Theorem modqexp 10643

Description: Exponentiation property of the modulo operation, see theorem 5.2(c) in [ApostolNT] p. 107. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 7-Sep-2024.)

Hypotheses

Ref	Expression
modqexp.a	|- ( ph -> A e. ZZ )
modqexp.b	|- ( ph -> B e. ZZ )
modqexp.c	|- ( ph -> C e. NN0 )
modqexp.dq	|- ( ph -> D e. QQ )
modqexp.dgt0	|- ( ph -> 0 < D )
modqexp.mod	|- ( ph -> ( A mod D ) = ( B mod D ) )

Assertion

Ref	Expression
modqexp	|- ( ph -> ( ( A ^ C ) mod D ) = ( ( B ^ C ) mod D ) )

Proof of Theorem modqexp

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

Step	Hyp	Ref	Expression
1		modqexp.c	. 2 |- ( ph -> C e. NN0 )
2		oveq2 5882	. . . . . 6 |- ( w = 0 -> ( A ^ w ) = ( A ^ 0 ) )
3	2	oveq1d 5889	. . . . 5 |- ( w = 0 -> ( ( A ^ w ) mod D ) = ( ( A ^ 0 ) mod D ) )
4		oveq2 5882	. . . . . 6 |- ( w = 0 -> ( B ^ w ) = ( B ^ 0 ) )
5	4	oveq1d 5889	. . . . 5 |- ( w = 0 -> ( ( B ^ w ) mod D ) = ( ( B ^ 0 ) mod D ) )
6	3, 5	eqeq12d 2192	. . . 4 |- ( w = 0 -> ( ( ( A ^ w ) mod D ) = ( ( B ^ w ) mod D ) <-> ( ( A ^ 0 ) mod D ) = ( ( B ^ 0 ) mod D ) ) )
7	6	imbi2d 230	. . 3 |- ( w = 0 -> ( ( ph -> ( ( A ^ w ) mod D ) = ( ( B ^ w ) mod D ) ) <-> ( ph -> ( ( A ^ 0 ) mod D ) = ( ( B ^ 0 ) mod D ) ) ) )
8		oveq2 5882	. . . . . 6 |- ( w = k -> ( A ^ w ) = ( A ^ k ) )
9	8	oveq1d 5889	. . . . 5 |- ( w = k -> ( ( A ^ w ) mod D ) = ( ( A ^ k ) mod D ) )
10		oveq2 5882	. . . . . 6 |- ( w = k -> ( B ^ w ) = ( B ^ k ) )
11	10	oveq1d 5889	. . . . 5 |- ( w = k -> ( ( B ^ w ) mod D ) = ( ( B ^ k ) mod D ) )
12	9, 11	eqeq12d 2192	. . . 4 |- ( w = k -> ( ( ( A ^ w ) mod D ) = ( ( B ^ w ) mod D ) <-> ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) )
13	12	imbi2d 230	. . 3 |- ( w = k -> ( ( ph -> ( ( A ^ w ) mod D ) = ( ( B ^ w ) mod D ) ) <-> ( ph -> ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) ) )
14		oveq2 5882	. . . . . 6 |- ( w = ( k + 1 ) -> ( A ^ w ) = ( A ^ ( k + 1 ) ) )
15	14	oveq1d 5889	. . . . 5 |- ( w = ( k + 1 ) -> ( ( A ^ w ) mod D ) = ( ( A ^ ( k + 1 ) ) mod D ) )
16		oveq2 5882	. . . . . 6 |- ( w = ( k + 1 ) -> ( B ^ w ) = ( B ^ ( k + 1 ) ) )
17	16	oveq1d 5889	. . . . 5 |- ( w = ( k + 1 ) -> ( ( B ^ w ) mod D ) = ( ( B ^ ( k + 1 ) ) mod D ) )
18	15, 17	eqeq12d 2192	. . . 4 |- ( w = ( k + 1 ) -> ( ( ( A ^ w ) mod D ) = ( ( B ^ w ) mod D ) <-> ( ( A ^ ( k + 1 ) ) mod D ) = ( ( B ^ ( k + 1 ) ) mod D ) ) )
19	18	imbi2d 230	. . 3 |- ( w = ( k + 1 ) -> ( ( ph -> ( ( A ^ w ) mod D ) = ( ( B ^ w ) mod D ) ) <-> ( ph -> ( ( A ^ ( k + 1 ) ) mod D ) = ( ( B ^ ( k + 1 ) ) mod D ) ) ) )
20		oveq2 5882	. . . . . 6 |- ( w = C -> ( A ^ w ) = ( A ^ C ) )
21	20	oveq1d 5889	. . . . 5 |- ( w = C -> ( ( A ^ w ) mod D ) = ( ( A ^ C ) mod D ) )
22		oveq2 5882	. . . . . 6 |- ( w = C -> ( B ^ w ) = ( B ^ C ) )
23	22	oveq1d 5889	. . . . 5 |- ( w = C -> ( ( B ^ w ) mod D ) = ( ( B ^ C ) mod D ) )
24	21, 23	eqeq12d 2192	. . . 4 |- ( w = C -> ( ( ( A ^ w ) mod D ) = ( ( B ^ w ) mod D ) <-> ( ( A ^ C ) mod D ) = ( ( B ^ C ) mod D ) ) )
25	24	imbi2d 230	. . 3 |- ( w = C -> ( ( ph -> ( ( A ^ w ) mod D ) = ( ( B ^ w ) mod D ) ) <-> ( ph -> ( ( A ^ C ) mod D ) = ( ( B ^ C ) mod D ) ) ) )
26		modqexp.a	. . . . . . 7 |- ( ph -> A e. ZZ )
27	26	zcnd 9374	. . . . . 6 |- ( ph -> A e. CC )
28		exp0 10521	. . . . . 6 |- ( A e. CC -> ( A ^ 0 ) = 1 )
29	27, 28	syl 14	. . . . 5 |- ( ph -> ( A ^ 0 ) = 1 )
30		modqexp.b	. . . . . . 7 |- ( ph -> B e. ZZ )
31	30	zcnd 9374	. . . . . 6 |- ( ph -> B e. CC )
32		exp0 10521	. . . . . 6 |- ( B e. CC -> ( B ^ 0 ) = 1 )
33	31, 32	syl 14	. . . . 5 |- ( ph -> ( B ^ 0 ) = 1 )
34	29, 33	eqtr4d 2213	. . . 4 |- ( ph -> ( A ^ 0 ) = ( B ^ 0 ) )
35	34	oveq1d 5889	. . 3 |- ( ph -> ( ( A ^ 0 ) mod D ) = ( ( B ^ 0 ) mod D ) )
36		zexpcl 10532	. . . . . . . . . 10 |- ( ( A e. ZZ /\ k e. NN0 ) -> ( A ^ k ) e. ZZ )
37	26, 36	sylan 283	. . . . . . . . 9 |- ( ( ph /\ k e. NN0 ) -> ( A ^ k ) e. ZZ )
38	37	adantr 276	. . . . . . . 8 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( A ^ k ) e. ZZ )
39		zexpcl 10532	. . . . . . . . . 10 |- ( ( B e. ZZ /\ k e. NN0 ) -> ( B ^ k ) e. ZZ )
40	30, 39	sylan 283	. . . . . . . . 9 |- ( ( ph /\ k e. NN0 ) -> ( B ^ k ) e. ZZ )
41	40	adantr 276	. . . . . . . 8 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( B ^ k ) e. ZZ )
42	26	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> A e. ZZ )
43	30	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> B e. ZZ )
44		modqexp.dq	. . . . . . . . 9 |- ( ph -> D e. QQ )
45	44	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> D e. QQ )
46		modqexp.dgt0	. . . . . . . . 9 |- ( ph -> 0 < D )
47	46	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> 0 < D )
48		simpr 110	. . . . . . . 8 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) )
49		modqexp.mod	. . . . . . . . 9 |- ( ph -> ( A mod D ) = ( B mod D ) )
50	49	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( A mod D ) = ( B mod D ) )
51	38, 41, 42, 43, 45, 47, 48, 50	modqmul12d 10375	. . . . . . 7 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( ( ( A ^ k ) x. A ) mod D ) = ( ( ( B ^ k ) x. B ) mod D ) )
52	27	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> A e. CC )
53		simpr 110	. . . . . . . . . 10 |- ( ( ph /\ k e. NN0 ) -> k e. NN0 )
54	53	adantr 276	. . . . . . . . 9 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> k e. NN0 )
55		expp1 10524	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
56	52, 54, 55	syl2anc 411	. . . . . . . 8 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
57	56	oveq1d 5889	. . . . . . 7 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( ( A ^ ( k + 1 ) ) mod D ) = ( ( ( A ^ k ) x. A ) mod D ) )
58	31	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> B e. CC )
59		expp1 10524	. . . . . . . . 9 |- ( ( B e. CC /\ k e. NN0 ) -> ( B ^ ( k + 1 ) ) = ( ( B ^ k ) x. B ) )
60	58, 54, 59	syl2anc 411	. . . . . . . 8 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( B ^ ( k + 1 ) ) = ( ( B ^ k ) x. B ) )
61	60	oveq1d 5889	. . . . . . 7 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( ( B ^ ( k + 1 ) ) mod D ) = ( ( ( B ^ k ) x. B ) mod D ) )
62	51, 57, 61	3eqtr4d 2220	. . . . . 6 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( ( A ^ ( k + 1 ) ) mod D ) = ( ( B ^ ( k + 1 ) ) mod D ) )
63	62	ex 115	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> ( ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) -> ( ( A ^ ( k + 1 ) ) mod D ) = ( ( B ^ ( k + 1 ) ) mod D ) ) )
64	63	expcom 116	. . . 4 |- ( k e. NN0 -> ( ph -> ( ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) -> ( ( A ^ ( k + 1 ) ) mod D ) = ( ( B ^ ( k + 1 ) ) mod D ) ) ) )
65	64	a2d 26	. . 3 |- ( k e. NN0 -> ( ( ph -> ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( ph -> ( ( A ^ ( k + 1 ) ) mod D ) = ( ( B ^ ( k + 1 ) ) mod D ) ) ) )
66	7, 13, 19, 25, 35, 65	nn0ind 9365	. 2 |- ( C e. NN0 -> ( ph -> ( ( A ^ C ) mod D ) = ( ( B ^ C ) mod D ) ) )
67	1, 66	mpcom 36	1 |- ( ph -> ( ( A ^ C ) mod D ) = ( ( B ^ C ) mod D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   class class class wbr 4003  (class class class)co 5874   CCcc 7808   0cc0 7810   1c1 7811   + caddc 7813   x. cmul 7815   < clt 7990   NN0cn0 9174   ZZcz 9251   QQcq 9617   mod cmo 10319   ^cexp 10516

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-n0 9175  df-z 9252  df-uz 9527  df-q 9618  df-rp 9652  df-fl 10267  df-mod 10320  df-seqfrec 10443  df-exp 10517

This theorem is referenced by:  dvdsmodexp  11797  odzdvds  12239  lgsmod  14320  lgsne0  14332