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Theorem expaddzaplem 10519
Description: Lemma for expaddzap 10520. (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 1016 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  A  e.  CC )
2 simp3 994 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  NN0 )
3 expcl 10494 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( A ^ N
)  e.  CC )
41, 2, 3syl2anc 409 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  CC )
5 simp2r 1019 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  NN )
65nnnn0d 9188 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  NN0 )
7 expcl 10494 . . . 4  |-  ( ( A  e.  CC  /\  -u M  e.  NN0 )  ->  ( A ^ -u M
)  e.  CC )
81, 6, 7syl2anc 409 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u M )  e.  CC )
9 simp1r 1017 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  A #  0 )
105nnzd 9333 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  ZZ )
11 expap0i 10508 . . . 4  |-  ( ( A  e.  CC  /\  A #  0  /\  -u M  e.  ZZ )  ->  ( A ^ -u M ) #  0 )
121, 9, 10, 11syl3anc 1233 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u M ) #  0 )
134, 8, 12divrecap2d 8711 . 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 1018 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  RR )
1514recnd 7948 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  CC )
1615negnegd 8221 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u -u M  =  M )
17 nnnegz 9215 . . . . . . . . . 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 2248 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  ZZ )
202nn0zd 9332 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  ZZ )
2119, 20zaddcld 9338 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( M  +  N )  e.  ZZ )
22 expclzap 10501 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  ( M  +  N )  e.  ZZ )  ->  ( A ^ ( M  +  N ) )  e.  CC )
231, 9, 21, 22syl3anc 1233 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^
( M  +  N
) )  e.  CC )
2423adantr 274 . . . . 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 274 . . . . 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 274 . . . . 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 8713 . . . 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 274 . . . . . . 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 274 . . . . . . 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 10518 . . . . . . 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 1233 . . . . . 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 9335 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( M  +  N )  e.  CC )
3433, 15negsubd 8236 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( ( M  +  N )  + 
-u M )  =  ( ( M  +  N )  -  M
) )
352nn0cnd 9190 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  CC )
3615, 35pncan2d 8232 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( ( M  +  N )  -  M )  =  N )
3734, 36eqtrd 2203 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( ( M  +  N )  + 
-u M )  =  N )
3837adantr 274 . . . . . . 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 5869 . . . . . 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 2205 . . . . 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 5868 . . . 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 2205 . . 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 274 . . . . 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 274 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  A #  0 )
4533adantr 274 . . . . 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 10485 . . . . 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 1234 . . . 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 9336 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u ( M  +  N )  e.  ZZ )
50 expclzap 10501 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A #  0  /\  -u ( M  +  N )  e.  ZZ )  ->  ( A ^ -u ( M  +  N ) )  e.  CC )
511, 9, 49, 50syl3anc 1233 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u ( M  +  N
) )  e.  CC )
5251adantr 274 . . . . . . . 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 274 . . . . . . . 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 10508 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  ZZ )  ->  ( A ^ N ) #  0 )
551, 9, 20, 54syl3anc 1233 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ N ) #  0 )
5655adantr 274 . . . . . . . 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 8713 . . . . . . 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 274 . . . . . . . . . 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 10518 . . . . . . . . . 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 1233 . . . . . . . . 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 8244 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u ( M  +  N )  =  (
-u M  -  N
) )
6261oveq1d 5868 . . . . . . . . . . . 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 8217 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  CC )
6463, 35npcand 8234 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( ( -u M  -  N )  +  N )  =  -u M )
6562, 64eqtrd 2203 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( -u ( M  +  N )  +  N )  =  -u M )
6665adantr 274 . . . . . . . . . 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 5869 . . . . . . . . 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 2205 . . . . . . . 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 5868 . . . . . . 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 2205 . . . . . 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 5869 . . . . 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 8724 . . . . . 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 274 . . . . 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 2203 . . . 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 2203 . . 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 9227 . . . . 5  |-  ( ( M  +  N )  e.  ZZ  <->  ( ( M  +  N )  e.  RR  /\  ( ( M  +  N )  e.  NN0  \/  -u ( M  +  N )  e.  NN0 ) ) )
7776simprbi 273 . . . 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 793 . 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 10485 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  CC  /\  -u M  e.  NN0 ) )  ->  ( A ^ M )  =  ( 1  /  ( A ^ -u M ) ) )
811, 9, 15, 6, 80syl22anc 1234 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ M )  =  ( 1  /  ( A ^ -u M ) ) )
8281oveq1d 5868 . 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 2213 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 703    /\ w3a 973    = wceq 1348    e. wcel 2141   class class class wbr 3989  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   1c1 7775    + caddc 7777    x. cmul 7779    - cmin 8090   -ucneg 8091   # cap 8500    / cdiv 8589   NNcn 8878   NN0cn0 9135   ZZcz 9212   ^cexp 10475
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-seqfrec 10402  df-exp 10476
This theorem is referenced by:  expaddzap  10520
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