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Theorem dvexp3 23741
Description: Derivative of an exponential of integer exponent. (Contributed by Mario Carneiro, 26-Feb-2015.)
Assertion
Ref Expression
dvexp3 |- ( N e. ZZ -> ( CC _D ( x e. ( CC \ { 0 } ) |-> ( x ^ N ) ) ) = ( x e. ( CC \ { 0 } ) |-> ( N x. ( x ^ ( N - 1 ) ) ) ) )
Distinct variable group:   x, N
Proof of Theorem dvexp3
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 elznn0nn 11391 . 2 |- ( N e. ZZ <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) )
2 cnelprrecn 10029 . . . . . 6 |- CC e. { RR , CC }
32a1i 11 . . . . 5 |- ( N e. NN0 -> CC e. { RR , CC } )
4 expcl 12878 . . . . . 6 |- ( ( x e. CC /\ N e. NN0 ) -> ( x ^ N ) e. CC )
54ancoms 469 . . . . 5 |- ( ( N e. NN0 /\ x e. CC ) -> ( x ^ N ) e. CC )
6 c0ex 10034 . . . . . . 7 |- 0 e. _V
7 ovex 6678 . . . . . . 7 |- ( N x. ( x ^ ( N - 1 ) ) ) e. _V
86, 7ifex 4156 . . . . . 6 |- if ( N = 0 , 0 , ( N x. ( x ^ ( N - 1 ) ) ) ) e. _V
98a1i 11 . . . . 5 |- ( ( N e. NN0 /\ x e. CC ) -> if ( N = 0 , 0 , ( N x. ( x ^ ( N - 1 ) ) ) ) e. _V )
10 dvexp2 23717 . . . . 5 |- ( N e. NN0 -> ( CC _D ( x e. CC |-> ( x ^ N ) ) ) = ( x e. CC |-> if ( N = 0 , 0 , ( N x. ( x ^ ( N - 1 ) ) ) ) ) )
11 difssd 3738 . . . . 5 |- ( N e. NN0 -> ( CC \ { 0 } ) C_ CC )
12 eqid 2622 . . . . . . . 8 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
1312cnfldtop 22587 . . . . . . 7 |- ( TopOpen ` ℂfld ) e. Top
1412cnfldtopon 22586 . . . . . . . . 9 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
1514toponunii 20721 . . . . . . . 8 |- CC = U. ( TopOpen ` ℂfld )
1615restid 16094 . . . . . . 7 |- ( ( TopOpen ` ℂfld ) e. Top -> ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld ) )
1713, 16ax-mp 5 . . . . . 6 |- ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld )
1817eqcomi 2631 . . . . 5 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
1912cnfldhaus 22588 . . . . . . . 8 |- ( TopOpen ` ℂfld ) e. Haus
20 0cn 10032 . . . . . . . 8 |- 0 e. CC
2115sncld 21175 . . . . . . . 8 |- ( ( ( TopOpen ` ℂfld ) e. Haus /\ 0 e. CC ) -> { 0 } e. ( Clsd ` ( TopOpen ` ℂfld ) ) )
2219, 20, 21mp2an 708 . . . . . . 7 |- { 0 } e. ( Clsd ` ( TopOpen ` ℂfld ) )
2315cldopn 20835 . . . . . . 7 |- ( { 0 } e. ( Clsd ` ( TopOpen ` ℂfld ) ) -> ( CC \ { 0 } ) e. ( TopOpen ` ℂfld ) )
2422, 23ax-mp 5 . . . . . 6 |- ( CC \ { 0 } ) e. ( TopOpen ` ℂfld )
2524a1i 11 . . . . 5 |- ( N e. NN0 -> ( CC \ { 0 } ) e. ( TopOpen ` ℂfld ) )
263, 5, 9, 10, 11, 18, 12, 25dvmptres 23726 . . . 4 |- ( N e. NN0 -> ( CC _D ( x e. ( CC \ { 0 } ) |-> ( x ^ N ) ) ) = ( x e. ( CC \ { 0 } ) |-> if ( N = 0 , 0 , ( N x. ( x ^ ( N - 1 ) ) ) ) ) )
27 ifid 4125 . . . . . 6 |- if ( N = 0 , ( N x. ( x ^ ( N - 1 ) ) ) , ( N x. ( x ^ ( N - 1 ) ) ) ) = ( N x. ( x ^ ( N - 1 ) ) )
28 id 22 . . . . . . . . 9 |- ( N = 0 -> N = 0 )
29 oveq1 6657 . . . . . . . . . 10 |- ( N = 0 -> ( N - 1 ) = ( 0 - 1 ) )
3029oveq2d 6666 . . . . . . . . 9 |- ( N = 0 -> ( x ^ ( N - 1 ) ) = ( x ^ ( 0 - 1 ) ) )
3128, 30oveq12d 6668 . . . . . . . 8 |- ( N = 0 -> ( N x. ( x ^ ( N - 1 ) ) ) = ( 0 x. ( x ^ ( 0 - 1 ) ) ) )
32 eldifsn 4317 . . . . . . . . . . 11 |- ( x e. ( CC \ { 0 } ) <-> ( x e. CC /\ x =/= 0 ) )
33 0z 11388 . . . . . . . . . . . . 13 |- 0 e. ZZ
34 peano2zm 11420 . . . . . . . . . . . . 13 |- ( 0 e. ZZ -> ( 0 - 1 ) e. ZZ )
3533, 34ax-mp 5 . . . . . . . . . . . 12 |- ( 0 - 1 ) e. ZZ
36 expclz 12885 . . . . . . . . . . . 12 |- ( ( x e. CC /\ x =/= 0 /\ ( 0 - 1 ) e. ZZ ) -> ( x ^ ( 0 - 1 ) ) e. CC )
3735, 36mp3an3 1413 . . . . . . . . . . 11 |- ( ( x e. CC /\ x =/= 0 ) -> ( x ^ ( 0 - 1 ) ) e. CC )
3832, 37sylbi 207 . . . . . . . . . 10 |- ( x e. ( CC \ { 0 } ) -> ( x ^ ( 0 - 1 ) ) e. CC )
3938adantl 482 . . . . . . . . 9 |- ( ( N e. NN0 /\ x e. ( CC \ { 0 } ) ) -> ( x ^ ( 0 - 1 ) ) e. CC )
4039mul02d 10234 . . . . . . . 8 |- ( ( N e. NN0 /\ x e. ( CC \ { 0 } ) ) -> ( 0 x. ( x ^ ( 0 - 1 ) ) ) = 0 )
4131, 40sylan9eqr 2678 . . . . . . 7 |- ( ( ( N e. NN0 /\ x e. ( CC \ { 0 } ) ) /\ N = 0 ) -> ( N x. ( x ^ ( N - 1 ) ) ) = 0 )
4241ifeq1da 4116 . . . . . 6 |- ( ( N e. NN0 /\ x e. ( CC \ { 0 } ) ) -> if ( N = 0 , ( N x. ( x ^ ( N - 1 ) ) ) , ( N x. ( x ^ ( N - 1 ) ) ) ) = if ( N = 0 , 0 , ( N x. ( x ^ ( N - 1 ) ) ) ) )
4327, 42syl5eqr 2670 . . . . 5 |- ( ( N e. NN0 /\ x e. ( CC \ { 0 } ) ) -> ( N x. ( x ^ ( N - 1 ) ) ) = if ( N = 0 , 0 , ( N x. ( x ^ ( N - 1 ) ) ) ) )
4443mpteq2dva 4744 . . . 4 |- ( N e. NN0 -> ( x e. ( CC \ { 0 } ) |-> ( N x. ( x ^ ( N - 1 ) ) ) ) = ( x e. ( CC \ { 0 } ) |-> if ( N = 0 , 0 , ( N x. ( x ^ ( N - 1 ) ) ) ) ) )
4526, 44eqtr4d 2659 . . 3 |- ( N e. NN0 -> ( CC _D ( x e. ( CC \ { 0 } ) |-> ( x ^ N ) ) ) = ( x e. ( CC \ { 0 } ) |-> ( N x. ( x ^ ( N - 1 ) ) ) ) )
46 eldifi 3732 . . . . . . . 8 |- ( x e. ( CC \ { 0 } ) -> x e. CC )
4746adantl 482 . . . . . . 7 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> x e. CC )
48 simpll 790 . . . . . . . 8 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> N e. RR )
4948recnd 10068 . . . . . . 7 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> N e. CC )
50 nnnn0 11299 . . . . . . . 8 |- ( -u N e. NN -> -u N e. NN0 )
5150ad2antlr 763 . . . . . . 7 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> -u N e. NN0 )
52 expneg2 12869 . . . . . . 7 |- ( ( x e. CC /\ N e. CC /\ -u N e. NN0 ) -> ( x ^ N ) = ( 1 / ( x ^ -u N ) ) )
5347, 49, 51, 52syl3anc 1326 . . . . . 6 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( x ^ N ) = ( 1 / ( x ^ -u N ) ) )
5453mpteq2dva 4744 . . . . 5 |- ( ( N e. RR /\ -u N e. NN ) -> ( x e. ( CC \ { 0 } ) |-> ( x ^ N ) ) = ( x e. ( CC \ { 0 } ) |-> ( 1 / ( x ^ -u N ) ) ) )
5554oveq2d 6666 . . . 4 |- ( ( N e. RR /\ -u N e. NN ) -> ( CC _D ( x e. ( CC \ { 0 } ) |-> ( x ^ N ) ) ) = ( CC _D ( x e. ( CC \ { 0 } ) |-> ( 1 / ( x ^ -u N ) ) ) ) )
562a1i 11 . . . . 5 |- ( ( N e. RR /\ -u N e. NN ) -> CC e. { RR , CC } )
57 eldifsni 4320 . . . . . . . 8 |- ( x e. ( CC \ { 0 } ) -> x =/= 0 )
5857adantl 482 . . . . . . 7 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> x =/= 0 )
59 nnz 11399 . . . . . . . 8 |- ( -u N e. NN -> -u N e. ZZ )
6059ad2antlr 763 . . . . . . 7 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> -u N e. ZZ )
6147, 58, 60expclzd 13013 . . . . . 6 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( x ^ -u N ) e. CC )
6247, 58, 60expne0d 13014 . . . . . 6 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( x ^ -u N ) =/= 0 )
63 eldifsn 4317 . . . . . 6 |- ( ( x ^ -u N ) e. ( CC \ { 0 } ) <-> ( ( x ^ -u N ) e. CC /\ ( x ^ -u N ) =/= 0 ) )
6461, 62, 63sylanbrc 698 . . . . 5 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( x ^ -u N ) e. ( CC \ { 0 } ) )
65 ovex 6678 . . . . . 6 |- ( -u N x. ( x ^ ( -u N - 1 ) ) ) e. _V
6665a1i 11 . . . . 5 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( -u N x. ( x ^ ( -u N - 1 ) ) ) e. _V )
67 simpr 477 . . . . . . 7 |- ( ( ( N e. RR /\ -u N e. NN ) /\ y e. ( CC \ { 0 } ) ) -> y e. ( CC \ { 0 } ) )
68 eldifsn 4317 . . . . . . 7 |- ( y e. ( CC \ { 0 } ) <-> ( y e. CC /\ y =/= 0 ) )
6967, 68sylib 208 . . . . . 6 |- ( ( ( N e. RR /\ -u N e. NN ) /\ y e. ( CC \ { 0 } ) ) -> ( y e. CC /\ y =/= 0 ) )
70 reccl 10692 . . . . . 6 |- ( ( y e. CC /\ y =/= 0 ) -> ( 1 / y ) e. CC )
7169, 70syl 17 . . . . 5 |- ( ( ( N e. RR /\ -u N e. NN ) /\ y e. ( CC \ { 0 } ) ) -> ( 1 / y ) e. CC )
72 negex 10279 . . . . . 6 |- -u ( 1 / ( y ^ 2 ) ) e. _V
7372a1i 11 . . . . 5 |- ( ( ( N e. RR /\ -u N e. NN ) /\ y e. ( CC \ { 0 } ) ) -> -u ( 1 / ( y ^ 2 ) ) e. _V )
74 simpr 477 . . . . . . 7 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. CC ) -> x e. CC )
7550ad2antlr 763 . . . . . . 7 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. CC ) -> -u N e. NN0 )
7674, 75expcld 13008 . . . . . 6 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. CC ) -> ( x ^ -u N ) e. CC )
7765a1i 11 . . . . . 6 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. CC ) -> ( -u N x. ( x ^ ( -u N - 1 ) ) ) e. _V )
78 dvexp 23716 . . . . . . 7 |- ( -u N e. NN -> ( CC _D ( x e. CC |-> ( x ^ -u N ) ) ) = ( x e. CC |-> ( -u N x. ( x ^ ( -u N - 1 ) ) ) ) )
7978adantl 482 . . . . . 6 |- ( ( N e. RR /\ -u N e. NN ) -> ( CC _D ( x e. CC |-> ( x ^ -u N ) ) ) = ( x e. CC |-> ( -u N x. ( x ^ ( -u N - 1 ) ) ) ) )
80 difssd 3738 . . . . . 6 |- ( ( N e. RR /\ -u N e. NN ) -> ( CC \ { 0 } ) C_ CC )
8124a1i 11 . . . . . 6 |- ( ( N e. RR /\ -u N e. NN ) -> ( CC \ { 0 } ) e. ( TopOpen ` ℂfld ) )
8256, 76, 77, 79, 80, 18, 12, 81dvmptres 23726 . . . . 5 |- ( ( N e. RR /\ -u N e. NN ) -> ( CC _D ( x e. ( CC \ { 0 } ) |-> ( x ^ -u N ) ) ) = ( x e. ( CC \ { 0 } ) |-> ( -u N x. ( x ^ ( -u N - 1 ) ) ) ) )
83 ax-1cn 9994 . . . . . 6 |- 1 e. CC
84 dvrec 23718 . . . . . 6 |- ( 1 e. CC -> ( CC _D ( y e. ( CC \ { 0 } ) |-> ( 1 / y ) ) ) = ( y e. ( CC \ { 0 } ) |-> -u ( 1 / ( y ^ 2 ) ) ) )
8583, 84mp1i 13 . . . . 5 |- ( ( N e. RR /\ -u N e. NN ) -> ( CC _D ( y e. ( CC \ { 0 } ) |-> ( 1 / y ) ) ) = ( y e. ( CC \ { 0 } ) |-> -u ( 1 / ( y ^ 2 ) ) ) )
86 oveq2 6658 . . . . 5 |- ( y = ( x ^ -u N ) -> ( 1 / y ) = ( 1 / ( x ^ -u N ) ) )
87 oveq1 6657 . . . . . . 7 |- ( y = ( x ^ -u N ) -> ( y ^ 2 ) = ( ( x ^ -u N ) ^ 2 ) )
8887oveq2d 6666 . . . . . 6 |- ( y = ( x ^ -u N ) -> ( 1 / ( y ^ 2 ) ) = ( 1 / ( ( x ^ -u N ) ^ 2 ) ) )
8988negeqd 10275 . . . . 5 |- ( y = ( x ^ -u N ) -> -u ( 1 / ( y ^ 2 ) ) = -u ( 1 / ( ( x ^ -u N ) ^ 2 ) ) )
9056, 56, 64, 66, 71, 73, 82, 85, 86, 89dvmptco 23735 . . . 4 |- ( ( N e. RR /\ -u N e. NN ) -> ( CC _D ( x e. ( CC \ { 0 } ) |-> ( 1 / ( x ^ -u N ) ) ) ) = ( x e. ( CC \ { 0 } ) |-> ( -u ( 1 / ( ( x ^ -u N ) ^ 2 ) ) x. ( -u N x. ( x ^ ( -u N - 1 ) ) ) ) ) )
91 2z 11409 . . . . . . . . . . . 12 |- 2 e. ZZ
9291a1i 11 . . . . . . . . . . 11 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> 2 e. ZZ )
93 expmulz 12906 . . . . . . . . . . 11 |- ( ( ( x e. CC /\ x =/= 0 ) /\ ( -u N e. ZZ /\ 2 e. ZZ ) ) -> ( x ^ ( -u N x. 2 ) ) = ( ( x ^ -u N ) ^ 2 ) )
9447, 58, 60, 92, 93syl22anc 1327 . . . . . . . . . 10 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( x ^ ( -u N x. 2 ) ) = ( ( x ^ -u N ) ^ 2 ) )
9594eqcomd 2628 . . . . . . . . 9 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( ( x ^ -u N ) ^ 2 ) = ( x ^ ( -u N x. 2 ) ) )
9695oveq2d 6666 . . . . . . . 8 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( 1 / ( ( x ^ -u N ) ^ 2 ) ) = ( 1 / ( x ^ ( -u N x. 2 ) ) ) )
9796negeqd 10275 . . . . . . 7 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> -u ( 1 / ( ( x ^ -u N ) ^ 2 ) ) = -u ( 1 / ( x ^ ( -u N x. 2 ) ) ) )
98 peano2zm 11420 . . . . . . . . . 10 |- ( -u N e. ZZ -> ( -u N - 1 ) e. ZZ )
9960, 98syl 17 . . . . . . . . 9 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( -u N - 1 ) e. ZZ )
10047, 58, 99expclzd 13013 . . . . . . . 8 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( x ^ ( -u N - 1 ) ) e. CC )
10149, 100mulneg1d 10483 . . . . . . 7 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( -u N x. ( x ^ ( -u N - 1 ) ) ) = -u ( N x. ( x ^ ( -u N - 1 ) ) ) )
10297, 101oveq12d 6668 . . . . . 6 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( -u ( 1 / ( ( x ^ -u N ) ^ 2 ) ) x. ( -u N x. ( x ^ ( -u N - 1 ) ) ) ) = ( -u ( 1 / ( x ^ ( -u N x. 2 ) ) ) x. -u ( N x. ( x ^ ( -u N - 1 ) ) ) ) )
103 zmulcl 11426 . . . . . . . . . 10 |- ( ( -u N e. ZZ /\ 2 e. ZZ ) -> ( -u N x. 2 ) e. ZZ )
10460, 91, 103sylancl 694 . . . . . . . . 9 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( -u N x. 2 ) e. ZZ )
10547, 58, 104expclzd 13013 . . . . . . . 8 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( x ^ ( -u N x. 2 ) ) e. CC )
10647, 58, 104expne0d 13014 . . . . . . . 8 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( x ^ ( -u N x. 2 ) ) =/= 0 )
107105, 106reccld 10794 . . . . . . 7 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( 1 / ( x ^ ( -u N x. 2 ) ) ) e. CC )
10849, 100mulcld 10060 . . . . . . 7 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( N x. ( x ^ ( -u N - 1 ) ) ) e. CC )
109107, 108mul2negd 10485 . . . . . 6 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( -u ( 1 / ( x ^ ( -u N x. 2 ) ) ) x. -u ( N x. ( x ^ ( -u N - 1 ) ) ) ) = ( ( 1 / ( x ^ ( -u N x. 2 ) ) ) x. ( N x. ( x ^ ( -u N - 1 ) ) ) ) )
110107, 49, 100mul12d 10245 . . . . . . 7 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( ( 1 / ( x ^ ( -u N x. 2 ) ) ) x. ( N x. ( x ^ ( -u N - 1 ) ) ) ) = ( N x. ( ( 1 / ( x ^ ( -u N x. 2 ) ) ) x. ( x ^ ( -u N - 1 ) ) ) ) )
11147, 58, 104, 99expsubd 13019 . . . . . . . . 9 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( x ^ ( ( -u N - 1 ) - ( -u N x. 2 ) ) ) = ( ( x ^ ( -u N - 1 ) ) / ( x ^ ( -u N x. 2 ) ) ) )
112 nncn 11028 . . . . . . . . . . . . 13 |- ( -u N e. NN -> -u N e. CC )
113112ad2antlr 763 . . . . . . . . . . . 12 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> -u N e. CC )
11483a1i 11 . . . . . . . . . . . 12 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> 1 e. CC )
115104zcnd 11483 . . . . . . . . . . . 12 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( -u N x. 2 ) e. CC )
116113, 114, 115sub32d 10424 . . . . . . . . . . 11 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( ( -u N - 1 ) - ( -u N x. 2 ) ) = ( ( -u N - ( -u N x. 2 ) ) - 1 ) )
117113times2d 11276 . . . . . . . . . . . . . . 15 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( -u N x. 2 ) = ( -u N + -u N ) )
118113, 49negsubd 10398 . . . . . . . . . . . . . . 15 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( -u N + -u N ) = ( -u N - N ) )
119117, 118eqtrd 2656 . . . . . . . . . . . . . 14 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( -u N x. 2 ) = ( -u N - N ) )
120119oveq2d 6666 . . . . . . . . . . . . 13 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( -u N - ( -u N x. 2 ) ) = ( -u N - ( -u N - N ) ) )
121113, 49nncand 10397 . . . . . . . . . . . . 13 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( -u N - ( -u N - N ) ) = N )
122120, 121eqtrd 2656 . . . . . . . . . . . 12 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( -u N - ( -u N x. 2 ) ) = N )
123122oveq1d 6665 . . . . . . . . . . 11 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( ( -u N - ( -u N x. 2 ) ) - 1 ) = ( N - 1 ) )
124116, 123eqtrd 2656 . . . . . . . . . 10 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( ( -u N - 1 ) - ( -u N x. 2 ) ) = ( N - 1 ) )
125124oveq2d 6666 . . . . . . . . 9 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( x ^ ( ( -u N - 1 ) - ( -u N x. 2 ) ) ) = ( x ^ ( N - 1 ) ) )
126100, 105, 106divrec2d 10805 . . . . . . . . 9 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( ( x ^ ( -u N - 1 ) ) / ( x ^ ( -u N x. 2 ) ) ) = ( ( 1 / ( x ^ ( -u N x. 2 ) ) ) x. ( x ^ ( -u N - 1 ) ) ) )
127111, 125, 1263eqtr3rd 2665 . . . . . . . 8 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( ( 1 / ( x ^ ( -u N x. 2 ) ) ) x. ( x ^ ( -u N - 1 ) ) ) = ( x ^ ( N - 1 ) ) )
128127oveq2d 6666 . . . . . . 7 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( N x. ( ( 1 / ( x ^ ( -u N x. 2 ) ) ) x. ( x ^ ( -u N - 1 ) ) ) ) = ( N x. ( x ^ ( N - 1 ) ) ) )
129110, 128eqtrd 2656 . . . . . 6 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( ( 1 / ( x ^ ( -u N x. 2 ) ) ) x. ( N x. ( x ^ ( -u N - 1 ) ) ) ) = ( N x. ( x ^ ( N - 1 ) ) ) )
130102, 109, 1293eqtrd 2660 . . . . 5 |- ( ( ( N e. RR /\ -u N e. NN ) /\ x e. ( CC \ { 0 } ) ) -> ( -u ( 1 / ( ( x ^ -u N ) ^ 2 ) ) x. ( -u N x. ( x ^ ( -u N - 1 ) ) ) ) = ( N x. ( x ^ ( N - 1 ) ) ) )
131130mpteq2dva 4744 . . . 4 |- ( ( N e. RR /\ -u N e. NN ) -> ( x e. ( CC \ { 0 } ) |-> ( -u ( 1 / ( ( x ^ -u N ) ^ 2 ) ) x. ( -u N x. ( x ^ ( -u N - 1 ) ) ) ) ) = ( x e. ( CC \ { 0 } ) |-> ( N x. ( x ^ ( N - 1 ) ) ) ) )
13255, 90, 1313eqtrd 2660 . . 3 |- ( ( N e. RR /\ -u N e. NN ) -> ( CC _D ( x e. ( CC \ { 0 } ) |-> ( x ^ N ) ) ) = ( x e. ( CC \ { 0 } ) |-> ( N x. ( x ^ ( N - 1 ) ) ) ) )
13345, 132jaoi 394 . 2 |- ( ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) -> ( CC _D ( x e. ( CC \ { 0 } ) |-> ( x ^ N ) ) ) = ( x e. ( CC \ { 0 } ) |-> ( N x. ( x ^ ( N - 1 ) ) ) ) )
1341, 133sylbi 207 1 |- ( N e. ZZ -> ( CC _D ( x e. ( CC \ { 0 } ) |-> ( x ^ N ) ) ) = ( x e. ( CC \ { 0 } ) |-> ( N x. ( x ^ ( N - 1 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   ifcif 4086   {csn 4177   {cpr 4179   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ^cexp 12860   ↾t crest 16081   TopOpenctopn 16082  ℂfldccnfld 19746   Topctop 20698   Clsdccld 20820   Hauscha 21112   _D cdv 23627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-t1 21118  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631
This theorem is referenced by: (None)
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