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Theorem expaddzaplem 9994
Description: Lemma for expaddzap 9995. (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 967 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  A  e.  CC )
2 simp3 945 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  NN0 )
3 expcl 9969 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( A ^ N
)  e.  CC )
41, 2, 3syl2anc 403 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  CC )
5 simp2r 970 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  NN )
65nnnn0d 8724 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  NN0 )
7 expcl 9969 . . . 4  |-  ( ( A  e.  CC  /\  -u M  e.  NN0 )  ->  ( A ^ -u M
)  e.  CC )
81, 6, 7syl2anc 403 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u M )  e.  CC )
9 simp1r 968 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  A #  0 )
105nnzd 8865 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  ZZ )
11 expap0i 9983 . . . 4  |-  ( ( A  e.  CC  /\  A #  0  /\  -u M  e.  ZZ )  ->  ( A ^ -u M ) #  0 )
121, 9, 10, 11syl3anc 1174 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u M ) #  0 )
134, 8, 12divrecap2d 8259 . 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 969 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  RR )
1514recnd 7514 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  CC )
1615negnegd 7782 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u -u M  =  M )
17 nnnegz 8751 . . . . . . . . . 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 2165 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  ZZ )
202nn0zd 8864 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  ZZ )
2119, 20zaddcld 8870 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( M  +  N )  e.  ZZ )
22 expclzap 9976 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  ( M  +  N )  e.  ZZ )  ->  ( A ^ ( M  +  N ) )  e.  CC )
231, 9, 21, 22syl3anc 1174 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^
( M  +  N
) )  e.  CC )
2423adantr 270 . . . . 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 270 . . . . 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 270 . . . . 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 8261 . . . 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 270 . . . . . . 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 108 . . . . . . 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 270 . . . . . . 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 9993 . . . . . . 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 1174 . . . . . 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 8867 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( M  +  N )  e.  CC )
3433, 15negsubd 7797 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( ( M  +  N )  + 
-u M )  =  ( ( M  +  N )  -  M
) )
352nn0cnd 8726 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  CC )
3615, 35pncan2d 7793 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( ( M  +  N )  -  M )  =  N )
3734, 36eqtrd 2120 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( ( M  +  N )  + 
-u M )  =  N )
3837adantr 270 . . . . . . 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 5668 . . . . . 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 2122 . . . . 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 5667 . . . 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 2122 . . 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 270 . . . . 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 270 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  A #  0 )
4533adantr 270 . . . . 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 108 . . . . 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 9960 . . . . 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 1175 . . . 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 8868 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u ( M  +  N )  e.  ZZ )
50 expclzap 9976 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A #  0  /\  -u ( M  +  N )  e.  ZZ )  ->  ( A ^ -u ( M  +  N ) )  e.  CC )
511, 9, 49, 50syl3anc 1174 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u ( M  +  N
) )  e.  CC )
5251adantr 270 . . . . . . . 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 270 . . . . . . . 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 9983 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  ZZ )  ->  ( A ^ N ) #  0 )
551, 9, 20, 54syl3anc 1174 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ N ) #  0 )
5655adantr 270 . . . . . . . 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 8261 . . . . . . 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 270 . . . . . . . . . 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 9993 . . . . . . . . . 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 1174 . . . . . . . . 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 7805 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u ( M  +  N )  =  (
-u M  -  N
) )
6261oveq1d 5667 . . . . . . . . . . . 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 7778 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  CC )
6463, 35npcand 7795 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( ( -u M  -  N )  +  N )  =  -u M )
6562, 64eqtrd 2120 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( -u ( M  +  N )  +  N )  =  -u M )
6665adantr 270 . . . . . . . . . 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 5668 . . . . . . . . 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 2122 . . . . . . . 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 5667 . . . . . . 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 2122 . . . . . 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 5668 . . . . 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 8272 . . . . . 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 270 . . . . 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 2120 . . . 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 2120 . . 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 8763 . . . . 5  |-  ( ( M  +  N )  e.  ZZ  <->  ( ( M  +  N )  e.  RR  /\  ( ( M  +  N )  e.  NN0  \/  -u ( M  +  N )  e.  NN0 ) ) )
7776simprbi 269 . . . 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 747 . 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 9960 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  CC  /\  -u M  e.  NN0 ) )  ->  ( A ^ M )  =  ( 1  /  ( A ^ -u M ) ) )
811, 9, 15, 6, 80syl22anc 1175 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ M )  =  ( 1  /  ( A ^ -u M ) ) )
8281oveq1d 5667 . 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 2130 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 102    \/ wo 664    /\ w3a 924    = wceq 1289    e. wcel 1438   class class class wbr 3845  (class class class)co 5652   CCcc 7346   RRcr 7347   0cc0 7348   1c1 7349    + caddc 7351    x. cmul 7353    - cmin 7651   -ucneg 7652   # cap 8056    / cdiv 8137   NNcn 8420   NN0cn0 8671   ZZcz 8748   ^cexp 9950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-mulrcl 7442  ax-addcom 7443  ax-mulcom 7444  ax-addass 7445  ax-mulass 7446  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-1rid 7450  ax-0id 7451  ax-rnegex 7452  ax-precex 7453  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459  ax-pre-mulgt0 7460  ax-pre-mulext 7461
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-reap 8050  df-ap 8057  df-div 8138  df-inn 8421  df-n0 8672  df-z 8749  df-uz 9018  df-iseq 9849  df-seq3 9850  df-exp 9951
This theorem is referenced by:  expaddzap  9995
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