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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 26808 | . . 3 ⊢ logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥))) | |
| 2 | ovexd 7466 | . . . 4 ⊢ (((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ (𝑥 ∈ (ℂ ∖ {0, 1}) ∧ 𝑦 ∈ (ℂ ∖ {0}))) → ((log‘𝑦) / (log‘𝑥)) ∈ V) | |
| 3 | 2 | ralrimivva 3202 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → ∀𝑥 ∈ (ℂ ∖ {0, 1})∀𝑦 ∈ (ℂ ∖ {0})((log‘𝑦) / (log‘𝑥)) ∈ V) |
| 4 | ax-1cn 11213 | . . . . . 6 ⊢ 1 ∈ ℂ | |
| 5 | ax-1ne0 11224 | . . . . . . 7 ⊢ 1 ≠ 0 | |
| 6 | elsng 4640 | . . . . . . . 8 ⊢ (1 ∈ ℂ → (1 ∈ {0} ↔ 1 = 0)) | |
| 7 | 4, 6 | ax-mp 5 | . . . . . . 7 ⊢ (1 ∈ {0} ↔ 1 = 0) |
| 8 | 5, 7 | nemtbir 3038 | . . . . . 6 ⊢ ¬ 1 ∈ {0} |
| 9 | eldif 3961 | . . . . . 6 ⊢ (1 ∈ (ℂ ∖ {0}) ↔ (1 ∈ ℂ ∧ ¬ 1 ∈ {0})) | |
| 10 | 4, 8, 9 | mpbir2an 711 | . . . . 5 ⊢ 1 ∈ (ℂ ∖ {0}) |
| 11 | 10 | ne0ii 4344 | . . . 4 ⊢ (ℂ ∖ {0}) ≠ ∅ |
| 12 | 11 | a1i 11 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (ℂ ∖ {0}) ≠ ∅) |
| 13 | cnex 11236 | . . . . 5 ⊢ ℂ ∈ V | |
| 14 | 13 | difexi 5330 | . . . 4 ⊢ (ℂ ∖ {0}) ∈ V |
| 15 | 14 | a1i 11 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (ℂ ∖ {0}) ∈ V) |
| 16 | eldifpr 4658 | . . . 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 8295 | . 2 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (curry logb ‘𝐵) = (𝑦 ∈ (ℂ ∖ {0}) ↦ ⦋𝐵 / 𝑥⦌((log‘𝑦) / (log‘𝑥)))) |
| 19 | csbov2g 7479 | . . . . 5 ⊢ (𝐵 ∈ ℂ → ⦋𝐵 / 𝑥⦌((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / ⦋𝐵 / 𝑥⦌(log‘𝑥))) | |
| 20 | csbfv 6956 | . . . . . . 7 ⊢ ⦋𝐵 / 𝑥⦌(log‘𝑥) = (log‘𝐵) | |
| 21 | 20 | a1i 11 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → ⦋𝐵 / 𝑥⦌(log‘𝑥) = (log‘𝐵)) |
| 22 | 21 | oveq2d 7447 | . . . . 5 ⊢ (𝐵 ∈ ℂ → ((log‘𝑦) / ⦋𝐵 / 𝑥⦌(log‘𝑥)) = ((log‘𝑦) / (log‘𝐵))) |
| 23 | 19, 22 | eqtrd 2777 | . . . 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 5244 | . 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 2777 | 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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ⦋csb 3899 ∖ cdif 3948 ∅c0 4333 {csn 4626 {cpr 4628 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 curry ccur 8290 ℂcc 11153 0cc0 11155 1c1 11156 / cdiv 11920 logclog 26596 logb clogb 26807 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-1ne0 11224 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-cur 8292 df-logb 26808 |
| This theorem is referenced by: logbf 26832 relogbf 26834 logblog 26835 |
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