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Theorem expaddzaplem 10287
Description: Lemma for expaddzap 10288. (Contributed by Jim Kingdon, 10-Jun-2020.)
Assertion
Ref Expression
expaddzaplem  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^
( M  +  N
) )  =  ( ( A ^ M
)  x.  ( A ^ N ) ) )

Proof of Theorem expaddzaplem
StepHypRef Expression
1 simp1l 988 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  A  e.  CC )
2 simp3 966 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  NN0 )
3 expcl 10262 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( A ^ N
)  e.  CC )
41, 2, 3syl2anc 406 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  CC )
5 simp2r 991 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  NN )
65nnnn0d 8984 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  NN0 )
7 expcl 10262 . . . 4  |-  ( ( A  e.  CC  /\  -u M  e.  NN0 )  ->  ( A ^ -u M
)  e.  CC )
81, 6, 7syl2anc 406 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u M )  e.  CC )
9 simp1r 989 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  A #  0 )
105nnzd 9126 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  ZZ )
11 expap0i 10276 . . . 4  |-  ( ( A  e.  CC  /\  A #  0  /\  -u M  e.  ZZ )  ->  ( A ^ -u M ) #  0 )
121, 9, 10, 11syl3anc 1199 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u M ) #  0 )
134, 8, 12divrecap2d 8517 . 2  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( ( A ^ N )  / 
( A ^ -u M
) )  =  ( ( 1  /  ( A ^ -u M ) )  x.  ( A ^ N ) ) )
14 simp2l 990 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  RR )
1514recnd 7758 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  CC )
1615negnegd 8028 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u -u M  =  M )
17 nnnegz 9011 . . . . . . . . . 10  |-  ( -u M  e.  NN  ->  -u -u M  e.  ZZ )
185, 17syl 14 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u -u M  e.  ZZ )
1916, 18eqeltrrd 2193 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  ZZ )
202nn0zd 9125 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  ZZ )
2119, 20zaddcld 9131 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( M  +  N )  e.  ZZ )
22 expclzap 10269 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  ( M  +  N )  e.  ZZ )  ->  ( A ^ ( M  +  N ) )  e.  CC )
231, 9, 21, 22syl3anc 1199 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^
( M  +  N
) )  e.  CC )
2423adantr 272 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  ( M  +  N )  e.  NN0 )  ->  ( A ^ ( M  +  N ) )  e.  CC )
258adantr 272 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  ( M  +  N )  e.  NN0 )  ->  ( A ^ -u M )  e.  CC )
2612adantr 272 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  ( M  +  N )  e.  NN0 )  ->  ( A ^ -u M ) #  0 )
2724, 25, 26divcanap4d 8519 . . . 4  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  ( M  +  N )  e.  NN0 )  ->  (
( ( A ^
( M  +  N
) )  x.  ( A ^ -u M ) )  /  ( A ^ -u M ) )  =  ( A ^ ( M  +  N ) ) )
281adantr 272 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  ( M  +  N )  e.  NN0 )  ->  A  e.  CC )
29 simpr 109 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  ( M  +  N )  e.  NN0 )  ->  ( M  +  N )  e.  NN0 )
306adantr 272 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  ( M  +  N )  e.  NN0 )  ->  -u M  e.  NN0 )
31 expadd 10286 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( M  +  N
)  e.  NN0  /\  -u M  e.  NN0 )  ->  ( A ^ (
( M  +  N
)  +  -u M
) )  =  ( ( A ^ ( M  +  N )
)  x.  ( A ^ -u M ) ) )
3228, 29, 30, 31syl3anc 1199 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  ( M  +  N )  e.  NN0 )  ->  ( A ^ ( ( M  +  N )  + 
-u M ) )  =  ( ( A ^ ( M  +  N ) )  x.  ( A ^ -u M
) ) )
3321zcnd 9128 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( M  +  N )  e.  CC )
3433, 15negsubd 8043 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( ( M  +  N )  + 
-u M )  =  ( ( M  +  N )  -  M
) )
352nn0cnd 8986 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  CC )
3615, 35pncan2d 8039 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( ( M  +  N )  -  M )  =  N )
3734, 36eqtrd 2148 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( ( M  +  N )  + 
-u M )  =  N )
3837adantr 272 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  ( M  +  N )  e.  NN0 )  ->  (
( M  +  N
)  +  -u M
)  =  N )
3938oveq2d 5756 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  ( M  +  N )  e.  NN0 )  ->  ( A ^ ( ( M  +  N )  + 
-u M ) )  =  ( A ^ N ) )
4032, 39eqtr3d 2150 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  ( M  +  N )  e.  NN0 )  ->  (
( A ^ ( M  +  N )
)  x.  ( A ^ -u M ) )  =  ( A ^ N ) )
4140oveq1d 5755 . . . 4  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  ( M  +  N )  e.  NN0 )  ->  (
( ( A ^
( M  +  N
) )  x.  ( A ^ -u M ) )  /  ( A ^ -u M ) )  =  ( ( A ^ N )  /  ( A ^ -u M ) ) )
4227, 41eqtr3d 2150 . . 3  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  ( M  +  N )  e.  NN0 )  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ N )  /  ( A ^ -u M ) ) )
431adantr 272 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  A  e.  CC )
449adantr 272 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  A #  0 )
4533adantr 272 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  ( M  +  N )  e.  CC )
46 simpr 109 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  -u ( M  +  N )  e.  NN0 )
47 expineg2 10253 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( ( M  +  N )  e.  CC  /\  -u ( M  +  N
)  e.  NN0 )
)  ->  ( A ^ ( M  +  N ) )  =  ( 1  /  ( A ^ -u ( M  +  N ) ) ) )
4843, 44, 45, 46, 47syl22anc 1200 . . . 4  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  ( A ^ ( M  +  N ) )  =  ( 1  /  ( A ^ -u ( M  +  N ) ) ) )
4921znegcld 9129 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u ( M  +  N )  e.  ZZ )
50 expclzap 10269 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A #  0  /\  -u ( M  +  N )  e.  ZZ )  ->  ( A ^ -u ( M  +  N ) )  e.  CC )
511, 9, 49, 50syl3anc 1199 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u ( M  +  N
) )  e.  CC )
5251adantr 272 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  ( A ^ -u ( M  +  N ) )  e.  CC )
534adantr 272 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  ( A ^ N )  e.  CC )
54 expap0i 10276 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  ZZ )  ->  ( A ^ N ) #  0 )
551, 9, 20, 54syl3anc 1199 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ N ) #  0 )
5655adantr 272 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  ( A ^ N ) #  0 )
5752, 53, 56divcanap4d 8519 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  (
( ( A ^ -u ( M  +  N
) )  x.  ( A ^ N ) )  /  ( A ^ N ) )  =  ( A ^ -u ( M  +  N )
) )
582adantr 272 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  N  e.  NN0 )
59 expadd 10286 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  -u ( M  +  N
)  e.  NN0  /\  N  e.  NN0 )  -> 
( A ^ ( -u ( M  +  N
)  +  N ) )  =  ( ( A ^ -u ( M  +  N )
)  x.  ( A ^ N ) ) )
6043, 46, 58, 59syl3anc 1199 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  ( A ^ ( -u ( M  +  N )  +  N ) )  =  ( ( A ^ -u ( M  +  N
) )  x.  ( A ^ N ) ) )
6115, 35negdi2d 8051 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u ( M  +  N )  =  (
-u M  -  N
) )
6261oveq1d 5755 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( -u ( M  +  N )  +  N )  =  ( ( -u M  -  N )  +  N
) )
6315negcld 8024 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  CC )
6463, 35npcand 8041 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( ( -u M  -  N )  +  N )  =  -u M )
6562, 64eqtrd 2148 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( -u ( M  +  N )  +  N )  =  -u M )
6665adantr 272 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  ( -u ( M  +  N
)  +  N )  =  -u M )
6766oveq2d 5756 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  ( A ^ ( -u ( M  +  N )  +  N ) )  =  ( A ^ -u M
) )
6860, 67eqtr3d 2150 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  (
( A ^ -u ( M  +  N )
)  x.  ( A ^ N ) )  =  ( A ^ -u M ) )
6968oveq1d 5755 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  (
( ( A ^ -u ( M  +  N
) )  x.  ( A ^ N ) )  /  ( A ^ N ) )  =  ( ( A ^ -u M )  /  ( A ^ N ) ) )
7057, 69eqtr3d 2150 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  ( A ^ -u ( M  +  N ) )  =  ( ( A ^ -u M )  /  ( A ^ N ) ) )
7170oveq2d 5756 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  (
1  /  ( A ^ -u ( M  +  N ) ) )  =  ( 1  /  ( ( A ^ -u M )  /  ( A ^ N ) ) ) )
728, 4, 12, 55recdivapd 8530 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( 1  / 
( ( A ^ -u M )  /  ( A ^ N ) ) )  =  ( ( A ^ N )  /  ( A ^ -u M ) ) )
7372adantr 272 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  (
1  /  ( ( A ^ -u M
)  /  ( A ^ N ) ) )  =  ( ( A ^ N )  /  ( A ^ -u M ) ) )
7471, 73eqtrd 2148 . . . 4  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  (
1  /  ( A ^ -u ( M  +  N ) ) )  =  ( ( A ^ N )  /  ( A ^ -u M ) ) )
7548, 74eqtrd 2148 . . 3  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ N )  /  ( A ^ -u M ) ) )
76 elznn0 9023 . . . . 5  |-  ( ( M  +  N )  e.  ZZ  <->  ( ( M  +  N )  e.  RR  /\  ( ( M  +  N )  e.  NN0  \/  -u ( M  +  N )  e.  NN0 ) ) )
7776simprbi 271 . . . 4  |-  ( ( M  +  N )  e.  ZZ  ->  (
( M  +  N
)  e.  NN0  \/  -u ( M  +  N
)  e.  NN0 )
)
7821, 77syl 14 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( ( M  +  N )  e. 
NN0  \/  -u ( M  +  N )  e. 
NN0 ) )
7942, 75, 78mpjaodan 770 . 2  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^
( M  +  N
) )  =  ( ( A ^ N
)  /  ( A ^ -u M ) ) )
80 expineg2 10253 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  CC  /\  -u M  e.  NN0 ) )  ->  ( A ^ M )  =  ( 1  /  ( A ^ -u M ) ) )
811, 9, 15, 6, 80syl22anc 1200 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ M )  =  ( 1  /  ( A ^ -u M ) ) )
8281oveq1d 5755 . 2  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( ( A ^ M )  x.  ( A ^ N
) )  =  ( ( 1  /  ( A ^ -u M ) )  x.  ( A ^ N ) ) )
8313, 79, 823eqtr4d 2158 1  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^
( M  +  N
) )  =  ( ( A ^ M
)  x.  ( A ^ N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 680    /\ w3a 945    = wceq 1314    e. wcel 1463   class class class wbr 3897  (class class class)co 5740   CCcc 7582   RRcr 7583   0cc0 7584   1c1 7585    + caddc 7587    x. cmul 7589    - cmin 7897   -ucneg 7898   # cap 8306    / cdiv 8395   NNcn 8680   NN0cn0 8931   ZZcz 9008   ^cexp 10243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8396  df-inn 8681  df-n0 8932  df-z 9009  df-uz 9279  df-seqfrec 10170  df-exp 10244
This theorem is referenced by:  expaddzap  10288
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