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Metamath Proof Explorer

Theorem expm1

Description: Value of a nonzero complex number raised to an integer power minus one. (Contributed by NM, 25-Dec-2008) (Revised by Mario Carneiro, 4-Jun-2014)

Ref Expression
Assertion expm1 
|- ( ( A e. CC /\ A =/= 0 /\ N e. ZZ ) -> ( A ^ ( N - 1 ) ) = ( ( A ^ N ) / A ) )

Proof

Step Hyp Ref Expression
1 1z 
 |-  1 e. ZZ
2 expsub 
 |-  ( ( ( A e. CC /\ A =/= 0 ) /\ ( N e. ZZ /\ 1 e. ZZ ) ) -> ( A ^ ( N - 1 ) ) = ( ( A ^ N ) / ( A ^ 1 ) ) )
3 1 2 mpanr2 
 |-  ( ( ( A e. CC /\ A =/= 0 ) /\ N e. ZZ ) -> ( A ^ ( N - 1 ) ) = ( ( A ^ N ) / ( A ^ 1 ) ) )
4 3 3impa 
 |-  ( ( A e. CC /\ A =/= 0 /\ N e. ZZ ) -> ( A ^ ( N - 1 ) ) = ( ( A ^ N ) / ( A ^ 1 ) ) )
5 exp1 
 |-  ( A e. CC -> ( A ^ 1 ) = A )
6 5 3ad2ant1 
 |-  ( ( A e. CC /\ A =/= 0 /\ N e. ZZ ) -> ( A ^ 1 ) = A )
7 6 oveq2d 
 |-  ( ( A e. CC /\ A =/= 0 /\ N e. ZZ ) -> ( ( A ^ N ) / ( A ^ 1 ) ) = ( ( A ^ N ) / A ) )
8 4 7 eqtrd 
 |-  ( ( A e. CC /\ A =/= 0 /\ N e. ZZ ) -> ( A ^ ( N - 1 ) ) = ( ( A ^ N ) / A ) )