Theorem modqexp 11053

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 6066	. . . . . 6 |- ( w = 0 -> ( A ^ w ) = ( A ^ 0 ) )
3	2	oveq1d 6073	. . . . 5 |- ( w = 0 -> ( ( A ^ w ) mod D ) = ( ( A ^ 0 ) mod D ) )
4		oveq2 6066	. . . . . 6 |- ( w = 0 -> ( B ^ w ) = ( B ^ 0 ) )
5	4	oveq1d 6073	. . . . 5 |- ( w = 0 -> ( ( B ^ w ) mod D ) = ( ( B ^ 0 ) mod D ) )
6	3, 5	eqeq12d 2249	. . . 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 6066	. . . . . 6 |- ( w = k -> ( A ^ w ) = ( A ^ k ) )
9	8	oveq1d 6073	. . . . 5 |- ( w = k -> ( ( A ^ w ) mod D ) = ( ( A ^ k ) mod D ) )
10		oveq2 6066	. . . . . 6 |- ( w = k -> ( B ^ w ) = ( B ^ k ) )
11	10	oveq1d 6073	. . . . 5 |- ( w = k -> ( ( B ^ w ) mod D ) = ( ( B ^ k ) mod D ) )
12	9, 11	eqeq12d 2249	. . . 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 6066	. . . . . 6 |- ( w = ( k + 1 ) -> ( A ^ w ) = ( A ^ ( k + 1 ) ) )
15	14	oveq1d 6073	. . . . 5 |- ( w = ( k + 1 ) -> ( ( A ^ w ) mod D ) = ( ( A ^ ( k + 1 ) ) mod D ) )
16		oveq2 6066	. . . . . 6 |- ( w = ( k + 1 ) -> ( B ^ w ) = ( B ^ ( k + 1 ) ) )
17	16	oveq1d 6073	. . . . 5 |- ( w = ( k + 1 ) -> ( ( B ^ w ) mod D ) = ( ( B ^ ( k + 1 ) ) mod D ) )
18	15, 17	eqeq12d 2249	. . . 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 6066	. . . . . 6 |- ( w = C -> ( A ^ w ) = ( A ^ C ) )
21	20	oveq1d 6073	. . . . 5 |- ( w = C -> ( ( A ^ w ) mod D ) = ( ( A ^ C ) mod D ) )
22		oveq2 6066	. . . . . 6 |- ( w = C -> ( B ^ w ) = ( B ^ C ) )
23	22	oveq1d 6073	. . . . 5 |- ( w = C -> ( ( B ^ w ) mod D ) = ( ( B ^ C ) mod D ) )
24	21, 23	eqeq12d 2249	. . . 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 9719	. . . . . 6 |- ( ph -> A e. CC )
28		exp0 10929	. . . . . 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 9719	. . . . . 6 |- ( ph -> B e. CC )
32		exp0 10929	. . . . . 6 |- ( B e. CC -> ( B ^ 0 ) = 1 )
33	31, 32	syl 14	. . . . 5 |- ( ph -> ( B ^ 0 ) = 1 )
34	29, 33	eqtr4d 2270	. . . 4 |- ( ph -> ( A ^ 0 ) = ( B ^ 0 ) )
35	34	oveq1d 6073	. . 3 |- ( ph -> ( ( A ^ 0 ) mod D ) = ( ( B ^ 0 ) mod D ) )
36		zexpcl 10940	. . . . . . . . . 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 10940	. . . . . . . . . 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 10764	. . . . . . 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 10932	. . . . . . . . 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 6073	. . . . . . 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 10932	. . . . . . . . 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 6073	. . . . . . 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 2277	. . . . . 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 9710	. 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 1398   e. wcel 2205   class class class wbr 4114  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   < clt 8324   NN0cn0 9513   ZZcz 9594   QQcq 9969   mod cmo 10708   ^cexp 10924

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925

This theorem is referenced by:  dvdsmodexp  12506  odzdvds  12968  lgsmod  16025  lgsne0  16037