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Metamath Proof Explorer

Theorem abscxp2

Description: Absolute value of a power, when the exponent is real. (Contributed by Mario Carneiro, 15-Sep-2014)

Ref Expression
Assertion abscxp2 
|- ( ( A e. CC /\ B e. RR ) -> ( abs ` ( A ^c B ) ) = ( ( abs ` A ) ^c B ) )

Proof

Step Hyp Ref Expression
1 0red 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> 0 e. RR )
2 0le0 
 |-  0 <_ 0
3 2 a1i 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> 0 <_ 0 )
4 simplr 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> B e. RR )
5 recxpcl 
 |-  ( ( 0 e. RR /\ 0 <_ 0 /\ B e. RR ) -> ( 0 ^c B ) e. RR )
6 1 3 4 5 syl3anc 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> ( 0 ^c B ) e. RR )
7 cxpge0 
 |-  ( ( 0 e. RR /\ 0 <_ 0 /\ B e. RR ) -> 0 <_ ( 0 ^c B ) )
8 1 3 4 7 syl3anc 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> 0 <_ ( 0 ^c B ) )
9 6 8 absidd 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> ( abs ` ( 0 ^c B ) ) = ( 0 ^c B ) )
10 simpr 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> A = 0 )
11 10 oveq1d 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> ( A ^c B ) = ( 0 ^c B ) )
12 11 fveq2d 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> ( abs ` ( A ^c B ) ) = ( abs ` ( 0 ^c B ) ) )
13 10 abs00bd 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> ( abs ` A ) = 0 )
14 13 oveq1d 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> ( ( abs ` A ) ^c B ) = ( 0 ^c B ) )
15 9 12 14 3eqtr4d 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> ( abs ` ( A ^c B ) ) = ( ( abs ` A ) ^c B ) )
16 simplr 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> B e. RR )
17 16 recnd 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> B e. CC )
18 logcl 
 |-  ( ( A e. CC /\ A =/= 0 ) -> ( log ` A ) e. CC )
19 18 adantlr 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( log ` A ) e. CC )
20 17 19 mulcld 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( B x. ( log ` A ) ) e. CC )
21 absef 
 |-  ( ( B x. ( log ` A ) ) e. CC -> ( abs ` ( exp ` ( B x. ( log ` A ) ) ) ) = ( exp ` ( Re ` ( B x. ( log ` A ) ) ) ) )
22 20 21 syl 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( abs ` ( exp ` ( B x. ( log ` A ) ) ) ) = ( exp ` ( Re ` ( B x. ( log ` A ) ) ) ) )
23 16 19 remul2d 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( Re ` ( B x. ( log ` A ) ) ) = ( B x. ( Re ` ( log ` A ) ) ) )
24 relog 
 |-  ( ( A e. CC /\ A =/= 0 ) -> ( Re ` ( log ` A ) ) = ( log ` ( abs ` A ) ) )
25 24 adantlr 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( Re ` ( log ` A ) ) = ( log ` ( abs ` A ) ) )
26 25 oveq2d 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( B x. ( Re ` ( log ` A ) ) ) = ( B x. ( log ` ( abs ` A ) ) ) )
27 23 26 eqtrd 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( Re ` ( B x. ( log ` A ) ) ) = ( B x. ( log ` ( abs ` A ) ) ) )
28 27 fveq2d 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( exp ` ( Re ` ( B x. ( log ` A ) ) ) ) = ( exp ` ( B x. ( log ` ( abs ` A ) ) ) ) )
29 22 28 eqtrd 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( abs ` ( exp ` ( B x. ( log ` A ) ) ) ) = ( exp ` ( B x. ( log ` ( abs ` A ) ) ) ) )
30 simpll 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> A e. CC )
31 simpr 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> A =/= 0 )
32 cxpef 
 |-  ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
33 30 31 17 32 syl3anc 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
34 33 fveq2d 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( abs ` ( A ^c B ) ) = ( abs ` ( exp ` ( B x. ( log ` A ) ) ) ) )
35 30 abscld 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( abs ` A ) e. RR )
36 35 recnd 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( abs ` A ) e. CC )
37 abs00 
 |-  ( A e. CC -> ( ( abs ` A ) = 0 <-> A = 0 ) )
38 37 adantr 
 |-  ( ( A e. CC /\ B e. RR ) -> ( ( abs ` A ) = 0 <-> A = 0 ) )
39 38 necon3bid 
 |-  ( ( A e. CC /\ B e. RR ) -> ( ( abs ` A ) =/= 0 <-> A =/= 0 ) )
40 39 biimpar 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( abs ` A ) =/= 0 )
41 cxpef 
 |-  ( ( ( abs ` A ) e. CC /\ ( abs ` A ) =/= 0 /\ B e. CC ) -> ( ( abs ` A ) ^c B ) = ( exp ` ( B x. ( log ` ( abs ` A ) ) ) ) )
42 36 40 17 41 syl3anc 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( ( abs ` A ) ^c B ) = ( exp ` ( B x. ( log ` ( abs ` A ) ) ) ) )
43 29 34 42 3eqtr4d 
 |-  ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( abs ` ( A ^c B ) ) = ( ( abs ` A ) ^c B ) )
44 15 43 pm2.61dane 
 |-  ( ( A e. CC /\ B e. RR ) -> ( abs ` ( A ^c B ) ) = ( ( abs ` A ) ^c B ) )