Theorem expval 14090

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 489	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → 𝑦 = 𝑁)
2	1	eqeq1d 2767	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (𝑦 = 0 ↔ 𝑁 = 0))
3	1	breq2d 5117	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (0 < 𝑦 ↔ 0 < 𝑁))
4		simpl 487	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → 𝑥 = 𝐴)
5	4	sneqd 4597	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → {𝑥} = {𝐴})
6	5	xpeq2d 5682	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (ℕ × {𝑥}) = (ℕ × {𝐴}))
7	6	seqeq3d 14036	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → seq1( · , (ℕ × {𝑥})) = seq1( · , (ℕ × {𝐴})))
8	7, 1	fveq12d 6878	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (seq1( · , (ℕ × {𝑥}))‘𝑦) = (seq1( · , (ℕ × {𝐴}))‘𝑁))
9	1	negeqd 11439	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → -𝑦 = -𝑁)
10	7, 9	fveq12d 6878	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (seq1( · , (ℕ × {𝑥}))‘-𝑦) = (seq1( · , (ℕ × {𝐴}))‘-𝑁))
11	10	oveq2d 7416	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)) = (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))
12	3, 8, 11	ifbieq12d 4512	. . 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 4510	. 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 14089	. 2 ⊢ ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))
15		1ex 11191	. . 3 ⊢ 1 ∈ V
16		fvex 6884	. . . 4 ⊢ (seq1( · , (ℕ × {𝐴}))‘𝑁) ∈ V
17		ovex 7433	. . . 4 ⊢ (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)) ∈ V
18	16, 17	ifex 4534	. . 3 ⊢ if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁))) ∈ V
19	15, 18	ifex 4534	. 2 ⊢ if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))) ∈ V
20	13, 14, 19	ovmpoa 7555	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 400   = wceq 1563   ∈ wcel 2145  ifcif 4483  {csn 4585   class class class wbr 5105   × cxp 5650  ‘cfv 6525  (class class class)co 7400  ℂcc 11086  0cc0 11088  1c1 11089   · cmul 11093   < clt 11231  -cneg 11430   / cdiv 11859  ℕcn 12224  ℤcz 12582  seqcseq 14028  ↑cexp 14088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-1cn 11146

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-neg 11432  df-seq 14029  df-exp 14089

This theorem is referenced by:  expnnval  14091  exp0  14092  expneg  14096