Theorem cxpmul2 24435

Description: Product of exponents law for complex exponentiation. Variation on cxpmul 24434 with more general conditions on A and B when C is an integer. (Contributed by Mario Carneiro, 9-Aug-2014.)

Assertion

Ref	Expression
cxpmul2	|- ( ( A e. CC /\ B e. CC /\ C e. NN0 ) -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^ C ) )

Proof of Theorem cxpmul2

Dummy variables x k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . . . 7 |- ( x = 0 -> ( B x. x ) = ( B x. 0 ) )
2	1	oveq2d 6666	. . . . . 6 |- ( x = 0 -> ( A ^c ( B x. x ) ) = ( A ^c ( B x. 0 ) ) )
3		oveq2 6658	. . . . . 6 |- ( x = 0 -> ( ( A ^c B ) ^ x ) = ( ( A ^c B ) ^ 0 ) )
4	2, 3	eqeq12d 2637	. . . . 5 |- ( x = 0 -> ( ( A ^c ( B x. x ) ) = ( ( A ^c B ) ^ x ) <-> ( A ^c ( B x. 0 ) ) = ( ( A ^c B ) ^ 0 ) ) )
5	4	imbi2d 330	. . . 4 |- ( x = 0 -> ( ( ( A e. CC /\ B e. CC ) -> ( A ^c ( B x. x ) ) = ( ( A ^c B ) ^ x ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( A ^c ( B x. 0 ) ) = ( ( A ^c B ) ^ 0 ) ) ) )
6		oveq2 6658	. . . . . . 7 |- ( x = k -> ( B x. x ) = ( B x. k ) )
7	6	oveq2d 6666	. . . . . 6 |- ( x = k -> ( A ^c ( B x. x ) ) = ( A ^c ( B x. k ) ) )
8		oveq2 6658	. . . . . 6 |- ( x = k -> ( ( A ^c B ) ^ x ) = ( ( A ^c B ) ^ k ) )
9	7, 8	eqeq12d 2637	. . . . 5 |- ( x = k -> ( ( A ^c ( B x. x ) ) = ( ( A ^c B ) ^ x ) <-> ( A ^c ( B x. k ) ) = ( ( A ^c B ) ^ k ) ) )
10	9	imbi2d 330	. . . 4 |- ( x = k -> ( ( ( A e. CC /\ B e. CC ) -> ( A ^c ( B x. x ) ) = ( ( A ^c B ) ^ x ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( A ^c ( B x. k ) ) = ( ( A ^c B ) ^ k ) ) ) )
11		oveq2 6658	. . . . . . 7 |- ( x = ( k + 1 ) -> ( B x. x ) = ( B x. ( k + 1 ) ) )
12	11	oveq2d 6666	. . . . . 6 |- ( x = ( k + 1 ) -> ( A ^c ( B x. x ) ) = ( A ^c ( B x. ( k + 1 ) ) ) )
13		oveq2 6658	. . . . . 6 |- ( x = ( k + 1 ) -> ( ( A ^c B ) ^ x ) = ( ( A ^c B ) ^ ( k + 1 ) ) )
14	12, 13	eqeq12d 2637	. . . . 5 |- ( x = ( k + 1 ) -> ( ( A ^c ( B x. x ) ) = ( ( A ^c B ) ^ x ) <-> ( A ^c ( B x. ( k + 1 ) ) ) = ( ( A ^c B ) ^ ( k + 1 ) ) ) )
15	14	imbi2d 330	. . . 4 |- ( x = ( k + 1 ) -> ( ( ( A e. CC /\ B e. CC ) -> ( A ^c ( B x. x ) ) = ( ( A ^c B ) ^ x ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( A ^c ( B x. ( k + 1 ) ) ) = ( ( A ^c B ) ^ ( k + 1 ) ) ) ) )
16		oveq2 6658	. . . . . . 7 |- ( x = C -> ( B x. x ) = ( B x. C ) )
17	16	oveq2d 6666	. . . . . 6 |- ( x = C -> ( A ^c ( B x. x ) ) = ( A ^c ( B x. C ) ) )
18		oveq2 6658	. . . . . 6 |- ( x = C -> ( ( A ^c B ) ^ x ) = ( ( A ^c B ) ^ C ) )
19	17, 18	eqeq12d 2637	. . . . 5 |- ( x = C -> ( ( A ^c ( B x. x ) ) = ( ( A ^c B ) ^ x ) <-> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^ C ) ) )
20	19	imbi2d 330	. . . 4 |- ( x = C -> ( ( ( A e. CC /\ B e. CC ) -> ( A ^c ( B x. x ) ) = ( ( A ^c B ) ^ x ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^ C ) ) ) )
21		cxp0 24416	. . . . . 6 |- ( A e. CC -> ( A ^c 0 ) = 1 )
22	21	adantr 481	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A ^c 0 ) = 1 )
23		mul01 10215	. . . . . . 7 |- ( B e. CC -> ( B x. 0 ) = 0 )
24	23	adantl 482	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( B x. 0 ) = 0 )
25	24	oveq2d 6666	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A ^c ( B x. 0 ) ) = ( A ^c 0 ) )
26		cxpcl 24420	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A ^c B ) e. CC )
27	26	exp0d 13002	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^c B ) ^ 0 ) = 1 )
28	22, 25, 27	3eqtr4d 2666	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A ^c ( B x. 0 ) ) = ( ( A ^c B ) ^ 0 ) )
29		oveq1 6657	. . . . . . 7 |- ( ( A ^c ( B x. k ) ) = ( ( A ^c B ) ^ k ) -> ( ( A ^c ( B x. k ) ) x. ( A ^c B ) ) = ( ( ( A ^c B ) ^ k ) x. ( A ^c B ) ) )
30		0cn 10032	. . . . . . . . . . . . 13 |- 0 e. CC
31		cxp0 24416	. . . . . . . . . . . . 13 |- ( 0 e. CC -> ( 0 ^c 0 ) = 1 )
32	30, 31	ax-mp 5	. . . . . . . . . . . 12 |- ( 0 ^c 0 ) = 1
33		1t1e1 11175	. . . . . . . . . . . 12 |- ( 1 x. 1 ) = 1
34	32, 33	eqtr4i 2647	. . . . . . . . . . 11 |- ( 0 ^c 0 ) = ( 1 x. 1 )
35		simplr 792	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B = 0 ) -> A = 0 )
36		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B = 0 ) -> B = 0 )
37	36	oveq1d 6665	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B = 0 ) -> ( B x. ( k + 1 ) ) = ( 0 x. ( k + 1 ) ) )
38		nn0p1nn 11332	. . . . . . . . . . . . . . . . 17 |- ( k e. NN0 -> ( k + 1 ) e. NN )
39	38	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> ( k + 1 ) e. NN )
40	39	nncnd 11036	. . . . . . . . . . . . . . 15 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> ( k + 1 ) e. CC )
41	40	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B = 0 ) -> ( k + 1 ) e. CC )
42	41	mul02d 10234	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B = 0 ) -> ( 0 x. ( k + 1 ) ) = 0 )
43	37, 42	eqtrd 2656	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B = 0 ) -> ( B x. ( k + 1 ) ) = 0 )
44	35, 43	oveq12d 6668	. . . . . . . . . . 11 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B = 0 ) -> ( A ^c ( B x. ( k + 1 ) ) ) = ( 0 ^c 0 ) )
45	36	oveq1d 6665	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B = 0 ) -> ( B x. k ) = ( 0 x. k ) )
46		nn0cn 11302	. . . . . . . . . . . . . . . . . 18 |- ( k e. NN0 -> k e. CC )
47	46	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> k e. CC )
48	47	ad2antrr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B = 0 ) -> k e. CC )
49	48	mul02d 10234	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B = 0 ) -> ( 0 x. k ) = 0 )
50	45, 49	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B = 0 ) -> ( B x. k ) = 0 )
51	35, 50	oveq12d 6668	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B = 0 ) -> ( A ^c ( B x. k ) ) = ( 0 ^c 0 ) )
52	51, 32	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B = 0 ) -> ( A ^c ( B x. k ) ) = 1 )
53	35, 36	oveq12d 6668	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B = 0 ) -> ( A ^c B ) = ( 0 ^c 0 ) )
54	53, 32	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B = 0 ) -> ( A ^c B ) = 1 )
55	52, 54	oveq12d 6668	. . . . . . . . . . 11 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B = 0 ) -> ( ( A ^c ( B x. k ) ) x. ( A ^c B ) ) = ( 1 x. 1 ) )
56	34, 44, 55	3eqtr4a 2682	. . . . . . . . . 10 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B = 0 ) -> ( A ^c ( B x. ( k + 1 ) ) ) = ( ( A ^c ( B x. k ) ) x. ( A ^c B ) ) )
57		simpll 790	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> A e. CC )
58	57	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B =/= 0 ) -> A e. CC )
59		simplr 792	. . . . . . . . . . . . . . 15 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> B e. CC )
60	59, 47	mulcld 10060	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> ( B x. k ) e. CC )
61	60	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B =/= 0 ) -> ( B x. k ) e. CC )
62		cxpcl 24420	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ ( B x. k ) e. CC ) -> ( A ^c ( B x. k ) ) e. CC )
63	58, 61, 62	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B =/= 0 ) -> ( A ^c ( B x. k ) ) e. CC )
64	63	mul01d 10235	. . . . . . . . . . 11 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B =/= 0 ) -> ( ( A ^c ( B x. k ) ) x. 0 ) = 0 )
65		simplr 792	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B =/= 0 ) -> A = 0 )
66	65	oveq1d 6665	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B =/= 0 ) -> ( A ^c B ) = ( 0 ^c B ) )
67	59	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B =/= 0 ) -> B e. CC )
68		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B =/= 0 ) -> B =/= 0 )
69		0cxp 24412	. . . . . . . . . . . . . 14 |- ( ( B e. CC /\ B =/= 0 ) -> ( 0 ^c B ) = 0 )
70	67, 68, 69	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B =/= 0 ) -> ( 0 ^c B ) = 0 )
71	66, 70	eqtrd 2656	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B =/= 0 ) -> ( A ^c B ) = 0 )
72	71	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B =/= 0 ) -> ( ( A ^c ( B x. k ) ) x. ( A ^c B ) ) = ( ( A ^c ( B x. k ) ) x. 0 ) )
73	65	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B =/= 0 ) -> ( A ^c ( B x. ( k + 1 ) ) ) = ( 0 ^c ( B x. ( k + 1 ) ) ) )
74	40	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B =/= 0 ) -> ( k + 1 ) e. CC )
75	67, 74	mulcld 10060	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B =/= 0 ) -> ( B x. ( k + 1 ) ) e. CC )
76	39	nnne0d 11065	. . . . . . . . . . . . . . 15 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> ( k + 1 ) =/= 0 )
77	76	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B =/= 0 ) -> ( k + 1 ) =/= 0 )
78	67, 74, 68, 77	mulne0d 10679	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B =/= 0 ) -> ( B x. ( k + 1 ) ) =/= 0 )
79		0cxp 24412	. . . . . . . . . . . . 13 |- ( ( ( B x. ( k + 1 ) ) e. CC /\ ( B x. ( k + 1 ) ) =/= 0 ) -> ( 0 ^c ( B x. ( k + 1 ) ) ) = 0 )
80	75, 78, 79	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B =/= 0 ) -> ( 0 ^c ( B x. ( k + 1 ) ) ) = 0 )
81	73, 80	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B =/= 0 ) -> ( A ^c ( B x. ( k + 1 ) ) ) = 0 )
82	64, 72, 81	3eqtr4rd 2667	. . . . . . . . . 10 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) /\ B =/= 0 ) -> ( A ^c ( B x. ( k + 1 ) ) ) = ( ( A ^c ( B x. k ) ) x. ( A ^c B ) ) )
83	56, 82	pm2.61dane 2881	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A = 0 ) -> ( A ^c ( B x. ( k + 1 ) ) ) = ( ( A ^c ( B x. k ) ) x. ( A ^c B ) ) )
84	59	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A =/= 0 ) -> B e. CC )
85	47	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A =/= 0 ) -> k e. CC )
86		1cnd 10056	. . . . . . . . . . . . 13 |- ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A =/= 0 ) -> 1 e. CC )
87	84, 85, 86	adddid 10064	. . . . . . . . . . . 12 |- ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A =/= 0 ) -> ( B x. ( k + 1 ) ) = ( ( B x. k ) + ( B x. 1 ) ) )
88	84	mulid1d 10057	. . . . . . . . . . . . 13 |- ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A =/= 0 ) -> ( B x. 1 ) = B )
89	88	oveq2d 6666	. . . . . . . . . . . 12 |- ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A =/= 0 ) -> ( ( B x. k ) + ( B x. 1 ) ) = ( ( B x. k ) + B ) )
90	87, 89	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A =/= 0 ) -> ( B x. ( k + 1 ) ) = ( ( B x. k ) + B ) )
91	90	oveq2d 6666	. . . . . . . . . 10 |- ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A =/= 0 ) -> ( A ^c ( B x. ( k + 1 ) ) ) = ( A ^c ( ( B x. k ) + B ) ) )
92	57	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A =/= 0 ) -> A e. CC )
93		simpr 477	. . . . . . . . . . 11 |- ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A =/= 0 ) -> A =/= 0 )
94	60	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A =/= 0 ) -> ( B x. k ) e. CC )
95		cxpadd 24425	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B x. k ) e. CC /\ B e. CC ) -> ( A ^c ( ( B x. k ) + B ) ) = ( ( A ^c ( B x. k ) ) x. ( A ^c B ) ) )
96	92, 93, 94, 84, 95	syl211anc 1332	. . . . . . . . . 10 |- ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A =/= 0 ) -> ( A ^c ( ( B x. k ) + B ) ) = ( ( A ^c ( B x. k ) ) x. ( A ^c B ) ) )
97	91, 96	eqtrd 2656	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ A =/= 0 ) -> ( A ^c ( B x. ( k + 1 ) ) ) = ( ( A ^c ( B x. k ) ) x. ( A ^c B ) ) )
98	83, 97	pm2.61dane 2881	. . . . . . . 8 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> ( A ^c ( B x. ( k + 1 ) ) ) = ( ( A ^c ( B x. k ) ) x. ( A ^c B ) ) )
99		expp1 12867	. . . . . . . . 9 |- ( ( ( A ^c B ) e. CC /\ k e. NN0 ) -> ( ( A ^c B ) ^ ( k + 1 ) ) = ( ( ( A ^c B ) ^ k ) x. ( A ^c B ) ) )
100	26, 99	sylan 488	. . . . . . . 8 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> ( ( A ^c B ) ^ ( k + 1 ) ) = ( ( ( A ^c B ) ^ k ) x. ( A ^c B ) ) )
101	98, 100	eqeq12d 2637	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> ( ( A ^c ( B x. ( k + 1 ) ) ) = ( ( A ^c B ) ^ ( k + 1 ) ) <-> ( ( A ^c ( B x. k ) ) x. ( A ^c B ) ) = ( ( ( A ^c B ) ^ k ) x. ( A ^c B ) ) ) )
102	29, 101	syl5ibr 236	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> ( ( A ^c ( B x. k ) ) = ( ( A ^c B ) ^ k ) -> ( A ^c ( B x. ( k + 1 ) ) ) = ( ( A ^c B ) ^ ( k + 1 ) ) ) )
103	102	expcom 451	. . . . 5 |- ( k e. NN0 -> ( ( A e. CC /\ B e. CC ) -> ( ( A ^c ( B x. k ) ) = ( ( A ^c B ) ^ k ) -> ( A ^c ( B x. ( k + 1 ) ) ) = ( ( A ^c B ) ^ ( k + 1 ) ) ) ) )
104	103	a2d 29	. . . 4 |- ( k e. NN0 -> ( ( ( A e. CC /\ B e. CC ) -> ( A ^c ( B x. k ) ) = ( ( A ^c B ) ^ k ) ) -> ( ( A e. CC /\ B e. CC ) -> ( A ^c ( B x. ( k + 1 ) ) ) = ( ( A ^c B ) ^ ( k + 1 ) ) ) ) )
105	5, 10, 15, 20, 28, 104	nn0ind 11472	. . 3 |- ( C e. NN0 -> ( ( A e. CC /\ B e. CC ) -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^ C ) ) )
106	105	com12 32	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( C e. NN0 -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^ C ) ) )
107	106	3impia 1261	1 |- ( ( A e. CC /\ B e. CC /\ C e. NN0 ) -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^ C ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   NNcn 11020   NN0cn0 11292   ^cexp 12860   ^c ccxp 24302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304

This theorem is referenced by:  cxproot  24436  cxpmul2z  24437  cxpmul2d  24455