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Theorem expaddzap 10520
Description: Sum of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 10-Jun-2020.)
Assertion
Ref Expression
expaddzap |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) )
Proof of Theorem expaddzap
StepHypRef Expression
1 elznn0nn 9226 . . 3 |- ( N e. ZZ <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) )
2 elznn0nn 9226 . . . 4 |- ( M e. ZZ <-> ( M e. NN0 \/ ( M e. RR /\ -u M e. NN ) ) )
3 expadd 10518 . . . . . . . 8 |- ( ( A e. CC /\ M e. NN0 /\ N e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) )
433expia 1200 . . . . . . 7 |- ( ( A e. CC /\ M e. NN0 ) -> ( N e. NN0 -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
54adantlr 474 . . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ M e. NN0 ) -> ( N e. NN0 -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
6 expaddzaplem 10519 . . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) )
763expia 1200 . . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) ) -> ( N e. NN0 -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
85, 7jaodan 792 . . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. NN0 \/ ( M e. RR /\ -u M e. NN ) ) ) -> ( N e. NN0 -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
9 expaddzaplem 10519 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> ( A ^ ( N + M ) ) = ( ( A ^ N ) x. ( A ^ M ) ) )
10 simp3 994 . . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> M e. NN0 )
1110nn0cnd 9190 . . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> M e. CC )
12 simp2l 1018 . . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> N e. RR )
1312recnd 7948 . . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> N e. CC )
1411, 13addcomd 8070 . . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> ( M + N ) = ( N + M ) )
1514oveq2d 5869 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> ( A ^ ( M + N ) ) = ( A ^ ( N + M ) ) )
16 simp1l 1016 . . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> A e. CC )
17 expcl 10494 . . . . . . . . . . 11 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ M ) e. CC )
1816, 10, 17syl2anc 409 . . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> ( A ^ M ) e. CC )
19 simp1r 1017 . . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> A # 0 )
2013negnegd 8221 . . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> -u -u N = N )
21 simp2r 1019 . . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> -u N e. NN )
2221nnnn0d 9188 . . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> -u N e. NN0 )
23 nn0negz 9246 . . . . . . . . . . . . 13 |- ( -u N e. NN0 -> -u -u N e. ZZ )
2422, 23syl 14 . . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> -u -u N e. ZZ )
2520, 24eqeltrrd 2248 . . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> N e. ZZ )
26 expclzap 10501 . . . . . . . . . . 11 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ N ) e. CC )
2716, 19, 25, 26syl3anc 1233 . . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> ( A ^ N ) e. CC )
2818, 27mulcomd 7941 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> ( ( A ^ M ) x. ( A ^ N ) ) = ( ( A ^ N ) x. ( A ^ M ) ) )
299, 15, 283eqtr4d 2213 . . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) )
30293expia 1200 . . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( M e. NN0 -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
3130impancom 258 . . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ M e. NN0 ) -> ( ( N e. RR /\ -u N e. NN ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
32 simp2l 1018 . . . . . . . . . . . . . . 15 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> M e. RR )
3332recnd 7948 . . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> M e. CC )
34 simp3l 1020 . . . . . . . . . . . . . . 15 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. RR )
3534recnd 7948 . . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. CC )
3633, 35negdid 8243 . . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u ( M + N ) = ( -u M + -u N ) )
3736oveq2d 5869 . . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u ( M + N ) ) = ( A ^ ( -u M + -u N ) ) )
38 simp1l 1016 . . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> A e. CC )
39 simp2r 1019 . . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u M e. NN )
4039nnnn0d 9188 . . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u M e. NN0 )
41 simp3r 1021 . . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. NN )
4241nnnn0d 9188 . . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. NN0 )
43 expadd 10518 . . . . . . . . . . . . 13 |- ( ( A e. CC /\ -u M e. NN0 /\ -u N e. NN0 ) -> ( A ^ ( -u M + -u N ) ) = ( ( A ^ -u M ) x. ( A ^ -u N ) ) )
4438, 40, 42, 43syl3anc 1233 . . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ ( -u M + -u N ) ) = ( ( A ^ -u M ) x. ( A ^ -u N ) ) )
4537, 44eqtrd 2203 . . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u ( M + N ) ) = ( ( A ^ -u M ) x. ( A ^ -u N ) ) )
4645oveq2d 5869 . . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( 1 / ( A ^ -u ( M + N ) ) ) = ( 1 / ( ( A ^ -u M ) x. ( A ^ -u N ) ) ) )
47 1t1e1 9030 . . . . . . . . . . 11 |- ( 1 x. 1 ) = 1
4847oveq1i 5863 . . . . . . . . . 10 |- ( ( 1 x. 1 ) / ( ( A ^ -u M ) x. ( A ^ -u N ) ) ) = ( 1 / ( ( A ^ -u M ) x. ( A ^ -u N ) ) )
4946, 48eqtr4di 2221 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( 1 / ( A ^ -u ( M + N ) ) ) = ( ( 1 x. 1 ) / ( ( A ^ -u M ) x. ( A ^ -u N ) ) ) )
50 expcl 10494 . . . . . . . . . . 11 |- ( ( A e. CC /\ -u M e. NN0 ) -> ( A ^ -u M ) e. CC )
5138, 40, 50syl2anc 409 . . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u M ) e. CC )
52 simp1r 1017 . . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> A # 0 )
5340nn0zd 9332 . . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u M e. ZZ )
54 expap0i 10508 . . . . . . . . . . 11 |- ( ( A e. CC /\ A # 0 /\ -u M e. ZZ ) -> ( A ^ -u M ) # 0 )
5538, 52, 53, 54syl3anc 1233 . . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u M ) # 0 )
56 expcl 10494 . . . . . . . . . . 11 |- ( ( A e. CC /\ -u N e. NN0 ) -> ( A ^ -u N ) e. CC )
5738, 42, 56syl2anc 409 . . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u N ) e. CC )
5842nn0zd 9332 . . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. ZZ )
59 expap0i 10508 . . . . . . . . . . 11 |- ( ( A e. CC /\ A # 0 /\ -u N e. ZZ ) -> ( A ^ -u N ) # 0 )
6038, 52, 58, 59syl3anc 1233 . . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u N ) # 0 )
61 ax-1cn 7867 . . . . . . . . . . 11 |- 1 e. CC
62 divmuldivap 8629 . . . . . . . . . . 11 |- ( ( ( 1 e. CC /\ 1 e. CC ) /\ ( ( ( A ^ -u M ) e. CC /\ ( A ^ -u M ) # 0 ) /\ ( ( A ^ -u N ) e. CC /\ ( A ^ -u N ) # 0 ) ) ) -> ( ( 1 / ( A ^ -u M ) ) x. ( 1 / ( A ^ -u N ) ) ) = ( ( 1 x. 1 ) / ( ( A ^ -u M ) x. ( A ^ -u N ) ) ) )
6361, 61, 62mpanl12 434 . . . . . . . . . 10 |- ( ( ( ( A ^ -u M ) e. CC /\ ( A ^ -u M ) # 0 ) /\ ( ( A ^ -u N ) e. CC /\ ( A ^ -u N ) # 0 ) ) -> ( ( 1 / ( A ^ -u M ) ) x. ( 1 / ( A ^ -u N ) ) ) = ( ( 1 x. 1 ) / ( ( A ^ -u M ) x. ( A ^ -u N ) ) ) )
6451, 55, 57, 60, 63syl22anc 1234 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( ( 1 / ( A ^ -u M ) ) x. ( 1 / ( A ^ -u N ) ) ) = ( ( 1 x. 1 ) / ( ( A ^ -u M ) x. ( A ^ -u N ) ) ) )
6549, 64eqtr4d 2206 . . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( 1 / ( A ^ -u ( M + N ) ) ) = ( ( 1 / ( A ^ -u M ) ) x. ( 1 / ( A ^ -u N ) ) ) )
6633, 35addcld 7939 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( M + N ) e. CC )
6740, 42nn0addcld 9192 . . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( -u M + -u N ) e. NN0 )
6836, 67eqeltrd 2247 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u ( M + N ) e. NN0 )
69 expineg2 10485 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( ( M + N ) e. CC /\ -u ( M + N ) e. NN0 ) ) -> ( A ^ ( M + N ) ) = ( 1 / ( A ^ -u ( M + N ) ) ) )
7038, 52, 66, 68, 69syl22anc 1234 . . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ ( M + N ) ) = ( 1 / ( A ^ -u ( M + N ) ) ) )
71 expineg2 10485 . . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. CC /\ -u M e. NN0 ) ) -> ( A ^ M ) = ( 1 / ( A ^ -u M ) ) )
7238, 52, 33, 40, 71syl22anc 1234 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ M ) = ( 1 / ( A ^ -u M ) ) )
73 expineg2 10485 . . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. CC /\ -u N e. NN0 ) ) -> ( A ^ N ) = ( 1 / ( A ^ -u N ) ) )
7438, 52, 35, 42, 73syl22anc 1234 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ N ) = ( 1 / ( A ^ -u N ) ) )
7572, 74oveq12d 5871 . . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( ( A ^ M ) x. ( A ^ N ) ) = ( ( 1 / ( A ^ -u M ) ) x. ( 1 / ( A ^ -u N ) ) ) )
7665, 70, 753eqtr4d 2213 . . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) )
77763expia 1200 . . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) ) -> ( ( N e. RR /\ -u N e. NN ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
7831, 77jaodan 792 . . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. NN0 \/ ( M e. RR /\ -u M e. NN ) ) ) -> ( ( N e. RR /\ -u N e. NN ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
798, 78jaod 712 . . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. NN0 \/ ( M e. RR /\ -u M e. NN ) ) ) -> ( ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
802, 79sylan2b 285 . . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ M e. ZZ ) -> ( ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
811, 80syl5bi 151 . 2 |- ( ( ( A e. CC /\ A # 0 ) /\ M e. ZZ ) -> ( N e. ZZ -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
8281impr 377 1 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( 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   -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:  m1expeven  10523  expsubap  10524  expp1zap  10525  pcaddlem  12292
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