![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 25351 | . . 3 ⊢ logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥))) | |
2 | ovexd 7170 | . . . 4 ⊢ (((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ (𝑥 ∈ (ℂ ∖ {0, 1}) ∧ 𝑦 ∈ (ℂ ∖ {0}))) → ((log‘𝑦) / (log‘𝑥)) ∈ V) | |
3 | 2 | ralrimivva 3156 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → ∀𝑥 ∈ (ℂ ∖ {0, 1})∀𝑦 ∈ (ℂ ∖ {0})((log‘𝑦) / (log‘𝑥)) ∈ V) |
4 | ax-1cn 10584 | . . . . . 6 ⊢ 1 ∈ ℂ | |
5 | ax-1ne0 10595 | . . . . . . 7 ⊢ 1 ≠ 0 | |
6 | elsng 4539 | . . . . . . . 8 ⊢ (1 ∈ ℂ → (1 ∈ {0} ↔ 1 = 0)) | |
7 | 4, 6 | ax-mp 5 | . . . . . . 7 ⊢ (1 ∈ {0} ↔ 1 = 0) |
8 | 5, 7 | nemtbir 3082 | . . . . . 6 ⊢ ¬ 1 ∈ {0} |
9 | eldif 3891 | . . . . . 6 ⊢ (1 ∈ (ℂ ∖ {0}) ↔ (1 ∈ ℂ ∧ ¬ 1 ∈ {0})) | |
10 | 4, 8, 9 | mpbir2an 710 | . . . . 5 ⊢ 1 ∈ (ℂ ∖ {0}) |
11 | 10 | ne0ii 4253 | . . . 4 ⊢ (ℂ ∖ {0}) ≠ ∅ |
12 | 11 | a1i 11 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (ℂ ∖ {0}) ≠ ∅) |
13 | cnex 10607 | . . . . 5 ⊢ ℂ ∈ V | |
14 | 13 | difexi 5196 | . . . 4 ⊢ (ℂ ∖ {0}) ∈ V |
15 | 14 | a1i 11 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (ℂ ∖ {0}) ∈ V) |
16 | eldifpr 4557 | . . . 4 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1)) | |
17 | 16 | biimpri 231 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → 𝐵 ∈ (ℂ ∖ {0, 1})) |
18 | 1, 3, 12, 15, 17 | mpocurryvald 7919 | . 2 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (curry logb ‘𝐵) = (𝑦 ∈ (ℂ ∖ {0}) ↦ ⦋𝐵 / 𝑥⦌((log‘𝑦) / (log‘𝑥)))) |
19 | csbov2g 7181 | . . . . 5 ⊢ (𝐵 ∈ ℂ → ⦋𝐵 / 𝑥⦌((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / ⦋𝐵 / 𝑥⦌(log‘𝑥))) | |
20 | csbfv 6690 | . . . . . . 7 ⊢ ⦋𝐵 / 𝑥⦌(log‘𝑥) = (log‘𝐵) | |
21 | 20 | a1i 11 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → ⦋𝐵 / 𝑥⦌(log‘𝑥) = (log‘𝐵)) |
22 | 21 | oveq2d 7151 | . . . . 5 ⊢ (𝐵 ∈ ℂ → ((log‘𝑦) / ⦋𝐵 / 𝑥⦌(log‘𝑥)) = ((log‘𝑦) / (log‘𝐵))) |
23 | 19, 22 | eqtrd 2833 | . . . 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 5126 | . 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 2833 | 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 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ⦋csb 3828 ∖ cdif 3878 ∅c0 4243 {csn 4525 {cpr 4527 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 curry ccur 7914 ℂcc 10524 0cc0 10526 1c1 10527 / cdiv 11286 logclog 25146 logb clogb 25350 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-1cn 10584 ax-1ne0 10595 |
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 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-cur 7916 df-logb 25351 |
This theorem is referenced by: logbf 25375 relogbf 25377 logblog 25378 |
Copyright terms: Public domain | W3C validator |