Theorem modexp 12999

Description: Exponentiation property of the modulo operation, see theorem 5.2(c) in [ApostolNT] p. 107. (Contributed by Mario Carneiro, 28-Feb-2014.)

Assertion

Ref	Expression
modexp	|- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. RR+ ) /\ ( A mod D ) = ( B mod D ) ) -> ( ( A ^ C ) mod D ) = ( ( B ^ C ) mod D ) )

Proof of Theorem modexp

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

Step	Hyp	Ref	Expression
1		simp2l 1087	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. RR+ ) /\ ( A mod D ) = ( B mod D ) ) -> C e. NN0 )
2		id 22	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) -> ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) )
3	2	3adant2l 1320	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. RR+ ) /\ ( A mod D ) = ( B mod D ) ) -> ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) )
4		oveq2 6658	. . . . . 6 |- ( x = 0 -> ( A ^ x ) = ( A ^ 0 ) )
5	4	oveq1d 6665	. . . . 5 |- ( x = 0 -> ( ( A ^ x ) mod D ) = ( ( A ^ 0 ) mod D ) )
6		oveq2 6658	. . . . . 6 |- ( x = 0 -> ( B ^ x ) = ( B ^ 0 ) )
7	6	oveq1d 6665	. . . . 5 |- ( x = 0 -> ( ( B ^ x ) mod D ) = ( ( B ^ 0 ) mod D ) )
8	5, 7	eqeq12d 2637	. . . 4 |- ( x = 0 -> ( ( ( A ^ x ) mod D ) = ( ( B ^ x ) mod D ) <-> ( ( A ^ 0 ) mod D ) = ( ( B ^ 0 ) mod D ) ) )
9	8	imbi2d 330	. . 3 |- ( x = 0 -> ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) -> ( ( A ^ x ) mod D ) = ( ( B ^ x ) mod D ) ) <-> ( ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) -> ( ( A ^ 0 ) mod D ) = ( ( B ^ 0 ) mod D ) ) ) )
10		oveq2 6658	. . . . . 6 |- ( x = k -> ( A ^ x ) = ( A ^ k ) )
11	10	oveq1d 6665	. . . . 5 |- ( x = k -> ( ( A ^ x ) mod D ) = ( ( A ^ k ) mod D ) )
12		oveq2 6658	. . . . . 6 |- ( x = k -> ( B ^ x ) = ( B ^ k ) )
13	12	oveq1d 6665	. . . . 5 |- ( x = k -> ( ( B ^ x ) mod D ) = ( ( B ^ k ) mod D ) )
14	11, 13	eqeq12d 2637	. . . 4 |- ( x = k -> ( ( ( A ^ x ) mod D ) = ( ( B ^ x ) mod D ) <-> ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) )
15	14	imbi2d 330	. . 3 |- ( x = k -> ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) -> ( ( A ^ x ) mod D ) = ( ( B ^ x ) mod D ) ) <-> ( ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) -> ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) ) )
16		oveq2 6658	. . . . . 6 |- ( x = ( k + 1 ) -> ( A ^ x ) = ( A ^ ( k + 1 ) ) )
17	16	oveq1d 6665	. . . . 5 |- ( x = ( k + 1 ) -> ( ( A ^ x ) mod D ) = ( ( A ^ ( k + 1 ) ) mod D ) )
18		oveq2 6658	. . . . . 6 |- ( x = ( k + 1 ) -> ( B ^ x ) = ( B ^ ( k + 1 ) ) )
19	18	oveq1d 6665	. . . . 5 |- ( x = ( k + 1 ) -> ( ( B ^ x ) mod D ) = ( ( B ^ ( k + 1 ) ) mod D ) )
20	17, 19	eqeq12d 2637	. . . 4 |- ( x = ( k + 1 ) -> ( ( ( A ^ x ) mod D ) = ( ( B ^ x ) mod D ) <-> ( ( A ^ ( k + 1 ) ) mod D ) = ( ( B ^ ( k + 1 ) ) mod D ) ) )
21	20	imbi2d 330	. . 3 |- ( x = ( k + 1 ) -> ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) -> ( ( A ^ x ) mod D ) = ( ( B ^ x ) mod D ) ) <-> ( ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) -> ( ( A ^ ( k + 1 ) ) mod D ) = ( ( B ^ ( k + 1 ) ) mod D ) ) ) )
22		oveq2 6658	. . . . . 6 |- ( x = C -> ( A ^ x ) = ( A ^ C ) )
23	22	oveq1d 6665	. . . . 5 |- ( x = C -> ( ( A ^ x ) mod D ) = ( ( A ^ C ) mod D ) )
24		oveq2 6658	. . . . . 6 |- ( x = C -> ( B ^ x ) = ( B ^ C ) )
25	24	oveq1d 6665	. . . . 5 |- ( x = C -> ( ( B ^ x ) mod D ) = ( ( B ^ C ) mod D ) )
26	23, 25	eqeq12d 2637	. . . 4 |- ( x = C -> ( ( ( A ^ x ) mod D ) = ( ( B ^ x ) mod D ) <-> ( ( A ^ C ) mod D ) = ( ( B ^ C ) mod D ) ) )
27	26	imbi2d 330	. . 3 |- ( x = C -> ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) -> ( ( A ^ x ) mod D ) = ( ( B ^ x ) mod D ) ) <-> ( ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) -> ( ( A ^ C ) mod D ) = ( ( B ^ C ) mod D ) ) ) )
28		zcn 11382	. . . . . . 7 |- ( A e. ZZ -> A e. CC )
29		exp0 12864	. . . . . . 7 |- ( A e. CC -> ( A ^ 0 ) = 1 )
30	28, 29	syl 17	. . . . . 6 |- ( A e. ZZ -> ( A ^ 0 ) = 1 )
31		zcn 11382	. . . . . . . 8 |- ( B e. ZZ -> B e. CC )
32		exp0 12864	. . . . . . . 8 |- ( B e. CC -> ( B ^ 0 ) = 1 )
33	31, 32	syl 17	. . . . . . 7 |- ( B e. ZZ -> ( B ^ 0 ) = 1 )
34	33	eqcomd 2628	. . . . . 6 |- ( B e. ZZ -> 1 = ( B ^ 0 ) )
35	30, 34	sylan9eq 2676	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A ^ 0 ) = ( B ^ 0 ) )
36	35	oveq1d 6665	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A ^ 0 ) mod D ) = ( ( B ^ 0 ) mod D ) )
37	36	3ad2ant1 1082	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) -> ( ( A ^ 0 ) mod D ) = ( ( B ^ 0 ) mod D ) )
38		simp21l 1178	. . . . . . . 8 |- ( ( k e. NN0 /\ ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> A e. ZZ )
39		simp1 1061	. . . . . . . 8 |- ( ( k e. NN0 /\ ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> k e. NN0 )
40		zexpcl 12875	. . . . . . . 8 |- ( ( A e. ZZ /\ k e. NN0 ) -> ( A ^ k ) e. ZZ )
41	38, 39, 40	syl2anc 693	. . . . . . 7 |- ( ( k e. NN0 /\ ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( A ^ k ) e. ZZ )
42		simp21r 1179	. . . . . . . 8 |- ( ( k e. NN0 /\ ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> B e. ZZ )
43		zexpcl 12875	. . . . . . . 8 |- ( ( B e. ZZ /\ k e. NN0 ) -> ( B ^ k ) e. ZZ )
44	42, 39, 43	syl2anc 693	. . . . . . 7 |- ( ( k e. NN0 /\ ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( B ^ k ) e. ZZ )
45		simp22 1095	. . . . . . 7 |- ( ( k e. NN0 /\ ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> D e. RR+ )
46		simp3 1063	. . . . . . 7 |- ( ( k e. NN0 /\ ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) )
47		simp23 1096	. . . . . . 7 |- ( ( k e. NN0 /\ ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( A mod D ) = ( B mod D ) )
48	41, 44, 38, 42, 45, 46, 47	modmul12d 12724	. . . . . 6 |- ( ( k e. NN0 /\ ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( ( ( A ^ k ) x. A ) mod D ) = ( ( ( B ^ k ) x. B ) mod D ) )
49	38	zcnd 11483	. . . . . . . 8 |- ( ( k e. NN0 /\ ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> A e. CC )
50		expp1 12867	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
51	49, 39, 50	syl2anc 693	. . . . . . 7 |- ( ( k e. NN0 /\ ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
52	51	oveq1d 6665	. . . . . 6 |- ( ( k e. NN0 /\ ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( ( A ^ ( k + 1 ) ) mod D ) = ( ( ( A ^ k ) x. A ) mod D ) )
53	42	zcnd 11483	. . . . . . . 8 |- ( ( k e. NN0 /\ ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> B e. CC )
54		expp1 12867	. . . . . . . 8 |- ( ( B e. CC /\ k e. NN0 ) -> ( B ^ ( k + 1 ) ) = ( ( B ^ k ) x. B ) )
55	53, 39, 54	syl2anc 693	. . . . . . 7 |- ( ( k e. NN0 /\ ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( B ^ ( k + 1 ) ) = ( ( B ^ k ) x. B ) )
56	55	oveq1d 6665	. . . . . 6 |- ( ( k e. NN0 /\ ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( ( B ^ ( k + 1 ) ) mod D ) = ( ( ( B ^ k ) x. B ) mod D ) )
57	48, 52, 56	3eqtr4d 2666	. . . . 5 |- ( ( k e. NN0 /\ ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( ( A ^ ( k + 1 ) ) mod D ) = ( ( B ^ ( k + 1 ) ) mod D ) )
58	57	3exp 1264	. . . 4 |- ( k e. NN0 -> ( ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) -> ( ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) -> ( ( A ^ ( k + 1 ) ) mod D ) = ( ( B ^ ( k + 1 ) ) mod D ) ) ) )
59	58	a2d 29	. . 3 |- ( k e. NN0 -> ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) -> ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) -> ( ( A ^ ( k + 1 ) ) mod D ) = ( ( B ^ ( k + 1 ) ) mod D ) ) ) )
60	9, 15, 21, 27, 37, 59	nn0ind 11472	. 2 |- ( C e. NN0 -> ( ( ( A e. ZZ /\ B e. ZZ ) /\ D e. RR+ /\ ( A mod D ) = ( B mod D ) ) -> ( ( A ^ C ) mod D ) = ( ( B ^ C ) mod D ) ) )
61	1, 3, 60	sylc 65	1 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. RR+ ) /\ ( A mod D ) = ( B mod D ) ) -> ( ( A ^ C ) mod D ) = ( ( B ^ C ) mod D ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   NN0cn0 11292   ZZcz 11377   RR+crp 11832   mod cmo 12668   ^cexp 12860

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-iun 4522  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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861

This theorem is referenced by:  fermltl  15489  odzdvds  15500  lgslem4  25025  lgsmod  25048  lgsne0  25060  fmtnoprmfac1lem  41476  sfprmdvdsmersenne  41520  41prothprmlem2  41535