Description: Define exponentiation to nonnegative integer powers. For example, ( 5 ^ 2 ) = 2 5 ( ex-exp ). Terminology: In general, "exponentiation" is the operation of raising a "base" x to the power of the "exponent" y , resulting in the "power" ( x ^ y ) , also called "x to the power of y". In this case, "integer exponentiation" is the operation of raising a complex "base" x to the power of an integer y , resulting in the "integer power" ( x ^ y ) .
This definition is not meant to be used directly; instead, exp0 and expp1 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 ( 0exp0e1 ).
4-Jun-2014: The definition was extended to include negative integer exponents. For example, ( -u 3 ^ -u 2 ) = ( 1 / 9 ) ( ex-exp ). The case x = 0 , y < 0 gives the value ( 1 / 0 ) , so we will avoid this case in our theorems.
For a definition of exponentiation including complex exponents see df-cxp (complex exponentiation). Both definitions are equivalent for integer exponents, see cxpexpz . (Contributed by Raph Levien, 20-May-2004) (Revised by NM, 15-Oct-2004)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | df-exp | ⊢ ↑ = ( 𝑥 ∈ ℂ , 𝑦 ∈ ℤ ↦ if ( 𝑦 = 0 , 1 , if ( 0 < 𝑦 , ( seq 1 ( · , ( ℕ × { 𝑥 } ) ) ‘ 𝑦 ) , ( 1 / ( seq 1 ( · , ( ℕ × { 𝑥 } ) ) ‘ - 𝑦 ) ) ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 0 | cexp | ⊢ ↑ | |
| 1 | vx | ⊢ 𝑥 | |
| 2 | cc | ⊢ ℂ | |
| 3 | vy | ⊢ 𝑦 | |
| 4 | cz | ⊢ ℤ | |
| 5 | 3 | cv | ⊢ 𝑦 |
| 6 | cc0 | ⊢ 0 | |
| 7 | 5 6 | wceq | ⊢ 𝑦 = 0 |
| 8 | c1 | ⊢ 1 | |
| 9 | clt | ⊢ < | |
| 10 | 6 5 9 | wbr | ⊢ 0 < 𝑦 |
| 11 | cmul | ⊢ · | |
| 12 | cn | ⊢ ℕ | |
| 13 | 1 | cv | ⊢ 𝑥 |
| 14 | 13 | csn | ⊢ { 𝑥 } |
| 15 | 12 14 | cxp | ⊢ ( ℕ × { 𝑥 } ) |
| 16 | 11 15 8 | cseq | ⊢ seq 1 ( · , ( ℕ × { 𝑥 } ) ) |
| 17 | 5 16 | cfv | ⊢ ( seq 1 ( · , ( ℕ × { 𝑥 } ) ) ‘ 𝑦 ) |
| 18 | cdiv | ⊢ / | |
| 19 | 5 | cneg | ⊢ - 𝑦 |
| 20 | 19 16 | cfv | ⊢ ( seq 1 ( · , ( ℕ × { 𝑥 } ) ) ‘ - 𝑦 ) |
| 21 | 8 20 18 | co | ⊢ ( 1 / ( seq 1 ( · , ( ℕ × { 𝑥 } ) ) ‘ - 𝑦 ) ) |
| 22 | 10 17 21 | cif | ⊢ if ( 0 < 𝑦 , ( seq 1 ( · , ( ℕ × { 𝑥 } ) ) ‘ 𝑦 ) , ( 1 / ( seq 1 ( · , ( ℕ × { 𝑥 } ) ) ‘ - 𝑦 ) ) ) |
| 23 | 7 8 22 | cif | ⊢ if ( 𝑦 = 0 , 1 , if ( 0 < 𝑦 , ( seq 1 ( · , ( ℕ × { 𝑥 } ) ) ‘ 𝑦 ) , ( 1 / ( seq 1 ( · , ( ℕ × { 𝑥 } ) ) ‘ - 𝑦 ) ) ) ) |
| 24 | 1 3 2 4 23 | cmpo | ⊢ ( 𝑥 ∈ ℂ , 𝑦 ∈ ℤ ↦ if ( 𝑦 = 0 , 1 , if ( 0 < 𝑦 , ( seq 1 ( · , ( ℕ × { 𝑥 } ) ) ‘ 𝑦 ) , ( 1 / ( seq 1 ( · , ( ℕ × { 𝑥 } ) ) ‘ - 𝑦 ) ) ) ) ) |
| 25 | 0 24 | wceq | ⊢ ↑ = ( 𝑥 ∈ ℂ , 𝑦 ∈ ℤ ↦ if ( 𝑦 = 0 , 1 , if ( 0 < 𝑦 , ( seq 1 ( · , ( ℕ × { 𝑥 } ) ) ‘ 𝑦 ) , ( 1 / ( seq 1 ( · , ( ℕ × { 𝑥 } ) ) ‘ - 𝑦 ) ) ) ) ) |