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| Mirrors > Home > MPE Home > Th. List > cxpval | Structured version Visualization version GIF version | ||
| Description: Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.) |
| Ref | Expression |
|---|---|
| cxpval | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴))))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpl 482 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴) | |
| 2 | 1 | eqeq1d 2739 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 0 ↔ 𝐴 = 0)) |
| 3 | simpr 484 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵) | |
| 4 | 3 | eqeq1d 2739 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 = 0 ↔ 𝐵 = 0)) |
| 5 | 4 | ifbid 4549 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = 0, 1, 0) = if(𝐵 = 0, 1, 0)) |
| 6 | 1 | fveq2d 6910 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (log‘𝑥) = (log‘𝐴)) |
| 7 | 3, 6 | oveq12d 7449 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 · (log‘𝑥)) = (𝐵 · (log‘𝐴))) |
| 8 | 7 | fveq2d 6910 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (exp‘(𝑦 · (log‘𝑥))) = (exp‘(𝐵 · (log‘𝐴)))) |
| 9 | 2, 5, 8 | ifbieq12d 4554 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑥 = 0, if(𝑦 = 0, 1, 0), (exp‘(𝑦 · (log‘𝑥)))) = if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴))))) |
| 10 | df-cxp 26599 | . 2 ⊢ ↑𝑐 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ if(𝑥 = 0, if(𝑦 = 0, 1, 0), (exp‘(𝑦 · (log‘𝑥))))) | |
| 11 | ax-1cn 11213 | . . . . 5 ⊢ 1 ∈ ℂ | |
| 12 | 0cn 11253 | . . . . 5 ⊢ 0 ∈ ℂ | |
| 13 | 11, 12 | ifcli 4573 | . . . 4 ⊢ if(𝐵 = 0, 1, 0) ∈ ℂ |
| 14 | 13 | elexi 3503 | . . 3 ⊢ if(𝐵 = 0, 1, 0) ∈ V |
| 15 | fvex 6919 | . . 3 ⊢ (exp‘(𝐵 · (log‘𝐴))) ∈ V | |
| 16 | 14, 15 | ifex 4576 | . 2 ⊢ if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴)))) ∈ V |
| 17 | 9, 10, 16 | ovmpoa 7588 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴))))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ifcif 4525 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 · cmul 11160 expce 16097 logclog 26596 ↑𝑐ccxp 26597 |
| 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-sep 5296 ax-nul 5306 ax-pr 5432 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-mulcl 11217 ax-i2m1 11223 |
| 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-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-cxp 26599 |
| This theorem is referenced by: cxpef 26707 0cxp 26708 cxpexp 26710 cxpcl 26716 recxpcl 26717 |
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