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Theorem logbmpt 25352
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 25329 . . 3 logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))
2 ovexd 7165 . . . 4 (((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ (𝑥 ∈ (ℂ ∖ {0, 1}) ∧ 𝑦 ∈ (ℂ ∖ {0}))) → ((log‘𝑦) / (log‘𝑥)) ∈ V)
32ralrimivva 3179 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → ∀𝑥 ∈ (ℂ ∖ {0, 1})∀𝑦 ∈ (ℂ ∖ {0})((log‘𝑦) / (log‘𝑥)) ∈ V)
4 ax-1cn 10572 . . . . . 6 1 ∈ ℂ
5 ax-1ne0 10583 . . . . . . 7 1 ≠ 0
6 elsng 4554 . . . . . . . 8 (1 ∈ ℂ → (1 ∈ {0} ↔ 1 = 0))
74, 6ax-mp 5 . . . . . . 7 (1 ∈ {0} ↔ 1 = 0)
85, 7nemtbir 3102 . . . . . 6 ¬ 1 ∈ {0}
9 eldif 3920 . . . . . 6 (1 ∈ (ℂ ∖ {0}) ↔ (1 ∈ ℂ ∧ ¬ 1 ∈ {0}))
104, 8, 9mpbir2an 710 . . . . 5 1 ∈ (ℂ ∖ {0})
1110ne0ii 4276 . . . 4 (ℂ ∖ {0}) ≠ ∅
1211a1i 11 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (ℂ ∖ {0}) ≠ ∅)
13 cnex 10595 . . . . 5 ℂ ∈ V
1413difexi 5205 . . . 4 (ℂ ∖ {0}) ∈ V
1514a1i 11 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (ℂ ∖ {0}) ∈ V)
16 eldifpr 4570 . . . 4 (𝐵 ∈ (ℂ ∖ {0, 1}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
1716biimpri 231 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → 𝐵 ∈ (ℂ ∖ {0, 1}))
181, 3, 12, 15, 17mpocurryvald 7911 . 2 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (curry logb𝐵) = (𝑦 ∈ (ℂ ∖ {0}) ↦ 𝐵 / 𝑥((log‘𝑦) / (log‘𝑥))))
19 csbov2g 7176 . . . . 5 (𝐵 ∈ ℂ → 𝐵 / 𝑥((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / 𝐵 / 𝑥(log‘𝑥)))
20 csbfv 6688 . . . . . . 7 𝐵 / 𝑥(log‘𝑥) = (log‘𝐵)
2120a1i 11 . . . . . 6 (𝐵 ∈ ℂ → 𝐵 / 𝑥(log‘𝑥) = (log‘𝐵))
2221oveq2d 7146 . . . . 5 (𝐵 ∈ ℂ → ((log‘𝑦) / 𝐵 / 𝑥(log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
2319, 22eqtrd 2856 . . . 4 (𝐵 ∈ ℂ → 𝐵 / 𝑥((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
24233ad2ant1 1130 . . 3 ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → 𝐵 / 𝑥((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
2524mpteq2dv 5135 . 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 2856 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3007  Vcvv 3471  csb 3857  cdif 3907  c0 4266  {csn 4540  {cpr 4542  cmpt 5119  cfv 6328  (class class class)co 7130  curry ccur 7906  cc 10512  0cc0 10514  1c1 10515   / cdiv 11274  logclog 25124   logb clogb 25328
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-1cn 10572  ax-1ne0 10583
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-cur 7908  df-logb 25329
This theorem is referenced by:  logbf  25353  relogbf  25355  logblog  25356
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