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Definition df-iexp 8909
Description: Define exponentiation to nonnegative integer powers. This definition is not meant to be used directly; instead, exp0 8913 and expp1 8916 provide the standard recursive definition. The up-arrow notation is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976) and is convenient for us since we don't have superscripts. 10-Jun-2005: The definition was extended to include zero exponents, so that  0 ^ 0  1 per the convention of Definition 10-4.1 of [Gleason] p. 134. 4-Jun-2014: The definition was extended to include negative integer exponents. The case  0 ,  <  0 gives the value  1  0, so we will avoid this case in our theorems. (Contributed by Jim Kingdon, 7-Jun-2020.)
Assertion
Ref Expression
df-iexp  ^  CC ,  ZZ  |->  if  0 ,  1 ,  if 0  <  ,  seq 1  x.  ,  NN 
X.  { } ,  CC `  ,  1  seq 1  x.  ,  NN  X.  { } ,  CC `  -u
Distinct variable group:   ,

Detailed syntax breakdown of Definition df-iexp
StepHypRef Expression
1 cexp 8908 . 2  ^
2 vx . . 3  setvar
3 vy . . 3  setvar
4 cc 6709 . . 3  CC
5 cz 8021 . . 3  ZZ
63cv 1241 . . . . 5
7 cc0 6711 . . . . 5  0
86, 7wceq 1242 . . . 4  0
9 c1 6712 . . . 4  1
10 clt 6857 . . . . . 6  <
117, 6, 10wbr 3755 . . . . 5  0  <
12 cmul 6716 . . . . . . 7  x.
13 cn 7695 . . . . . . . 8  NN
142cv 1241 . . . . . . . . 9
1514csn 3367 . . . . . . . 8  { }
1613, 15cxp 4286 . . . . . . 7  NN 
X.  { }
1712, 4, 16, 9cseq 8892 . . . . . 6  seq 1  x.  ,  NN 
X.  { } ,  CC
186, 17cfv 4845 . . . . 5  seq 1  x.  ,  NN  X.  { } ,  CC `
196cneg 6980 . . . . . . 7  -u
2019, 17cfv 4845 . . . . . 6  seq 1  x.  ,  NN  X.  { } ,  CC `  -u
21 cdiv 7433 . . . . . 6
229, 20, 21co 5455 . . . . 5  1  seq 1  x.  ,  NN 
X.  { } ,  CC `  -u
2311, 18, 22cif 3325 . . . 4  if 0  <  ,  seq 1  x.  ,  NN 
X.  { } ,  CC `  ,  1  seq 1  x.  ,  NN  X.  { } ,  CC `  -u
248, 9, 23cif 3325 . . 3  if  0 ,  1 ,  if 0  <  ,  seq 1  x.  ,  NN 
X.  { } ,  CC `  ,  1  seq 1  x.  ,  NN  X.  { } ,  CC `  -u
252, 3, 4, 5, 24cmpt2 5457 . 2  CC ,  ZZ  |->  if  0 ,  1 ,  if 0  <  ,  seq 1  x.  ,  NN 
X.  { } ,  CC `  ,  1  seq 1  x.  ,  NN  X.  { } ,  CC `  -u
261, 25wceq 1242 1  ^  CC ,  ZZ  |->  if  0 ,  1 ,  if 0  <  ,  seq 1  x.  ,  NN 
X.  { } ,  CC `  ,  1  seq 1  x.  ,  NN  X.  { } ,  CC `  -u
Colors of variables: wff set class
This definition is referenced by:  expival  8911
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