))) → ((log‘𝑦) / (log‘𝑥)) ∈ V)
32ralrimivva 3199 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → ∀𝑥 ∈ (ℂ ∖ {0, 1})∀𝑦 ∈ (ℂ ∖ {0})((log‘𝑦) / (log‘𝑥)) ∈ V)
4 ax-1cn 11210 . . . . . 6 1 ∈ ℂ
5 ax-1ne0 11221 . . . . . . 7 1 ≠ 0
6 elsng 4644 . . . . . . . 8 (1 ∈ ℂ → (1 ∈ {0} ↔ 1 = 0))
74, 6ax-mp 5 . . . . . . 7 (1 ∈ {0} ↔ 1 = 0)
85, 7nemtbir 3035 . . . . . 6 ¬ 1 ∈ {0}
9 eldif 3972 . . . . . 6 (1 ∈ (ℂ ∖ {0}) ↔ (1 ∈ ℂ ∧ ¬ 1 ∈ {0}))
104, 8, 9mpbir2an 711 . . . . 5 1 ∈ (ℂ ∖ {0})
1110ne0ii 4349 . . . 4 (ℂ ∖ {0}) ≠ ∅
1211a1i 11 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (ℂ ∖ {0}) ≠ ∅)
13 cnex 11233 . . . . 5 ℂ ∈ V
1413difexi 5335 . . . 4 (ℂ ∖ {0}) ∈ V
1514a1i 11 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (ℂ ∖ {0}) ∈ V)
16 eldifpr 4662 . . . 4 (𝐵 ∈ (ℂ ∖ {0, 1}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
1716biimpri 228 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → 𝐵 ∈ (ℂ ∖ {0, 1}))
181, 3, 12, 15, 17mpocurryvald 8293 . 2 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (curry logb𝐵) = (𝑦 ∈ (ℂ ∖ {0}) ↦ 𝐵 / 𝑥((log‘𝑦) / (log‘𝑥))))
19 csbov2g 7478 . . . . 5 (𝐵 ∈ ℂ → 𝐵 / 𝑥((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / 𝐵 / 𝑥(log‘𝑥)))
20 csbfv 6956 . . . . . . 7 𝐵 / 𝑥(log‘𝑥) = (log‘𝐵)
2120a1i 11 . . . . . 6 (𝐵 ∈ ℂ → 𝐵 / 𝑥(log‘𝑥) = (log‘𝐵))
2221oveq2d 7446 . . . . 5 (𝐵 ∈ ℂ → ((log‘𝑦) / 𝐵 / 𝑥(log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
2319, 22eqtrd 2774 . . . 4 (𝐵 ∈ ℂ → 𝐵 / 𝑥((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
24233ad2ant1 1132 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → 𝐵 / 𝑥((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
2524mpteq2dv 5249 . 2 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (𝑦 ∈ (ℂ ∖ {0}) ↦ 𝐵 / 𝑥((log‘𝑦) / (log‘𝑥))) = (𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝐵))))
2618, 25eqtrd 2774 1 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (curry logb𝐵) = (𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wne 2937  Vcvv 3477  csb 3907  cdif 3959  c0 4338  {csn 4630  {cpr 4632  cmpt 5230  cfv 6562  (class class class)co 7430  curry ccur 8288  cc 11150  0cc0 11152  1c1 11153   / cdiv 11917  logclog 26610   logb clogb 26821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-1cn 11210  ax-1ne0 11221
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-cur 8290  df-logb 26822
This theorem is referenced by:  logbf  26846  relogbf  26848  logblog  26849
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