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Theorem expval 14105
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 2738 . . 3 ((𝑥 = 𝐴𝑦 = 𝑁) → (𝑦 = 0 ↔ 𝑁 = 0))
31breq2d 5154 . . . 4 ((𝑥 = 𝐴𝑦 = 𝑁) → (0 < 𝑦 ↔ 0 < 𝑁))
4 simpl 482 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝑁) → 𝑥 = 𝐴)
54sneqd 4637 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝑁) → {𝑥} = {𝐴})
65xpeq2d 5714 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝑁) → (ℕ × {𝑥}) = (ℕ × {𝐴}))
76seqeq3d 14051 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝑁) → seq1( · , (ℕ × {𝑥})) = seq1( · , (ℕ × {𝐴})))
87, 1fveq12d 6912 . . . 4 ((𝑥 = 𝐴𝑦 = 𝑁) → (seq1( · , (ℕ × {𝑥}))‘𝑦) = (seq1( · , (ℕ × {𝐴}))‘𝑁))
91negeqd 11503 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝑁) → -𝑦 = -𝑁)
107, 9fveq12d 6912 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝑁) → (seq1( · , (ℕ × {𝑥}))‘-𝑦) = (seq1( · , (ℕ × {𝐴}))‘-𝑁))
1110oveq2d 7448 . . . 4 ((𝑥 = 𝐴𝑦 = 𝑁) → (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)) = (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))
123, 8, 11ifbieq12d 4553 . . 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 4551 . 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 14104 . 2 ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))
15 1ex 11258 . . 3 1 ∈ V
16 fvex 6918 . . . 4 (seq1( · , (ℕ × {𝐴}))‘𝑁) ∈ V
17 ovex 7465 . . . 4 (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)) ∈ V
1816, 17ifex 4575 . . 3 if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁))) ∈ V
1915, 18ifex 4575 . 2 if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))) ∈ V
2013, 14, 19ovmpoa 7589 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 1539  wcel 2107  ifcif 4524  {csn 4625   class class class wbr 5142   × cxp 5682  cfv 6560  (class class class)co 7432  cc 11154  0cc0 11156  1c1 11157   · cmul 11161   < clt 11296  -cneg 11494   / cdiv 11921  cn 12267  cz 12615  seqcseq 14043  cexp 14103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-1cn 11214
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-neg 11496  df-seq 14044  df-exp 14104
This theorem is referenced by:  expnnval  14106  exp0  14107  expneg  14111
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