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Theorem logbmpt 26831
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 26808 . . 3 logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))
2 ovexd 7466 . . . 4 (((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ (𝑥 ∈ (ℂ ∖ {0, 1}) ∧ 𝑦 ∈ (ℂ ∖ {0}))) → ((log‘𝑦) / (log‘𝑥)) ∈ V)
32ralrimivva 3202 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → ∀𝑥 ∈ (ℂ ∖ {0, 1})∀𝑦 ∈ (ℂ ∖ {0})((log‘𝑦) / (log‘𝑥)) ∈ V)
4 ax-1cn 11213 . . . . . 6 1 ∈ ℂ
5 ax-1ne0 11224 . . . . . . 7 1 ≠ 0
6 elsng 4640 . . . . . . . 8 (1 ∈ ℂ → (1 ∈ {0} ↔ 1 = 0))
74, 6ax-mp 5 . . . . . . 7 (1 ∈ {0} ↔ 1 = 0)
85, 7nemtbir 3038 . . . . . 6 ¬ 1 ∈ {0}
9 eldif 3961 . . . . . 6 (1 ∈ (ℂ ∖ {0}) ↔ (1 ∈ ℂ ∧ ¬ 1 ∈ {0}))
104, 8, 9mpbir2an 711 . . . . 5 1 ∈ (ℂ ∖ {0})
1110ne0ii 4344 . . . 4 (ℂ ∖ {0}) ≠ ∅
1211a1i 11 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (ℂ ∖ {0}) ≠ ∅)
13 cnex 11236 . . . . 5 ℂ ∈ V
1413difexi 5330 . . . 4 (ℂ ∖ {0}) ∈ V
1514a1i 11 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (ℂ ∖ {0}) ∈ V)
16 eldifpr 4658 . . . 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 8295 . 2 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (curry logb𝐵) = (𝑦 ∈ (ℂ ∖ {0}) ↦ 𝐵 / 𝑥((log‘𝑦) / (log‘𝑥))))
19 csbov2g 7479 . . . . 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 7447 . . . . 5 (𝐵 ∈ ℂ → ((log‘𝑦) / 𝐵 / 𝑥(log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
2319, 22eqtrd 2777 . . . 4 (𝐵 ∈ ℂ → 𝐵 / 𝑥((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
24233ad2ant1 1134 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → 𝐵 / 𝑥((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
2524mpteq2dv 5244 . 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 2777 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 1087   = wceq 1540  wcel 2108  wne 2940  Vcvv 3480  csb 3899  cdif 3948  c0 4333  {csn 4626  {cpr 4628  cmpt 5225  cfv 6561  (class class class)co 7431  curry ccur 8290  cc 11153  0cc0 11155  1c1 11156   / cdiv 11920  logclog 26596   logb clogb 26807
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-1cn 11213  ax-1ne0 11224
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-cur 8292  df-logb 26808
This theorem is referenced by:  logbf  26832  relogbf  26834  logblog  26835
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