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

Theorem expval 14036
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 484 . . . 4 ((𝑥 = 𝐴𝑦 = 𝑁) → 𝑦 = 𝑁)
21eqeq1d 2733 . . 3 ((𝑥 = 𝐴𝑦 = 𝑁) → (𝑦 = 0 ↔ 𝑁 = 0))
31breq2d 5160 . . . 4 ((𝑥 = 𝐴𝑦 = 𝑁) → (0 < 𝑦 ↔ 0 < 𝑁))
4 simpl 482 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝑁) → 𝑥 = 𝐴)
54sneqd 4640 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝑁) → {𝑥} = {𝐴})
65xpeq2d 5706 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝑁) → (ℕ × {𝑥}) = (ℕ × {𝐴}))
76seqeq3d 13981 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝑁) → seq1( · , (ℕ × {𝑥})) = seq1( · , (ℕ × {𝐴})))
87, 1fveq12d 6898 . . . 4 ((𝑥 = 𝐴𝑦 = 𝑁) → (seq1( · , (ℕ × {𝑥}))‘𝑦) = (seq1( · , (ℕ × {𝐴}))‘𝑁))
91negeqd 11461 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝑁) → -𝑦 = -𝑁)
107, 9fveq12d 6898 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝑁) → (seq1( · , (ℕ × {𝑥}))‘-𝑦) = (seq1( · , (ℕ × {𝐴}))‘-𝑁))
1110oveq2d 7428 . . . 4 ((𝑥 = 𝐴𝑦 = 𝑁) → (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)) = (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))
123, 8, 11ifbieq12d 4556 . . 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 4554 . 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 14035 . 2 ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))
15 1ex 11217 . . 3 1 ∈ V
16 fvex 6904 . . . 4 (seq1( · , (ℕ × {𝐴}))‘𝑁) ∈ V
17 ovex 7445 . . . 4 (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)) ∈ V
1816, 17ifex 4578 . . 3 if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁))) ∈ V
1915, 18ifex 4578 . 2 if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))) ∈ V
2013, 14, 19ovmpoa 7566 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 395   = wceq 1540  wcel 2105  ifcif 4528  {csn 4628   class class class wbr 5148   × cxp 5674  cfv 6543  (class class class)co 7412  cc 11114  0cc0 11116  1c1 11117   · cmul 11121   < clt 11255  -cneg 11452   / cdiv 11878  cn 12219  cz 12565  seqcseq 13973  cexp 14034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-1cn 11174
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-neg 11454  df-seq 13974  df-exp 14035
This theorem is referenced by:  expnnval  14037  exp0  14038  expneg  14042
  Copyright terms: Public domain W3C validator