Theorem expval 14114

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) → (𝐴↑𝑁) = 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.

Step	Hyp	Ref	Expression
1		simpr 484	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → 𝑦 = 𝑁)
2	1	eqeq1d 2742	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (𝑦 = 0 ↔ 𝑁 = 0))
3	1	breq2d 5178	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (0 < 𝑦 ↔ 0 < 𝑁))
4		simpl 482	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → 𝑥 = 𝐴)
5	4	sneqd 4660	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → {𝑥} = {𝐴})
6	5	xpeq2d 5730	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (ℕ × {𝑥}) = (ℕ × {𝐴}))
7	6	seqeq3d 14060	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → seq1( · , (ℕ × {𝑥})) = seq1( · , (ℕ × {𝐴})))
8	7, 1	fveq12d 6927	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (seq1( · , (ℕ × {𝑥}))‘𝑦) = (seq1( · , (ℕ × {𝐴}))‘𝑁))
9	1	negeqd 11530	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → -𝑦 = -𝑁)
10	7, 9	fveq12d 6927	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (seq1( · , (ℕ × {𝑥}))‘-𝑦) = (seq1( · , (ℕ × {𝐴}))‘-𝑁))
11	10	oveq2d 7464	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)) = (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))
12	3, 8, 11	ifbieq12d 4576	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 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))))
13	2, 12	ifbieq2d 4574	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 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 14113	. 2 ⊢ ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))
15		1ex 11286	. . 3 ⊢ 1 ∈ V
16		fvex 6933	. . . 4 ⊢ (seq1( · , (ℕ × {𝐴}))‘𝑁) ∈ V
17		ovex 7481	. . . 4 ⊢ (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)) ∈ V
18	16, 17	ifex 4598	. . 3 ⊢ if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁))) ∈ V
19	15, 18	ifex 4598	. 2 ⊢ if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))) ∈ V
20	13, 14, 19	ovmpoa 7605	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ifcif 4548  {csn 4648   class class class wbr 5166   × cxp 5698  ‘cfv 6573  (class class class)co 7448  ℂcc 11182  0cc0 11184  1c1 11185   · cmul 11189   < clt 11324  -cneg 11521   / cdiv 11947  ℕcn 12293  ℤcz 12639  seqcseq 14052  ↑cexp 14112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-1cn 11242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-neg 11523  df-seq 14053  df-exp 14113

This theorem is referenced by:  expnnval  14115  exp0  14116  expneg  14120