MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  expaddzlem Structured version   Unicode version

Theorem expaddzlem 11415
Description: Lemma for expaddz 11416. (Contributed by Mario Carneiro, 4-Jun-2014.)
Assertion
Ref Expression
expaddzlem  |-  ( ( ( 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 expaddzlem
StepHypRef Expression
1 simp1l 981 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  A  e.  CC )
2 simp3 959 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  NN0 )
3 expcl 11391 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( A ^ N
)  e.  CC )
41, 2, 3syl2anc 643 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  CC )
5 simp2r 984 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  NN )
65nnnn0d 10266 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  NN0 )
7 expcl 11391 . . . 4  |-  ( ( A  e.  CC  /\  -u M  e.  NN0 )  ->  ( A ^ -u M
)  e.  CC )
81, 6, 7syl2anc 643 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u M )  e.  CC )
9 simp1r 982 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  A  =/=  0 )
105nnzd 10366 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  ZZ )
11 expne0i 11404 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  -u M  e.  ZZ )  ->  ( A ^ -u M )  =/=  0 )
121, 9, 10, 11syl3anc 1184 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u M )  =/=  0 )
134, 8, 12divrec2d 9786 . 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 983 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  RR )
1514recnd 9106 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  CC )
1615negnegd 9394 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u -u M  =  M )
17 nnnegz 10277 . . . . . . . . . 10  |-  ( -u M  e.  NN  ->  -u -u M  e.  ZZ )
185, 17syl 16 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u -u M  e.  ZZ )
1916, 18eqeltrrd 2510 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  ZZ )
202nn0zd 10365 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  ZZ )
2119, 20zaddcld 10371 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( M  +  N )  e.  ZZ )
22 expclz 11398 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  ( M  +  N )  e.  ZZ )  ->  ( A ^ ( M  +  N ) )  e.  CC )
231, 9, 21, 22syl3anc 1184 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ ( M  +  N ) )  e.  CC )
2423adantr 452 . . . . 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 452 . . . . 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 452 . . . . 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, 26divcan4d 9788 . . . 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 452 . . . . . . 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 448 . . . . . . 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 452 . . . . . . 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 11414 . . . . . . 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 1184 . . . . . 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 10368 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( M  +  N )  e.  CC )
3433, 15negsubd 9409 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  (
( M  +  N
)  +  -u M
)  =  ( ( M  +  N )  -  M ) )
352nn0cnd 10268 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  CC )
3615, 35pncan2d 9405 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  (
( M  +  N
)  -  M )  =  N )
3734, 36eqtrd 2467 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  (
( M  +  N
)  +  -u M
)  =  N )
3837adantr 452 . . . . . . 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 6089 . . . . . 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 2469 . . . . 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 6088 . . . 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 2469 . . 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 452 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  A  e.  CC )
4433adantr 452 . . . . 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 )
45 simpr 448 . . . . 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 )
46 expneg2 11382 . . . . 5  |-  ( ( A  e.  CC  /\  ( M  +  N
)  e.  CC  /\  -u ( M  +  N
)  e.  NN0 )  ->  ( A ^ ( M  +  N )
)  =  ( 1  /  ( A ^ -u ( M  +  N
) ) ) )
4743, 44, 45, 46syl3anc 1184 . . . 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 ) ) ) )
4821znegcld 10369 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u ( M  +  N )  e.  ZZ )
49 expclz 11398 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  -u ( M  +  N )  e.  ZZ )  ->  ( A ^ -u ( M  +  N ) )  e.  CC )
501, 9, 48, 49syl3anc 1184 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u ( M  +  N ) )  e.  CC )
5150adantr 452 . . . . . . . 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 )
524adantr 452 . . . . . . . 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 )
53 expne0i 11404 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ N )  =/=  0 )
541, 9, 20, 53syl3anc 1184 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ N )  =/=  0 )
5554adantr 452 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  ( A ^ N )  =/=  0 )
5651, 52, 55divcan4d 9788 . . . . . . 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 )
) )
572adantr 452 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  /\  -u ( M  +  N )  e.  NN0 )  ->  N  e.  NN0 )
58 expadd 11414 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  -u ( M  +  N
)  e.  NN0  /\  N  e.  NN0 )  -> 
( A ^ ( -u ( M  +  N
)  +  N ) )  =  ( ( A ^ -u ( M  +  N )
)  x.  ( A ^ N ) ) )
5943, 45, 57, 58syl3anc 1184 . . . . . . . . 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 ) ) )
6015, 35negdi2d 9417 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u ( M  +  N )  =  ( -u M  -  N ) )
6160oveq1d 6088 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( -u ( M  +  N
)  +  N )  =  ( ( -u M  -  N )  +  N ) )
6215negcld 9390 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  CC )
6362, 35npcand 9407 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  (
( -u M  -  N
)  +  N )  =  -u M )
6461, 63eqtrd 2467 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( -u ( M  +  N
)  +  N )  =  -u M )
6564adantr 452 . . . . . . . . . 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 )
6665oveq2d 6089 . . . . . . . . 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
) )
6759, 66eqtr3d 2469 . . . . . . . 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 ) )
6867oveq1d 6088 . . . . . . 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 ) ) )
6956, 68eqtr3d 2469 . . . . . 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 ) ) )
7069oveq2d 6089 . . . . 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 ) ) ) )
718, 4, 12, 54recdivd 9799 . . . . . 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 ) ) )
7271adantr 452 . . . . 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 ) ) )
7370, 72eqtrd 2467 . . . 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 ) ) )
7447, 73eqtrd 2467 . . 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 ) ) )
75 elznn0 10288 . . . . 5  |-  ( ( M  +  N )  e.  ZZ  <->  ( ( M  +  N )  e.  RR  /\  ( ( M  +  N )  e.  NN0  \/  -u ( M  +  N )  e.  NN0 ) ) )
7675simprbi 451 . . . 4  |-  ( ( M  +  N )  e.  ZZ  ->  (
( M  +  N
)  e.  NN0  \/  -u ( M  +  N
)  e.  NN0 )
)
7721, 76syl 16 . . 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 )
)
7842, 74, 77mpjaodan 762 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ N )  /  ( A ^ -u M ) ) )
79 expneg2 11382 . . . 4  |-  ( ( A  e.  CC  /\  M  e.  CC  /\  -u M  e.  NN0 )  ->  ( A ^ M )  =  ( 1  /  ( A ^ -u M ) ) )
801, 15, 6, 79syl3anc 1184 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ M )  =  ( 1  /  ( A ^ -u M ) ) )
8180oveq1d 6088 . 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
) ) )
8213, 78, 813eqtr4d 2477 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    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    - cmin 9283   -ucneg 9284    / cdiv 9669   NNcn 9992   NN0cn0 10213   ZZcz 10274   ^cexp 11374
This theorem is referenced by:  expaddz  11416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-seq 11316  df-exp 11375
  Copyright terms: Public domain W3C validator