Theorem dvdsmodexp 12321

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 12771). (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 12318	. . 3 |- ( N || A -> ( N e. ZZ /\ A e. ZZ ) )
2		dvdsmod0 12319	. . . . . . . . 9 |- ( ( N e. NN /\ N || A ) -> ( A mod N ) = 0 )
3	2	3ad2antl2 1184	. . . . . . . 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 1026	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> B e. NN )
6	5	0expd 10923	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> ( 0 ^ B ) = 0 )
7	6	oveq1d 6022	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> ( ( 0 ^ B ) mod N ) = ( 0 mod N ) )
8		simpl1 1024	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> A e. ZZ )
9		0zd 9469	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> 0 e. ZZ )
10		nnnn0 9387	. . . . . . . . . . . 12 |- ( B e. NN -> B e. NN0 )
11	10	3ad2ant3 1044	. . . . . . . . . . 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 1025	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> N e. NN )
14		nnq 9840	. . . . . . . . . . 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 9871	. . . . . . . . . . . . 13 |- ( N e. NN -> N e. RR+ )
17	16	3ad2ant2 1043	. . . . . . . . . . . 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 9907	. . . . . . . . . 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 10589	. . . . . . . . . . . 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 2265	. . . . . . . . . 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 10900	. . . . . . . . 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 2272	. . . . . . . 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 1226	. . . . 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 1220	1 |- ( ( N e. NN /\ B e. NN /\ N || A ) -> ( ( A ^ B ) mod N ) = ( A mod N ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4083  (class class class)co 6007   0cc0 8010   < clt 8192   NNcn 9121   NN0cn0 9380   ZZcz 9457   QQcq 9826   RR+crp 9861   mod cmo 10556   ^cexp 10772   || cdvds 12313

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-fl 10502  df-mod 10557  df-seqfrec 10682  df-exp 10773  df-dvds 12314

This theorem is referenced by:  fermltl  12771