Theorem logbmpt 24526

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 logb ‘𝐵) = (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.

Step	Hyp	Ref	Expression
1		df-logb 24503	. . 3 ⊢ logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))
2		ovexd 6680	. . . 4 ⊢ (((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ (𝑥 ∈ (ℂ ∖ {0, 1}) ∧ 𝑦 ∈ (ℂ ∖ {0}))) → ((log‘𝑦) / (log‘𝑥)) ∈ V)
3	2	ralrimivva 2971	. . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → ∀𝑥 ∈ (ℂ ∖ {0, 1})∀𝑦 ∈ (ℂ ∖ {0})((log‘𝑦) / (log‘𝑥)) ∈ V)
4		ax-1cn 9994	. . . . . 6 ⊢ 1 ∈ ℂ
5		ax-1ne0 10005	. . . . . . 7 ⊢ 1 ≠ 0
6		elsng 4191	. . . . . . . 8 ⊢ (1 ∈ ℂ → (1 ∈ {0} ↔ 1 = 0))
7	4, 6	ax-mp 5	. . . . . . 7 ⊢ (1 ∈ {0} ↔ 1 = 0)
8	5, 7	nemtbir 2889	. . . . . 6 ⊢ ¬ 1 ∈ {0}
9		eldif 3584	. . . . . 6 ⊢ (1 ∈ (ℂ ∖ {0}) ↔ (1 ∈ ℂ ∧ ¬ 1 ∈ {0}))
10	4, 8, 9	mpbir2an 955	. . . . 5 ⊢ 1 ∈ (ℂ ∖ {0})
11	10	ne0ii 3923	. . . 4 ⊢ (ℂ ∖ {0}) ≠ ∅
12	11	a1i 11	. . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (ℂ ∖ {0}) ≠ ∅)
13		cnex 10017	. . . . 5 ⊢ ℂ ∈ V
14		difexg 4808	. . . . 5 ⊢ (ℂ ∈ V → (ℂ ∖ {0}) ∈ V)
15	13, 14	ax-mp 5	. . . 4 ⊢ (ℂ ∖ {0}) ∈ V
16	15	a1i 11	. . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (ℂ ∖ {0}) ∈ V)
17		eldifpr 4204	. . . 4 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
18	17	biimpri 218	. . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → 𝐵 ∈ (ℂ ∖ {0, 1}))
19	1, 3, 12, 16, 18	mpt2curryvald 7396	. 2 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (curry logb ‘𝐵) = (𝑦 ∈ (ℂ ∖ {0}) ↦ ⦋𝐵 / 𝑥⦌((log‘𝑦) / (log‘𝑥))))
20		csbov2g 6691	. . . . 5 ⊢ (𝐵 ∈ ℂ → ⦋𝐵 / 𝑥⦌((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / ⦋𝐵 / 𝑥⦌(log‘𝑥)))
21		csbfv 6233	. . . . . . 7 ⊢ ⦋𝐵 / 𝑥⦌(log‘𝑥) = (log‘𝐵)
22	21	a1i 11	. . . . . 6 ⊢ (𝐵 ∈ ℂ → ⦋𝐵 / 𝑥⦌(log‘𝑥) = (log‘𝐵))
23	22	oveq2d 6666	. . . . 5 ⊢ (𝐵 ∈ ℂ → ((log‘𝑦) / ⦋𝐵 / 𝑥⦌(log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
24	20, 23	eqtrd 2656	. . . 4 ⊢ (𝐵 ∈ ℂ → ⦋𝐵 / 𝑥⦌((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
25	24	3ad2ant1 1082	. . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → ⦋𝐵 / 𝑥⦌((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
26	25	mpteq2dv 4745	. 2 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (𝑦 ∈ (ℂ ∖ {0}) ↦ ⦋𝐵 / 𝑥⦌((log‘𝑦) / (log‘𝑥))) = (𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝐵))))
27	19, 26	eqtrd 2656	1 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (curry logb ‘𝐵) = (𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝐵))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200  ⦋csb 3533   ∖ cdif 3571  ∅c0 3915  {csn 4177  {cpr 4179   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650  curry ccur 7391  ℂcc 9934  0cc0 9936  1c1 9937   / cdiv 10684  logclog 24301   log_b clogb 24502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-1cn 9994  ax-1ne0 10005

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-cur 7393  df-logb 24503

This theorem is referenced by:  logbf  24527  relogbf  24529  logblog  24530