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Theorem expaddzaplem 9457
Description: Lemma for expaddzap 9458. (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 939 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  A  e.  CC )
2 simp3 917 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  NN0 )
3 expcl 9432 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( A ^ N
)  e.  CC )
41, 2, 3syl2anc 397 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  CC )
5 simp2r 942 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  NN )
65nnnn0d 8291 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  NN0 )
7 expcl 9432 . . . 4  |-  ( ( A  e.  CC  /\  -u M  e.  NN0 )  ->  ( A ^ -u M
)  e.  CC )
81, 6, 7syl2anc 397 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u M )  e.  CC )
9 simp1r 940 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  A #  0 )
105nnzd 8417 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  ZZ )
11 expap0i 9446 . . . 4  |-  ( ( A  e.  CC  /\  A #  0  /\  -u M  e.  ZZ )  ->  ( A ^ -u M ) #  0 )
121, 9, 10, 11syl3anc 1146 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u M ) #  0 )
134, 8, 12divrecap2d 7843 . 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 941 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  RR )
1514recnd 7112 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  CC )
1615negnegd 7375 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u -u M  =  M )
17 nnnegz 8304 . . . . . . . . . 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 2131 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  ZZ )
202nn0zd 8416 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  ZZ )
2119, 20zaddcld 8422 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( M  +  N )  e.  ZZ )
22 expclzap 9439 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  ( M  +  N )  e.  ZZ )  ->  ( A ^ ( M  +  N ) )  e.  CC )
231, 9, 21, 22syl3anc 1146 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^
( M  +  N
) )  e.  CC )
2423adantr 265 . . . . 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 265 . . . . 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 265 . . . . 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 7845 . . . 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 265 . . . . . . 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 107 . . . . . . 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 265 . . . . . . 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 9456 . . . . . . 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 1146 . . . . . 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 8419 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( M  +  N )  e.  CC )
3433, 15negsubd 7390 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( ( M  +  N )  + 
-u M )  =  ( ( M  +  N )  -  M
) )
352nn0cnd 8293 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  CC )
3615, 35pncan2d 7386 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( ( M  +  N )  -  M )  =  N )
3734, 36eqtrd 2088 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( ( M  +  N )  + 
-u M )  =  N )
3837adantr 265 . . . . . . 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 5555 . . . . . 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 2090 . . . . 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 5554 . . . 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 2090 . . 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 265 . . . . 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 265 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  A #  0 )
4533adantr 265 . . . . 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 107 . . . . 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 9423 . . . . 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 1147 . . . 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 8420 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u ( M  +  N )  e.  ZZ )
50 expclzap 9439 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A #  0  /\  -u ( M  +  N )  e.  ZZ )  ->  ( A ^ -u ( M  +  N ) )  e.  CC )
511, 9, 49, 50syl3anc 1146 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u ( M  +  N
) )  e.  CC )
5251adantr 265 . . . . . . . 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 265 . . . . . . . 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 9446 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  ZZ )  ->  ( A ^ N ) #  0 )
551, 9, 20, 54syl3anc 1146 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ N ) #  0 )
5655adantr 265 . . . . . . . 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 7845 . . . . . . 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 265 . . . . . . . . . 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 9456 . . . . . . . . . 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 1146 . . . . . . . . 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 7398 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u ( M  +  N )  =  (
-u M  -  N
) )
6261oveq1d 5554 . . . . . . . . . . . 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 7371 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  CC )
6463, 35npcand 7388 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( ( -u M  -  N )  +  N )  =  -u M )
6562, 64eqtrd 2088 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( -u ( M  +  N )  +  N )  =  -u M )
6665adantr 265 . . . . . . . . . 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 5555 . . . . . . . . 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 2090 . . . . . . . 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 5554 . . . . . . 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 2090 . . . . . 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 5555 . . . . 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 7856 . . . . . 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 265 . . . . 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 2088 . . . 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 2088 . . 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 8316 . . . . 5  |-  ( ( M  +  N )  e.  ZZ  <->  ( ( M  +  N )  e.  RR  /\  ( ( M  +  N )  e.  NN0  \/  -u ( M  +  N )  e.  NN0 ) ) )
7776simprbi 264 . . . 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 722 . 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 9423 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  CC  /\  -u M  e.  NN0 ) )  ->  ( A ^ M )  =  ( 1  /  ( A ^ -u M ) ) )
811, 9, 15, 6, 80syl22anc 1147 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ M )  =  ( 1  /  ( A ^ -u M ) ) )
8281oveq1d 5554 . 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 2098 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 101    \/ wo 639    /\ w3a 896    = wceq 1259    e. wcel 1409   class class class wbr 3791  (class class class)co 5539   CCcc 6944   RRcr 6945   0cc0 6946   1c1 6947    + caddc 6949    x. cmul 6951    - cmin 7244   -ucneg 7245   # cap 7645    / cdiv 7724   NNcn 7989   NN0cn0 8238   ZZcz 8301   ^cexp 9413
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338  ax-cnex 7032  ax-resscn 7033  ax-1cn 7034  ax-1re 7035  ax-icn 7036  ax-addcl 7037  ax-addrcl 7038  ax-mulcl 7039  ax-mulrcl 7040  ax-addcom 7041  ax-mulcom 7042  ax-addass 7043  ax-mulass 7044  ax-distr 7045  ax-i2m1 7046  ax-1rid 7048  ax-0id 7049  ax-rnegex 7050  ax-precex 7051  ax-cnre 7052  ax-pre-ltirr 7053  ax-pre-ltwlin 7054  ax-pre-lttrn 7055  ax-pre-apti 7056  ax-pre-ltadd 7057  ax-pre-mulgt0 7058  ax-pre-mulext 7059
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-if 3359  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-riota 5495  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-frec 6008  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-i1p 6622  df-iplp 6623  df-iltp 6625  df-enr 6868  df-nr 6869  df-ltr 6872  df-0r 6873  df-1r 6874  df-0 6953  df-1 6954  df-r 6956  df-lt 6959  df-pnf 7120  df-mnf 7121  df-xr 7122  df-ltxr 7123  df-le 7124  df-sub 7246  df-neg 7247  df-reap 7639  df-ap 7646  df-div 7725  df-inn 7990  df-n0 8239  df-z 8302  df-uz 8569  df-iseq 9370  df-iexp 9414
This theorem is referenced by:  expaddzap  9458
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