Theorem logbmpt 26752

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 26729	. . 3 ⊢ logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))
2		ovexd 7402	. . . 4 ⊢ (((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ (𝑥 ∈ (ℂ ∖ {0, 1}) ∧ 𝑦 ∈ (ℂ ∖ {0}))) → ((log‘𝑦) / (log‘𝑥)) ∈ V)
3	2	ralrimivva 3180	. . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → ∀𝑥 ∈ (ℂ ∖ {0, 1})∀𝑦 ∈ (ℂ ∖ {0})((log‘𝑦) / (log‘𝑥)) ∈ V)
4		ax-1cn 11096	. . . . . 6 ⊢ 1 ∈ ℂ
5		ax-1ne0 11107	. . . . . . 7 ⊢ 1 ≠ 0
6		elsng 4581	. . . . . . . 8 ⊢ (1 ∈ ℂ → (1 ∈ {0} ↔ 1 = 0))
7	4, 6	ax-mp 5	. . . . . . 7 ⊢ (1 ∈ {0} ↔ 1 = 0)
8	5, 7	nemtbir 3028	. . . . . 6 ⊢ ¬ 1 ∈ {0}
9		eldif 3899	. . . . . 6 ⊢ (1 ∈ (ℂ ∖ {0}) ↔ (1 ∈ ℂ ∧ ¬ 1 ∈ {0}))
10	4, 8, 9	mpbir2an 712	. . . . 5 ⊢ 1 ∈ (ℂ ∖ {0})
11	10	ne0ii 4284	. . . 4 ⊢ (ℂ ∖ {0}) ≠ ∅
12	11	a1i 11	. . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (ℂ ∖ {0}) ≠ ∅)
13		cnex 11119	. . . . 5 ⊢ ℂ ∈ V
14	13	difexi 5271	. . . 4 ⊢ (ℂ ∖ {0}) ∈ V
15	14	a1i 11	. . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (ℂ ∖ {0}) ∈ V)
16		eldifpr 4602	. . . 4 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
17	16	biimpri 228	. . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → 𝐵 ∈ (ℂ ∖ {0, 1}))
18	1, 3, 12, 15, 17	mpocurryvald 8220	. 2 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (curry logb ‘𝐵) = (𝑦 ∈ (ℂ ∖ {0}) ↦ ⦋𝐵 / 𝑥⦌((log‘𝑦) / (log‘𝑥))))
19		csbov2g 7415	. . . . 5 ⊢ (𝐵 ∈ ℂ → ⦋𝐵 / 𝑥⦌((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / ⦋𝐵 / 𝑥⦌(log‘𝑥)))
20		csbfv 6887	. . . . . . 7 ⊢ ⦋𝐵 / 𝑥⦌(log‘𝑥) = (log‘𝐵)
21	20	a1i 11	. . . . . 6 ⊢ (𝐵 ∈ ℂ → ⦋𝐵 / 𝑥⦌(log‘𝑥) = (log‘𝐵))
22	21	oveq2d 7383	. . . . 5 ⊢ (𝐵 ∈ ℂ → ((log‘𝑦) / ⦋𝐵 / 𝑥⦌(log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
23	19, 22	eqtrd 2771	. . . 4 ⊢ (𝐵 ∈ ℂ → ⦋𝐵 / 𝑥⦌((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
24	23	3ad2ant1 1134	. . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → ⦋𝐵 / 𝑥⦌((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
25	24	mpteq2dv 5179	. 2 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (𝑦 ∈ (ℂ ∖ {0}) ↦ ⦋𝐵 / 𝑥⦌((log‘𝑦) / (log‘𝑥))) = (𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝐵))))
26	18, 25	eqtrd 2771	1 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (curry logb ‘𝐵) = (𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝐵))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  Vcvv 3429  ⦋csb 3837   ∖ cdif 3886  ∅c0 4273  {csn 4567  {cpr 4569   ↦ cmpt 5166  ‘cfv 6498  (class class class)co 7367  curry ccur 8215  ℂcc 11036  0cc0 11038  1c1 11039   / cdiv 11807  logclog 26518   log_b clogb 26728

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-1cn 11096  ax-1ne0 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-cur 8217  df-logb 26729

This theorem is referenced by:  logbf  26753  relogbf  26755  logblog  26756