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Theorem expaddzap 10675
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 9340 . . 3 |- ( N e. ZZ <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) )
2 elznn0nn 9340 . . . 4 |- ( M e. ZZ <-> ( M e. NN0 \/ ( M e. RR /\ -u M e. NN ) ) )
3 expadd 10673 . . . . . . . 8 |- ( ( A e. CC /\ M e. NN0 /\ N e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) )
433expia 1207 . . . . . . 7 |- ( ( A e. CC /\ M e. NN0 ) -> ( N e. NN0 -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
54adantlr 477 . . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ M e. NN0 ) -> ( N e. NN0 -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
6 expaddzaplem 10674 . . . . . . 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 1207 . . . . . 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 798 . . . . 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 10674 . . . . . . . . 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 1001 . . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> M e. NN0 )
1110nn0cnd 9304 . . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> M e. CC )
12 simp2l 1025 . . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> N e. RR )
1312recnd 8055 . . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> N e. CC )
1411, 13addcomd 8177 . . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> ( M + N ) = ( N + M ) )
1514oveq2d 5938 . . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> ( A ^ ( M + N ) ) = ( A ^ ( N + M ) ) )
16 simp1l 1023 . . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> A e. CC )
17 expcl 10649 . . . . . . . . . . 11 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ M ) e. CC )
1816, 10, 17syl2anc 411 . . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> ( A ^ M ) e. CC )
19 simp1r 1024 . . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> A # 0 )
2013negnegd 8328 . . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> -u -u N = N )
21 simp2r 1026 . . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> -u N e. NN )
2221nnnn0d 9302 . . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> -u N e. NN0 )
23 nn0negz 9360 . . . . . . . . . . . . 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 2274 . . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> N e. ZZ )
26 expclzap 10656 . . . . . . . . . . 11 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ N ) e. CC )
2716, 19, 25, 26syl3anc 1249 . . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) /\ M e. NN0 ) -> ( A ^ N ) e. CC )
2818, 27mulcomd 8048 . . . . . . . . 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 2239 . . . . . . . 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 1207 . . . . . . 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 260 . . . . . 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 1025 . . . . . . . . . . . . . . 15 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> M e. RR )
3332recnd 8055 . . . . . . . . . . . . . 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 1027 . . . . . . . . . . . . . . 15 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. RR )
3534recnd 8055 . . . . . . . . . . . . . 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 8350 . . . . . . . . . . . . 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 5938 . . . . . . . . . . . 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 1023 . . . . . . . . . . . . 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 1026 . . . . . . . . . . . . . 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 9302 . . . . . . . . . . . . 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 1028 . . . . . . . . . . . . . 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 9302 . . . . . . . . . . . . 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 10673 . . . . . . . . . . . . 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 1249 . . . . . . . . . . . 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 2229 . . . . . . . . . . 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 5938 . . . . . . . . . 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 9143 . . . . . . . . . . 11 |- ( 1 x. 1 ) = 1
4847oveq1i 5932 . . . . . . . . . 10 |- ( ( 1 x. 1 ) / ( ( A ^ -u M ) x. ( A ^ -u N ) ) ) = ( 1 / ( ( A ^ -u M ) x. ( A ^ -u N ) ) )
4946, 48eqtr4di 2247 . . . . . . . . 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 10649 . . . . . . . . . . 11 |- ( ( A e. CC /\ -u M e. NN0 ) -> ( A ^ -u M ) e. CC )
5138, 40, 50syl2anc 411 . . . . . . . . . 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 1024 . . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> A # 0 )
5340nn0zd 9446 . . . . . . . . . . 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 10663 . . . . . . . . . . 11 |- ( ( A e. CC /\ A # 0 /\ -u M e. ZZ ) -> ( A ^ -u M ) # 0 )
5538, 52, 53, 54syl3anc 1249 . . . . . . . . . 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 10649 . . . . . . . . . . 11 |- ( ( A e. CC /\ -u N e. NN0 ) -> ( A ^ -u N ) e. CC )
5738, 42, 56syl2anc 411 . . . . . . . . . 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 9446 . . . . . . . . . . 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 10663 . . . . . . . . . . 11 |- ( ( A e. CC /\ A # 0 /\ -u N e. ZZ ) -> ( A ^ -u N ) # 0 )
6038, 52, 58, 59syl3anc 1249 . . . . . . . . . 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 7972 . . . . . . . . . . 11 |- 1 e. CC
62 divmuldivap 8739 . . . . . . . . . . 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 436 . . . . . . . . . 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 1250 . . . . . . . . 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 2232 . . . . . . . 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 8046 . . . . . . . . 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 9306 . . . . . . . . . 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 2273 . . . . . . . . 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 10640 . . . . . . . . 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 1250 . . . . . . . 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 10640 . . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. CC /\ -u M e. NN0 ) ) -> ( A ^ M ) = ( 1 / ( A ^ -u M ) ) )
7238, 52, 33, 40, 71syl22anc 1250 . . . . . . . . 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 10640 . . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. CC /\ -u N e. NN0 ) ) -> ( A ^ N ) = ( 1 / ( A ^ -u N ) ) )
7438, 52, 35, 42, 73syl22anc 1250 . . . . . . . . 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 5940 . . . . . . . 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 2239 . . . . . . 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 1207 . . . . . 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 798 . . . . 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 718 . . . 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 287 . . 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, 80biimtrid 152 . 2 |- ( ( ( A e. CC /\ A # 0 ) /\ M e. ZZ ) -> ( N e. ZZ -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
8281impr 379 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 104   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2167   class class class wbr 4033  (class class class)co 5922   CCcc 7877   RRcr 7878   0cc0 7879   1c1 7880   + caddc 7882   x. cmul 7884   -ucneg 8198   # cap 8608   / cdiv 8699   NNcn 8990   NN0cn0 9249   ZZcz 9326   ^cexp 10630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602  df-seqfrec 10540  df-exp 10631
This theorem is referenced by:  m1expeven  10678  expsubap  10679  expp1zap  10680  pcaddlem  12508  expghmap  14163  lgseisenlem4  15314  lgsquadlem1  15318  lgsquad2lem1  15322
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