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Theorem logbmpt 25293
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 25270 . . 3 logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))
2 ovexd 7180 . . . 4 (((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ (𝑥 ∈ (ℂ ∖ {0, 1}) ∧ 𝑦 ∈ (ℂ ∖ {0}))) → ((log‘𝑦) / (log‘𝑥)) ∈ V)
32ralrimivva 3188 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → ∀𝑥 ∈ (ℂ ∖ {0, 1})∀𝑦 ∈ (ℂ ∖ {0})((log‘𝑦) / (log‘𝑥)) ∈ V)
4 ax-1cn 10583 . . . . . 6 1 ∈ ℂ
5 ax-1ne0 10594 . . . . . . 7 1 ≠ 0
6 elsng 4571 . . . . . . . 8 (1 ∈ ℂ → (1 ∈ {0} ↔ 1 = 0))
74, 6ax-mp 5 . . . . . . 7 (1 ∈ {0} ↔ 1 = 0)
85, 7nemtbir 3109 . . . . . 6 ¬ 1 ∈ {0}
9 eldif 3943 . . . . . 6 (1 ∈ (ℂ ∖ {0}) ↔ (1 ∈ ℂ ∧ ¬ 1 ∈ {0}))
104, 8, 9mpbir2an 707 . . . . 5 1 ∈ (ℂ ∖ {0})
1110ne0ii 4300 . . . 4 (ℂ ∖ {0}) ≠ ∅
1211a1i 11 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (ℂ ∖ {0}) ≠ ∅)
13 cnex 10606 . . . . 5 ℂ ∈ V
1413difexi 5223 . . . 4 (ℂ ∖ {0}) ∈ V
1514a1i 11 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (ℂ ∖ {0}) ∈ V)
16 eldifpr 4587 . . . 4 (𝐵 ∈ (ℂ ∖ {0, 1}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
1716biimpri 229 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → 𝐵 ∈ (ℂ ∖ {0, 1}))
181, 3, 12, 15, 17mpocurryvald 7925 . 2 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (curry logb𝐵) = (𝑦 ∈ (ℂ ∖ {0}) ↦ 𝐵 / 𝑥((log‘𝑦) / (log‘𝑥))))
19 csbov2g 7191 . . . . 5 (𝐵 ∈ ℂ → 𝐵 / 𝑥((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / 𝐵 / 𝑥(log‘𝑥)))
20 csbfv 6708 . . . . . . 7 𝐵 / 𝑥(log‘𝑥) = (log‘𝐵)
2120a1i 11 . . . . . 6 (𝐵 ∈ ℂ → 𝐵 / 𝑥(log‘𝑥) = (log‘𝐵))
2221oveq2d 7161 . . . . 5 (𝐵 ∈ ℂ → ((log‘𝑦) / 𝐵 / 𝑥(log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
2319, 22eqtrd 2853 . . . 4 (𝐵 ∈ ℂ → 𝐵 / 𝑥((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
24233ad2ant1 1125 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → 𝐵 / 𝑥((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
2524mpteq2dv 5153 . 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 2853 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  Vcvv 3492  csb 3880  cdif 3930  c0 4288  {csn 4557  {cpr 4559  cmpt 5137  cfv 6348  (class class class)co 7145  curry ccur 7920  cc 10523  0cc0 10525  1c1 10526   / cdiv 11285  logclog 25065   logb clogb 25269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-1cn 10583  ax-1ne0 10594
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-cur 7922  df-logb 25270
This theorem is referenced by:  logbf  25294  relogbf  25296  logblog  25297
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