Theorem expvalt 4677

Description: Value of exponentiation to natural number powers. ℕ × {A} is the constant function with value A. The seq operation produces the sequence A, A · A, (A · A) · A,... that we evaluate at index B.

Assertion

Ref	Expression
expvalt	⊢ ((A ∈ ℂ ∧ B ∈ ℕ) → (A↑B) = (( · seq(ℕ × {A})) ‘B))

Proof of Theorem expvalt

Step	Hyp	Ref	Expression
1		fvex 2838	. 2 ⊢ (( · seq(ℕ × {A})) ‘B) ∈ V
2		sneq 1816	. . . . 5 ⊢ (x = A → {x} = {A})
3		xpeq2 2441	. . . . 5 ⊢ ({x} = {A} → (ℕ × {x}) = (ℕ × {A}))
4	2, 3	syl 12	. . . 4 ⊢ (x = A → (ℕ × {x}) = (ℕ × {A}))
5	4	opreq2d 3013	. . 3 ⊢ (x = A → ( · seq(ℕ × {x})) = ( · seq(ℕ × {A})))
6	5	fveq1d 2834	. 2 ⊢ (x = A → (( · seq(ℕ × {x})) ‘y) = (( · seq(ℕ × {A})) ‘y))
7		fveq2 2832	. 2 ⊢ (y = B → (( · seq(ℕ × {A})) ‘y) = (( · seq(ℕ × {A})) ‘B))
8		df-exp 4676	. 2 ⊢ ↑ = {⟨⟨x, y⟩, z⟩∣((x ∈ ℂ ∧ y ∈ ℕ) ∧ z = (( · seq(ℕ × {x})) ‘y))}
9	1, 6, 7, 8	oprabval2 3051	1 ⊢ ((A ∈ ℂ ∧ B ∈ ℕ) → (A↑B) = (( · seq(ℕ × {A})) ‘B))

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  {csn 1808   × cxp 2408   ‘cfv 2422  (class class class)co 3001  ℂcc 4026   · cmulc 4032  ℕcn 4093  seqcseq 4660  ↑cexp 4675

This theorem is referenced by:  expp1t 4678  exp1t 4679

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-opr 3003  df-oprab 3004  df-exp 4676

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