Theorem modqexp 10537

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 5829	. . . . . 6 |- ( w = 0 -> ( A ^ w ) = ( A ^ 0 ) )
3	2	oveq1d 5836	. . . . 5 |- ( w = 0 -> ( ( A ^ w ) mod D ) = ( ( A ^ 0 ) mod D ) )
4		oveq2 5829	. . . . . 6 |- ( w = 0 -> ( B ^ w ) = ( B ^ 0 ) )
5	4	oveq1d 5836	. . . . 5 |- ( w = 0 -> ( ( B ^ w ) mod D ) = ( ( B ^ 0 ) mod D ) )
6	3, 5	eqeq12d 2172	. . . 4 |- ( w = 0 -> ( ( ( A ^ w ) mod D ) = ( ( B ^ w ) mod D ) <-> ( ( A ^ 0 ) mod D ) = ( ( B ^ 0 ) mod D ) ) )
7	6	imbi2d 229	. . 3 |- ( w = 0 -> ( ( ph -> ( ( A ^ w ) mod D ) = ( ( B ^ w ) mod D ) ) <-> ( ph -> ( ( A ^ 0 ) mod D ) = ( ( B ^ 0 ) mod D ) ) ) )
8		oveq2 5829	. . . . . 6 |- ( w = k -> ( A ^ w ) = ( A ^ k ) )
9	8	oveq1d 5836	. . . . 5 |- ( w = k -> ( ( A ^ w ) mod D ) = ( ( A ^ k ) mod D ) )
10		oveq2 5829	. . . . . 6 |- ( w = k -> ( B ^ w ) = ( B ^ k ) )
11	10	oveq1d 5836	. . . . 5 |- ( w = k -> ( ( B ^ w ) mod D ) = ( ( B ^ k ) mod D ) )
12	9, 11	eqeq12d 2172	. . . 4 |- ( w = k -> ( ( ( A ^ w ) mod D ) = ( ( B ^ w ) mod D ) <-> ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) )
13	12	imbi2d 229	. . 3 |- ( w = k -> ( ( ph -> ( ( A ^ w ) mod D ) = ( ( B ^ w ) mod D ) ) <-> ( ph -> ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) ) )
14		oveq2 5829	. . . . . 6 |- ( w = ( k + 1 ) -> ( A ^ w ) = ( A ^ ( k + 1 ) ) )
15	14	oveq1d 5836	. . . . 5 |- ( w = ( k + 1 ) -> ( ( A ^ w ) mod D ) = ( ( A ^ ( k + 1 ) ) mod D ) )
16		oveq2 5829	. . . . . 6 |- ( w = ( k + 1 ) -> ( B ^ w ) = ( B ^ ( k + 1 ) ) )
17	16	oveq1d 5836	. . . . 5 |- ( w = ( k + 1 ) -> ( ( B ^ w ) mod D ) = ( ( B ^ ( k + 1 ) ) mod D ) )
18	15, 17	eqeq12d 2172	. . . 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 229	. . 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 5829	. . . . . 6 |- ( w = C -> ( A ^ w ) = ( A ^ C ) )
21	20	oveq1d 5836	. . . . 5 |- ( w = C -> ( ( A ^ w ) mod D ) = ( ( A ^ C ) mod D ) )
22		oveq2 5829	. . . . . 6 |- ( w = C -> ( B ^ w ) = ( B ^ C ) )
23	22	oveq1d 5836	. . . . 5 |- ( w = C -> ( ( B ^ w ) mod D ) = ( ( B ^ C ) mod D ) )
24	21, 23	eqeq12d 2172	. . . 4 |- ( w = C -> ( ( ( A ^ w ) mod D ) = ( ( B ^ w ) mod D ) <-> ( ( A ^ C ) mod D ) = ( ( B ^ C ) mod D ) ) )
25	24	imbi2d 229	. . 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 9281	. . . . . 6 |- ( ph -> A e. CC )
28		exp0 10416	. . . . . 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 9281	. . . . . 6 |- ( ph -> B e. CC )
32		exp0 10416	. . . . . 6 |- ( B e. CC -> ( B ^ 0 ) = 1 )
33	31, 32	syl 14	. . . . 5 |- ( ph -> ( B ^ 0 ) = 1 )
34	29, 33	eqtr4d 2193	. . . 4 |- ( ph -> ( A ^ 0 ) = ( B ^ 0 ) )
35	34	oveq1d 5836	. . 3 |- ( ph -> ( ( A ^ 0 ) mod D ) = ( ( B ^ 0 ) mod D ) )
36		zexpcl 10427	. . . . . . . . . 10 |- ( ( A e. ZZ /\ k e. NN0 ) -> ( A ^ k ) e. ZZ )
37	26, 36	sylan 281	. . . . . . . . 9 |- ( ( ph /\ k e. NN0 ) -> ( A ^ k ) e. ZZ )
38	37	adantr 274	. . . . . . . 8 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( A ^ k ) e. ZZ )
39		zexpcl 10427	. . . . . . . . . 10 |- ( ( B e. ZZ /\ k e. NN0 ) -> ( B ^ k ) e. ZZ )
40	30, 39	sylan 281	. . . . . . . . 9 |- ( ( ph /\ k e. NN0 ) -> ( B ^ k ) e. ZZ )
41	40	adantr 274	. . . . . . . 8 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( B ^ k ) e. ZZ )
42	26	ad2antrr 480	. . . . . . . 8 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> A e. ZZ )
43	30	ad2antrr 480	. . . . . . . 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 480	. . . . . . . 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 480	. . . . . . . 8 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> 0 < D )
48		simpr 109	. . . . . . . 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 480	. . . . . . . 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 10270	. . . . . . 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 480	. . . . . . . . 9 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> A e. CC )
53		simpr 109	. . . . . . . . . 10 |- ( ( ph /\ k e. NN0 ) -> k e. NN0 )
54	53	adantr 274	. . . . . . . . 9 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> k e. NN0 )
55		expp1 10419	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
56	52, 54, 55	syl2anc 409	. . . . . . . 8 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
57	56	oveq1d 5836	. . . . . . 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 480	. . . . . . . . 9 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> B e. CC )
59		expp1 10419	. . . . . . . . 9 |- ( ( B e. CC /\ k e. NN0 ) -> ( B ^ ( k + 1 ) ) = ( ( B ^ k ) x. B ) )
60	58, 54, 59	syl2anc 409	. . . . . . . 8 |- ( ( ( ph /\ k e. NN0 ) /\ ( ( A ^ k ) mod D ) = ( ( B ^ k ) mod D ) ) -> ( B ^ ( k + 1 ) ) = ( ( B ^ k ) x. B ) )
61	60	oveq1d 5836	. . . . . . 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 2200	. . . . . 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 114	. . . . 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 115	. . . 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 9272	. 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 103   = wceq 1335   e. wcel 2128   class class class wbr 3965  (class class class)co 5821   CCcc 7724   0cc0 7726   1c1 7727   + caddc 7729   x. cmul 7731   < clt 7906   NN0cn0 9084   ZZcz 9161   QQcq 9521   mod cmo 10214   ^cexp 10411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546  ax-cnex 7817  ax-resscn 7818  ax-1cn 7819  ax-1re 7820  ax-icn 7821  ax-addcl 7822  ax-addrcl 7823  ax-mulcl 7824  ax-mulrcl 7825  ax-addcom 7826  ax-mulcom 7827  ax-addass 7828  ax-mulass 7829  ax-distr 7830  ax-i2m1 7831  ax-0lt1 7832  ax-1rid 7833  ax-0id 7834  ax-rnegex 7835  ax-precex 7836  ax-cnre 7837  ax-pre-ltirr 7838  ax-pre-ltwlin 7839  ax-pre-lttrn 7840  ax-pre-apti 7841  ax-pre-ltadd 7842  ax-pre-mulgt0 7843  ax-pre-mulext 7844  ax-arch 7845

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-riota 5777  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-frec 6335  df-pnf 7908  df-mnf 7909  df-xr 7910  df-ltxr 7911  df-le 7912  df-sub 8042  df-neg 8043  df-reap 8444  df-ap 8451  df-div 8540  df-inn 8828  df-n0 9085  df-z 9162  df-uz 9434  df-q 9522  df-rp 9554  df-fl 10162  df-mod 10215  df-seqfrec 10338  df-exp 10412

This theorem is referenced by:  dvdsmodexp  11684