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Theorem expval 13074
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 477 . . . 4 ((𝑥 = 𝐴𝑦 = 𝑁) → 𝑦 = 𝑁)
21eqeq1d 2767 . . 3 ((𝑥 = 𝐴𝑦 = 𝑁) → (𝑦 = 0 ↔ 𝑁 = 0))
31breq2d 4823 . . . 4 ((𝑥 = 𝐴𝑦 = 𝑁) → (0 < 𝑦 ↔ 0 < 𝑁))
4 simpl 474 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝑁) → 𝑥 = 𝐴)
54sneqd 4348 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝑁) → {𝑥} = {𝐴})
65xpeq2d 5309 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝑁) → (ℕ × {𝑥}) = (ℕ × {𝐴}))
76seqeq3d 13021 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝑁) → seq1( · , (ℕ × {𝑥})) = seq1( · , (ℕ × {𝐴})))
87, 1fveq12d 6386 . . . 4 ((𝑥 = 𝐴𝑦 = 𝑁) → (seq1( · , (ℕ × {𝑥}))‘𝑦) = (seq1( · , (ℕ × {𝐴}))‘𝑁))
91negeqd 10533 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝑁) → -𝑦 = -𝑁)
107, 9fveq12d 6386 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝑁) → (seq1( · , (ℕ × {𝑥}))‘-𝑦) = (seq1( · , (ℕ × {𝐴}))‘-𝑁))
1110oveq2d 6862 . . . 4 ((𝑥 = 𝐴𝑦 = 𝑁) → (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)) = (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))
123, 8, 11ifbieq12d 4272 . . 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 4270 . 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 13073 . 2 ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))
15 1ex 10293 . . 3 1 ∈ V
16 fvex 6392 . . . 4 (seq1( · , (ℕ × {𝐴}))‘𝑁) ∈ V
17 ovex 6878 . . . 4 (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)) ∈ V
1816, 17ifex 4293 . . 3 if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁))) ∈ V
1915, 18ifex 4293 . 2 if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))) ∈ V
2013, 14, 19ovmpt2a 6993 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 384   = wceq 1652  wcel 2155  ifcif 4245  {csn 4336   class class class wbr 4811   × cxp 5277  cfv 6070  (class class class)co 6846  cc 10191  0cc0 10193  1c1 10194   · cmul 10198   < clt 10332  -cneg 10525   / cdiv 10942  cn 11278  cz 11628  seqcseq 13013  cexp 13072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064  ax-1cn 10251
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-iota 6033  df-fun 6072  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-neg 10527  df-seq 13014  df-exp 13073
This theorem is referenced by:  expnnval  13075  exp0  13076  expneg  13080
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