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Mirrors > Home > MPE Home > Th. List > logbmpt | Structured version Visualization version GIF version |
Description: The general logarithm to a fixed base regarded as mapping. (Contributed by AV, 11-Jun-2020.) |
Ref | Expression |
---|---|
logbmpt | ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (curry logb ‘𝐵) = (𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-logb 26742 | . . 3 ⊢ logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥))) | |
2 | ovexd 7454 | . . . 4 ⊢ (((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ (𝑥 ∈ (ℂ ∖ {0, 1}) ∧ 𝑦 ∈ (ℂ ∖ {0}))) → ((log‘𝑦) / (log‘𝑥)) ∈ V) | |
3 | 2 | ralrimivva 3190 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → ∀𝑥 ∈ (ℂ ∖ {0, 1})∀𝑦 ∈ (ℂ ∖ {0})((log‘𝑦) / (log‘𝑥)) ∈ V) |
4 | ax-1cn 11198 | . . . . . 6 ⊢ 1 ∈ ℂ | |
5 | ax-1ne0 11209 | . . . . . . 7 ⊢ 1 ≠ 0 | |
6 | elsng 4644 | . . . . . . . 8 ⊢ (1 ∈ ℂ → (1 ∈ {0} ↔ 1 = 0)) | |
7 | 4, 6 | ax-mp 5 | . . . . . . 7 ⊢ (1 ∈ {0} ↔ 1 = 0) |
8 | 5, 7 | nemtbir 3027 | . . . . . 6 ⊢ ¬ 1 ∈ {0} |
9 | eldif 3954 | . . . . . 6 ⊢ (1 ∈ (ℂ ∖ {0}) ↔ (1 ∈ ℂ ∧ ¬ 1 ∈ {0})) | |
10 | 4, 8, 9 | mpbir2an 709 | . . . . 5 ⊢ 1 ∈ (ℂ ∖ {0}) |
11 | 10 | ne0ii 4337 | . . . 4 ⊢ (ℂ ∖ {0}) ≠ ∅ |
12 | 11 | a1i 11 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (ℂ ∖ {0}) ≠ ∅) |
13 | cnex 11221 | . . . . 5 ⊢ ℂ ∈ V | |
14 | 13 | difexi 5331 | . . . 4 ⊢ (ℂ ∖ {0}) ∈ V |
15 | 14 | a1i 11 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (ℂ ∖ {0}) ∈ V) |
16 | eldifpr 4662 | . . . 4 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1)) | |
17 | 16 | biimpri 227 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → 𝐵 ∈ (ℂ ∖ {0, 1})) |
18 | 1, 3, 12, 15, 17 | mpocurryvald 8276 | . 2 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (curry logb ‘𝐵) = (𝑦 ∈ (ℂ ∖ {0}) ↦ ⦋𝐵 / 𝑥⦌((log‘𝑦) / (log‘𝑥)))) |
19 | csbov2g 7466 | . . . . 5 ⊢ (𝐵 ∈ ℂ → ⦋𝐵 / 𝑥⦌((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / ⦋𝐵 / 𝑥⦌(log‘𝑥))) | |
20 | csbfv 6946 | . . . . . . 7 ⊢ ⦋𝐵 / 𝑥⦌(log‘𝑥) = (log‘𝐵) | |
21 | 20 | a1i 11 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → ⦋𝐵 / 𝑥⦌(log‘𝑥) = (log‘𝐵)) |
22 | 21 | oveq2d 7435 | . . . . 5 ⊢ (𝐵 ∈ ℂ → ((log‘𝑦) / ⦋𝐵 / 𝑥⦌(log‘𝑥)) = ((log‘𝑦) / (log‘𝐵))) |
23 | 19, 22 | eqtrd 2765 | . . . 4 ⊢ (𝐵 ∈ ℂ → ⦋𝐵 / 𝑥⦌((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵))) |
24 | 23 | 3ad2ant1 1130 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → ⦋𝐵 / 𝑥⦌((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵))) |
25 | 24 | mpteq2dv 5251 | . 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 2765 | 1 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (curry logb ‘𝐵) = (𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 Vcvv 3461 ⦋csb 3889 ∖ cdif 3941 ∅c0 4322 {csn 4630 {cpr 4632 ↦ cmpt 5232 ‘cfv 6549 (class class class)co 7419 curry ccur 8271 ℂcc 11138 0cc0 11140 1c1 11141 / cdiv 11903 logclog 26533 logb clogb 26741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-1cn 11198 ax-1ne0 11209 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-cur 8273 df-logb 26742 |
This theorem is referenced by: logbf 26766 relogbf 26768 logblog 26769 |
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