Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 486 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴) | |
2 | 1 | eqeq1d 2739 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 0 ↔ 𝐴 = 0)) |
3 | simpr 488 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵) | |
4 | 3 | eqeq1d 2739 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 = 0 ↔ 𝐵 = 0)) |
5 | 4 | ifbid 4462 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = 0, 1, 0) = if(𝐵 = 0, 1, 0)) |
6 | 1 | fveq2d 6721 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (log‘𝑥) = (log‘𝐴)) |
7 | 3, 6 | oveq12d 7231 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 · (log‘𝑥)) = (𝐵 · (log‘𝐴))) |
8 | 7 | fveq2d 6721 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (exp‘(𝑦 · (log‘𝑥))) = (exp‘(𝐵 · (log‘𝐴)))) |
9 | 2, 5, 8 | ifbieq12d 4467 | . 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 25446 | . 2 ⊢ ↑𝑐 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ if(𝑥 = 0, if(𝑦 = 0, 1, 0), (exp‘(𝑦 · (log‘𝑥))))) | |
11 | ax-1cn 10787 | . . . . 5 ⊢ 1 ∈ ℂ | |
12 | 0cn 10825 | . . . . 5 ⊢ 0 ∈ ℂ | |
13 | 11, 12 | ifcli 4486 | . . . 4 ⊢ if(𝐵 = 0, 1, 0) ∈ ℂ |
14 | 13 | elexi 3427 | . . 3 ⊢ if(𝐵 = 0, 1, 0) ∈ V |
15 | fvex 6730 | . . 3 ⊢ (exp‘(𝐵 · (log‘𝐴))) ∈ V | |
16 | 14, 15 | ifex 4489 | . 2 ⊢ if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴)))) ∈ V |
17 | 9, 10, 16 | ovmpoa 7364 | 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 399 = wceq 1543 ∈ wcel 2110 ifcif 4439 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 0cc0 10729 1c1 10730 · cmul 10734 expce 15623 logclog 25443 ↑𝑐ccxp 25444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-mulcl 10791 ax-i2m1 10797 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-iota 6338 df-fun 6382 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-cxp 25446 |
This theorem is referenced by: cxpef 25553 0cxp 25554 cxpexp 25556 cxpcl 25562 recxpcl 25563 |
Copyright terms: Public domain | W3C validator |