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Theorem expval 13426
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 2828 . . 3 ((𝑥 = 𝐴𝑦 = 𝑁) → (𝑦 = 0 ↔ 𝑁 = 0))
31breq2d 5075 . . . 4 ((𝑥 = 𝐴𝑦 = 𝑁) → (0 < 𝑦 ↔ 0 < 𝑁))
4 simpl 483 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝑁) → 𝑥 = 𝐴)
54sneqd 4576 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝑁) → {𝑥} = {𝐴})
65xpeq2d 5584 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝑁) → (ℕ × {𝑥}) = (ℕ × {𝐴}))
76seqeq3d 13372 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝑁) → seq1( · , (ℕ × {𝑥})) = seq1( · , (ℕ × {𝐴})))
87, 1fveq12d 6676 . . . 4 ((𝑥 = 𝐴𝑦 = 𝑁) → (seq1( · , (ℕ × {𝑥}))‘𝑦) = (seq1( · , (ℕ × {𝐴}))‘𝑁))
91negeqd 10874 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝑁) → -𝑦 = -𝑁)
107, 9fveq12d 6676 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝑁) → (seq1( · , (ℕ × {𝑥}))‘-𝑦) = (seq1( · , (ℕ × {𝐴}))‘-𝑁))
1110oveq2d 7166 . . . 4 ((𝑥 = 𝐴𝑦 = 𝑁) → (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)) = (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))
123, 8, 11ifbieq12d 4497 . . 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 4495 . 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 13425 . 2 ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))
15 1ex 10631 . . 3 1 ∈ V
16 fvex 6682 . . . 4 (seq1( · , (ℕ × {𝐴}))‘𝑁) ∈ V
17 ovex 7183 . . . 4 (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)) ∈ V
1816, 17ifex 4518 . . 3 if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁))) ∈ V
1915, 18ifex 4518 . 2 if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))) ∈ V
2013, 14, 19ovmpoa 7299 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 1530  wcel 2107  ifcif 4470  {csn 4564   class class class wbr 5063   × cxp 5552  cfv 6354  (class class class)co 7150  cc 10529  0cc0 10531  1c1 10532   · cmul 10536   < clt 10669  -cneg 10865   / cdiv 11291  cn 11632  cz 11975  seqcseq 13364  cexp 13424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326  ax-1cn 10589
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-iota 6313  df-fun 6356  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-neg 10867  df-seq 13365  df-exp 13425
This theorem is referenced by:  expnnval  13427  exp0  13428  expneg  13432
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