MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  expval Structured version   Visualization version   GIF version

Theorem expval 13419
Description: Value of exponentiation to integer powers. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.)
Assertion
Ref Expression
expval ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (𝐴𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))))

Proof of Theorem expval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 485 . . . 4 ((𝑥 = 𝐴𝑦 = 𝑁) → 𝑦 = 𝑁)
21eqeq1d 2820 . . 3 ((𝑥 = 𝐴𝑦 = 𝑁) → (𝑦 = 0 ↔ 𝑁 = 0))
31breq2d 5069 . . . 4 ((𝑥 = 𝐴𝑦 = 𝑁) → (0 < 𝑦 ↔ 0 < 𝑁))
4 simpl 483 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝑁) → 𝑥 = 𝐴)
54sneqd 4569 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝑁) → {𝑥} = {𝐴})
65xpeq2d 5578 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝑁) → (ℕ × {𝑥}) = (ℕ × {𝐴}))
76seqeq3d 13365 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝑁) → seq1( · , (ℕ × {𝑥})) = seq1( · , (ℕ × {𝐴})))
87, 1fveq12d 6670 . . . 4 ((𝑥 = 𝐴𝑦 = 𝑁) → (seq1( · , (ℕ × {𝑥}))‘𝑦) = (seq1( · , (ℕ × {𝐴}))‘𝑁))
91negeqd 10868 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝑁) → -𝑦 = -𝑁)
107, 9fveq12d 6670 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝑁) → (seq1( · , (ℕ × {𝑥}))‘-𝑦) = (seq1( · , (ℕ × {𝐴}))‘-𝑁))
1110oveq2d 7161 . . . 4 ((𝑥 = 𝐴𝑦 = 𝑁) → (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)) = (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))
123, 8, 11ifbieq12d 4490 . . 3 ((𝑥 = 𝐴𝑦 = 𝑁) → if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦))) = if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁))))
132, 12ifbieq2d 4488 . 2 ((𝑥 = 𝐴𝑦 = 𝑁) → if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))))
14 df-exp 13418 . 2 ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))
15 1ex 10625 . . 3 1 ∈ V
16 fvex 6676 . . . 4 (seq1( · , (ℕ × {𝐴}))‘𝑁) ∈ V
17 ovex 7178 . . . 4 (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)) ∈ V
1816, 17ifex 4511 . . 3 if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁))) ∈ V
1915, 18ifex 4511 . 2 if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))) ∈ V
2013, 14, 19ovmpoa 7294 1 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (𝐴𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  ifcif 4463  {csn 4557   class class class wbr 5057   × cxp 5546  cfv 6348  (class class class)co 7145  cc 10523  0cc0 10525  1c1 10526   · cmul 10530   < clt 10663  -cneg 10859   / cdiv 11285  cn 11626  cz 11969  seqcseq 13357  cexp 13417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-1cn 10583
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-neg 10861  df-seq 13358  df-exp 13418
This theorem is referenced by:  expnnval  13420  exp0  13421  expneg  13425
  Copyright terms: Public domain W3C validator