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Theorem logbmpt 24820
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 24797 . . 3 logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))
2 ovexd 6878 . . . 4 (((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ (𝑥 ∈ (ℂ ∖ {0, 1}) ∧ 𝑦 ∈ (ℂ ∖ {0}))) → ((log‘𝑦) / (log‘𝑥)) ∈ V)
32ralrimivva 3118 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → ∀𝑥 ∈ (ℂ ∖ {0, 1})∀𝑦 ∈ (ℂ ∖ {0})((log‘𝑦) / (log‘𝑥)) ∈ V)
4 ax-1cn 10249 . . . . . 6 1 ∈ ℂ
5 ax-1ne0 10260 . . . . . . 7 1 ≠ 0
6 elsng 4350 . . . . . . . 8 (1 ∈ ℂ → (1 ∈ {0} ↔ 1 = 0))
74, 6ax-mp 5 . . . . . . 7 (1 ∈ {0} ↔ 1 = 0)
85, 7nemtbir 3032 . . . . . 6 ¬ 1 ∈ {0}
9 eldif 3744 . . . . . 6 (1 ∈ (ℂ ∖ {0}) ↔ (1 ∈ ℂ ∧ ¬ 1 ∈ {0}))
104, 8, 9mpbir2an 702 . . . . 5 1 ∈ (ℂ ∖ {0})
1110ne0ii 4090 . . . 4 (ℂ ∖ {0}) ≠ ∅
1211a1i 11 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (ℂ ∖ {0}) ≠ ∅)
13 cnex 10272 . . . . 5 ℂ ∈ V
14 difexg 4971 . . . . 5 (ℂ ∈ V → (ℂ ∖ {0}) ∈ V)
1513, 14ax-mp 5 . . . 4 (ℂ ∖ {0}) ∈ V
1615a1i 11 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (ℂ ∖ {0}) ∈ V)
17 eldifpr 4364 . . . 4 (𝐵 ∈ (ℂ ∖ {0, 1}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
1817biimpri 219 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → 𝐵 ∈ (ℂ ∖ {0, 1}))
191, 3, 12, 16, 18mpt2curryvald 7601 . 2 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (curry logb𝐵) = (𝑦 ∈ (ℂ ∖ {0}) ↦ 𝐵 / 𝑥((log‘𝑦) / (log‘𝑥))))
20 csbov2g 6889 . . . . 5 (𝐵 ∈ ℂ → 𝐵 / 𝑥((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / 𝐵 / 𝑥(log‘𝑥)))
21 csbfv 6423 . . . . . . 7 𝐵 / 𝑥(log‘𝑥) = (log‘𝐵)
2221a1i 11 . . . . . 6 (𝐵 ∈ ℂ → 𝐵 / 𝑥(log‘𝑥) = (log‘𝐵))
2322oveq2d 6860 . . . . 5 (𝐵 ∈ ℂ → ((log‘𝑦) / 𝐵 / 𝑥(log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
2420, 23eqtrd 2799 . . . 4 (𝐵 ∈ ℂ → 𝐵 / 𝑥((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
25243ad2ant1 1163 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → 𝐵 / 𝑥((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
2625mpteq2dv 4906 . 2 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (𝑦 ∈ (ℂ ∖ {0}) ↦ 𝐵 / 𝑥((log‘𝑦) / (log‘𝑥))) = (𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝐵))))
2719, 26eqtrd 2799 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  Vcvv 3350  csb 3693  cdif 3731  c0 4081  {csn 4336  {cpr 4338  cmpt 4890  cfv 6070  (class class class)co 6844  curry ccur 7596  cc 10189  0cc0 10191  1c1 10192   / cdiv 10940  logclog 24595   logb clogb 24796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-cnex 10247  ax-1cn 10249  ax-1ne0 10260
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-1st 7368  df-2nd 7369  df-cur 7598  df-logb 24797
This theorem is referenced by:  logbf  24821  relogbf  24823  logblog  24824
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