Theorem dvdsmodexp 11804

Description: If a positive integer divides another integer, this other integer is equal to its positive powers modulo the positive integer. (Formerly part of the proof for fermltl 12236). (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by AV, 19-Mar-2022.)

Assertion

Ref	Expression
dvdsmodexp	|- ( ( N e. NN /\ B e. NN /\ N || A ) -> ( ( A ^ B ) mod N ) = ( A mod N ) )

Proof of Theorem dvdsmodexp

Step	Hyp	Ref	Expression
1		dvdszrcl 11801	. . 3 |- ( N || A -> ( N e. ZZ /\ A e. ZZ ) )
2		dvdsmod0 11802	. . . . . . . . 9 |- ( ( N e. NN /\ N || A ) -> ( A mod N ) = 0 )
3	2	3ad2antl2 1160	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ N || A ) -> ( A mod N ) = 0 )
4	3	ex 115	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN /\ B e. NN ) -> ( N || A -> ( A mod N ) = 0 ) )
5		simpl3 1002	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> B e. NN )
6	5	0expd 10672	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> ( 0 ^ B ) = 0 )
7	6	oveq1d 5892	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> ( ( 0 ^ B ) mod N ) = ( 0 mod N ) )
8		simpl1 1000	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> A e. ZZ )
9		0zd 9267	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> 0 e. ZZ )
10		nnnn0 9185	. . . . . . . . . . . 12 |- ( B e. NN -> B e. NN0 )
11	10	3ad2ant3 1020	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ N e. NN /\ B e. NN ) -> B e. NN0 )
12	11	adantr 276	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> B e. NN0 )
13		simpl2 1001	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> N e. NN )
14		nnq 9635	. . . . . . . . . . 11 |- ( N e. NN -> N e. QQ )
15	13, 14	syl 14	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> N e. QQ )
16		nnrp 9665	. . . . . . . . . . . . 13 |- ( N e. NN -> N e. RR+ )
17	16	3ad2ant2 1019	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ N e. NN /\ B e. NN ) -> N e. RR+ )
18	17	adantr 276	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> N e. RR+ )
19	18	rpgt0d 9701	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> 0 < N )
20		simpr 110	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> ( A mod N ) = 0 )
21		q0mod 10357	. . . . . . . . . . . 12 |- ( ( N e. QQ /\ 0 < N ) -> ( 0 mod N ) = 0 )
22	15, 19, 21	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> ( 0 mod N ) = 0 )
23	20, 22	eqtr4d 2213	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> ( A mod N ) = ( 0 mod N ) )
24	8, 9, 12, 15, 19, 23	modqexp 10649	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> ( ( A ^ B ) mod N ) = ( ( 0 ^ B ) mod N ) )
25	7, 24, 23	3eqtr4d 2220	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> ( ( A ^ B ) mod N ) = ( A mod N ) )
26	25	ex 115	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN /\ B e. NN ) -> ( ( A mod N ) = 0 -> ( ( A ^ B ) mod N ) = ( A mod N ) ) )
27	4, 26	syld 45	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN /\ B e. NN ) -> ( N || A -> ( ( A ^ B ) mod N ) = ( A mod N ) ) )
28	27	3exp 1202	. . . . 5 |- ( A e. ZZ -> ( N e. NN -> ( B e. NN -> ( N || A -> ( ( A ^ B ) mod N ) = ( A mod N ) ) ) ) )
29	28	com24 87	. . . 4 |- ( A e. ZZ -> ( N || A -> ( B e. NN -> ( N e. NN -> ( ( A ^ B ) mod N ) = ( A mod N ) ) ) ) )
30	29	adantl 277	. . 3 |- ( ( N e. ZZ /\ A e. ZZ ) -> ( N || A -> ( B e. NN -> ( N e. NN -> ( ( A ^ B ) mod N ) = ( A mod N ) ) ) ) )
31	1, 30	mpcom 36	. 2 |- ( N || A -> ( B e. NN -> ( N e. NN -> ( ( A ^ B ) mod N ) = ( A mod N ) ) ) )
32	31	3imp31 1196	1 |- ( ( N e. NN /\ B e. NN /\ N || A ) -> ( ( A ^ B ) mod N ) = ( A mod N ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   class class class wbr 4005  (class class class)co 5877   0cc0 7813   < clt 7994   NNcn 8921   NN0cn0 9178   ZZcz 9255   QQcq 9621   RR+crp 9655   mod cmo 10324   ^cexp 10521   || cdvds 11796

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 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fl 10272  df-mod 10325  df-seqfrec 10448  df-exp 10522  df-dvds 11797

This theorem is referenced by:  fermltl  12236