Theorem expmulz 11381

Description: Product of exponents law for integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 7-Jul-2014.)

Assertion

Ref	Expression
expmulz	|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) )

Proof of Theorem expmulz

Step	Hyp	Ref	Expression
1		elznn0nn 10251	. . 3 |- ( N e. ZZ <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) )
2		elznn0nn 10251	. . . 4 |- ( M e. ZZ <-> ( M e. NN0 \/ ( M e. RR /\ -u M e. NN ) ) )
3		expmul 11380	. . . . . . . 8 |- ( ( A e. CC /\ M e. NN0 /\ N e. NN0 ) -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) )
4	3	3expia 1155	. . . . . . 7 |- ( ( A e. CC /\ M e. NN0 ) -> ( N e. NN0 -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) )
5	4	adantlr 696	. . . . . 6 |- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 ) -> ( N e. NN0 -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) )
6		simp2l 983	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> M e. RR )
7	6	recnd 9070	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> M e. CC )
8		simp3 959	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> N e. NN0 )
9	8	nn0cnd 10232	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> N e. CC )
10	7, 9	mulneg1d 9442	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( -u M x. N ) = -u ( M x. N ) )
11	10	oveq2d 6056	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ ( -u M x. N ) ) = ( A ^ -u ( M x. N ) ) )
12		simp1l 981	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> A e. CC )
13		simp2r 984	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. NN )
14	13	nnnn0d 10230	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. NN0 )
15		expmul 11380	. . . . . . . . . . . 12 |- ( ( A e. CC /\ -u M e. NN0 /\ N e. NN0 ) -> ( A ^ ( -u M x. N ) ) = ( ( A ^ -u M ) ^ N ) )
16	12, 14, 8, 15	syl3anc 1184	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ ( -u M x. N ) ) = ( ( A ^ -u M ) ^ N ) )
17	11, 16	eqtr3d 2438	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ -u ( M x. N ) ) = ( ( A ^ -u M ) ^ N ) )
18	17	oveq2d 6056	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( 1 / ( A ^ -u ( M x. N ) ) ) = ( 1 / ( ( A ^ -u M ) ^ N ) ) )
19		expcl 11354	. . . . . . . . . . 11 |- ( ( A e. CC /\ -u M e. NN0 ) -> ( A ^ -u M ) e. CC )
20	12, 14, 19	syl2anc 643	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ -u M ) e. CC )
21		simp1r 982	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> A =/= 0 )
22	13	nnzd 10330	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. ZZ )
23		expne0i 11367	. . . . . . . . . . 11 |- ( ( A e. CC /\ A =/= 0 /\ -u M e. ZZ ) -> ( A ^ -u M ) =/= 0 )
24	12, 21, 22, 23	syl3anc 1184	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ -u M ) =/= 0 )
25	8	nn0zd 10329	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> N e. ZZ )
26		exprec 11376	. . . . . . . . . 10 |- ( ( ( A ^ -u M ) e. CC /\ ( A ^ -u M ) =/= 0 /\ N e. ZZ ) -> ( ( 1 / ( A ^ -u M ) ) ^ N ) = ( 1 / ( ( A ^ -u M ) ^ N ) ) )
27	20, 24, 25, 26	syl3anc 1184	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( ( 1 / ( A ^ -u M ) ) ^ N ) = ( 1 / ( ( A ^ -u M ) ^ N ) ) )
28	18, 27	eqtr4d 2439	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( 1 / ( A ^ -u ( M x. N ) ) ) = ( ( 1 / ( A ^ -u M ) ) ^ N ) )
29	7, 9	mulcld 9064	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( M x. N ) e. CC )
30	14, 8	nn0mulcld 10235	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( -u M x. N ) e. NN0 )
31	10, 30	eqeltrrd 2479	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u ( M x. N ) e. NN0 )
32		expneg2 11345	. . . . . . . . 9 |- ( ( A e. CC /\ ( M x. N ) e. CC /\ -u ( M x. N ) e. NN0 ) -> ( A ^ ( M x. N ) ) = ( 1 / ( A ^ -u ( M x. N ) ) ) )
33	12, 29, 31, 32	syl3anc 1184	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ ( M x. N ) ) = ( 1 / ( A ^ -u ( M x. N ) ) ) )
34		expneg2 11345	. . . . . . . . . 10 |- ( ( A e. CC /\ M e. CC /\ -u M e. NN0 ) -> ( A ^ M ) = ( 1 / ( A ^ -u M ) ) )
35	12, 7, 14, 34	syl3anc 1184	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ M ) = ( 1 / ( A ^ -u M ) ) )
36	35	oveq1d 6055	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( ( A ^ M ) ^ N ) = ( ( 1 / ( A ^ -u M ) ) ^ N ) )
37	28, 33, 36	3eqtr4d 2446	. . . . . . 7 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) )
38	37	3expia 1155	. . . . . 6 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) ) -> ( N e. NN0 -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) )
39	5, 38	jaodan 761	. . . . 5 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. NN0 \/ ( M e. RR /\ -u M e. NN ) ) ) -> ( N e. NN0 -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) )
40		simp2 958	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> M e. NN0 )
41	40	nn0cnd 10232	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> M e. CC )
42		simp3l 985	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> N e. RR )
43	42	recnd 9070	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> N e. CC )
44	41, 43	mulneg2d 9443	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> ( M x. -u N ) = -u ( M x. N ) )
45	44	oveq2d 6056	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ ( M x. -u N ) ) = ( A ^ -u ( M x. N ) ) )
46		simp1l 981	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> A e. CC )
47		simp3r 986	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. NN )
48	47	nnnn0d 10230	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. NN0 )
49		expmul 11380	. . . . . . . . . . 11 |- ( ( A e. CC /\ M e. NN0 /\ -u N e. NN0 ) -> ( A ^ ( M x. -u N ) ) = ( ( A ^ M ) ^ -u N ) )
50	46, 40, 48, 49	syl3anc 1184	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ ( M x. -u N ) ) = ( ( A ^ M ) ^ -u N ) )
51	45, 50	eqtr3d 2438	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u ( M x. N ) ) = ( ( A ^ M ) ^ -u N ) )
52	51	oveq2d 6056	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> ( 1 / ( A ^ -u ( M x. N ) ) ) = ( 1 / ( ( A ^ M ) ^ -u N ) ) )
53	41, 43	mulcld 9064	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> ( M x. N ) e. CC )
54	40, 48	nn0mulcld 10235	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> ( M x. -u N ) e. NN0 )
55	44, 54	eqeltrrd 2479	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> -u ( M x. N ) e. NN0 )
56	46, 53, 55, 32	syl3anc 1184	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ ( M x. N ) ) = ( 1 / ( A ^ -u ( M x. N ) ) ) )
57		expcl 11354	. . . . . . . . . 10 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ M ) e. CC )
58	46, 40, 57	syl2anc 643	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ M ) e. CC )
59		expneg2 11345	. . . . . . . . 9 |- ( ( ( A ^ M ) e. CC /\ N e. CC /\ -u N e. NN0 ) -> ( ( A ^ M ) ^ N ) = ( 1 / ( ( A ^ M ) ^ -u N ) ) )
60	58, 43, 48, 59	syl3anc 1184	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> ( ( A ^ M ) ^ N ) = ( 1 / ( ( A ^ M ) ^ -u N ) ) )
61	52, 56, 60	3eqtr4d 2446	. . . . . . 7 |- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) )
62	61	3expia 1155	. . . . . 6 |- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 ) -> ( ( N e. RR /\ -u N e. NN ) -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) )
63		simp1l 981	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> A e. CC )
64		simp2l 983	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> M e. RR )
65	64	recnd 9070	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> M e. CC )
66		simp2r 984	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u M e. NN )
67	66	nnnn0d 10230	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u M e. NN0 )
68	63, 65, 67, 34	syl3anc 1184	. . . . . . . . 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 ) ) )
69	68	oveq1d 6055	. . . . . . . 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 ) )
70	63, 67, 19	syl2anc 643	. . . . . . . . . 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 )
71		simp1r 982	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> A =/= 0 )
72	66	nnzd 10330	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u M e. ZZ )
73	63, 71, 72, 23	syl3anc 1184	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u M ) =/= 0 )
74	70, 73	reccld 9739	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( 1 / ( A ^ -u M ) ) e. CC )
75		simp3l 985	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. RR )
76	75	recnd 9070	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. CC )
77		simp3r 986	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. NN )
78	77	nnnn0d 10230	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. NN0 )
79		expneg2 11345	. . . . . . . . 9 |- ( ( ( 1 / ( A ^ -u M ) ) e. CC /\ N e. CC /\ -u N e. NN0 ) -> ( ( 1 / ( A ^ -u M ) ) ^ N ) = ( 1 / ( ( 1 / ( A ^ -u M ) ) ^ -u N ) ) )
80	74, 76, 78, 79	syl3anc 1184	. . . . . . . 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 / ( ( 1 / ( A ^ -u M ) ) ^ -u N ) ) )
81	77	nnzd 10330	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. ZZ )
82		exprec 11376	. . . . . . . . . . 11 |- ( ( ( A ^ -u M ) e. CC /\ ( A ^ -u M ) =/= 0 /\ -u N e. ZZ ) -> ( ( 1 / ( A ^ -u M ) ) ^ -u N ) = ( 1 / ( ( A ^ -u M ) ^ -u N ) ) )
83	70, 73, 81, 82	syl3anc 1184	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( ( 1 / ( A ^ -u M ) ) ^ -u N ) = ( 1 / ( ( A ^ -u M ) ^ -u N ) ) )
84	83	oveq2d 6056	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( 1 / ( ( 1 / ( A ^ -u M ) ) ^ -u N ) ) = ( 1 / ( 1 / ( ( A ^ -u M ) ^ -u N ) ) ) )
85		expcl 11354	. . . . . . . . . . 11 |- ( ( ( A ^ -u M ) e. CC /\ -u N e. NN0 ) -> ( ( A ^ -u M ) ^ -u N ) e. CC )
86	70, 78, 85	syl2anc 643	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( ( A ^ -u M ) ^ -u N ) e. CC )
87		expne0i 11367	. . . . . . . . . . 11 |- ( ( ( A ^ -u M ) e. CC /\ ( A ^ -u M ) =/= 0 /\ -u N e. ZZ ) -> ( ( A ^ -u M ) ^ -u N ) =/= 0 )
88	70, 73, 81, 87	syl3anc 1184	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( ( A ^ -u M ) ^ -u N ) =/= 0 )
89	86, 88	recrecd 9743	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( 1 / ( 1 / ( ( A ^ -u M ) ^ -u N ) ) ) = ( ( A ^ -u M ) ^ -u N ) )
90		expmul 11380	. . . . . . . . . . 11 |- ( ( A e. CC /\ -u M e. NN0 /\ -u N e. NN0 ) -> ( A ^ ( -u M x. -u N ) ) = ( ( A ^ -u M ) ^ -u N ) )
91	63, 67, 78, 90	syl3anc 1184	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ ( -u M x. -u N ) ) = ( ( A ^ -u M ) ^ -u N ) )
92	65, 76	mul2negd 9444	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( -u M x. -u N ) = ( M x. N ) )
93	92	oveq2d 6056	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ ( -u M x. -u N ) ) = ( A ^ ( M x. N ) ) )
94	91, 93	eqtr3d 2438	. . . . . . . . 9 |- ( ( ( 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 ^ ( M x. N ) ) )
95	84, 89, 94	3eqtrd 2440	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( 1 / ( ( 1 / ( A ^ -u M ) ) ^ -u N ) ) = ( A ^ ( M x. N ) ) )
96	69, 80, 95	3eqtrrd 2441	. . . . . . 7 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) )
97	96	3expia 1155	. . . . . 6 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) ) -> ( ( N e. RR /\ -u N e. NN ) -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) )
98	62, 97	jaodan 761	. . . . 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 x. N ) ) = ( ( A ^ M ) ^ N ) ) )
99	39, 98	jaod 370	. . . 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 x. N ) ) = ( ( A ^ M ) ^ N ) ) )
100	2, 99	sylan2b 462	. . 3 |- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. ZZ ) -> ( ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) )
101	1, 100	syl5bi 209	. 2 |- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. ZZ ) -> ( N e. ZZ -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) )
102	101	impr 603	1 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) )

Syntax hints:   -> wi 4   \/ wo 358   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   =/= wne 2567  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947   x. cmul 8951   -ucneg 9248   / cdiv 9633   NNcn 9956   NN0cn0 10177   ZZcz 10238   ^cexp 11337

This theorem is referenced by:  iexpcyc  11440  iseraltlem2  12431  iseraltlem3  12432  dvexp3  19815  cxpeq  20594  atantayl2  20731  basellem3  20818  lgseisenlem1  21086  lgseisenlem4  21089  lgsquadlem1  21091  lgsquad2lem1  21095  m1lgs  21099  jm2.21  26955

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-seq 11279  df-exp 11338