Theorem dvdsmodexp 12581

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 13035). (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 12578	. . 3 |- ( N || A -> ( N e. ZZ /\ A e. ZZ ) )
2		dvdsmod0 12579	. . . . . . . . 9 |- ( ( N e. NN /\ N || A ) -> ( A mod N ) = 0 )
3	2	3ad2antl2 1191	. . . . . . . 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 1033	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> B e. NN )
6	5	0expd 11142	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> ( 0 ^ B ) = 0 )
7	6	oveq1d 6100	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> ( ( 0 ^ B ) mod N ) = ( 0 mod N ) )
8		simpl1 1031	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> A e. ZZ )
9		0zd 9661	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> 0 e. ZZ )
10		nnnn0 9575	. . . . . . . . . . . 12 |- ( B e. NN -> B e. NN0 )
11	10	3ad2ant3 1051	. . . . . . . . . . 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 1032	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> N e. NN )
14		nnq 10043	. . . . . . . . . . 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 10075	. . . . . . . . . . . . 13 |- ( N e. NN -> N e. RR+ )
17	16	3ad2ant2 1050	. . . . . . . . . . . 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 10111	. . . . . . . . . 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 10807	. . . . . . . . . . . 12 |- ( ( N e. QQ /\ 0 < N ) -> ( 0 mod N ) = 0 )
22	15, 19, 21	syl2anc 415	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. NN /\ B e. NN ) /\ ( A mod N ) = 0 ) -> ( 0 mod N ) = 0 )
23	20, 22	eqtr4d 2274	. . . . . . . . . 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 11119	. . . . . . . . 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 2281	. . . . . . . 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 1233	. . . . 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 1227	1 |- ( ( N e. NN /\ B e. NN /\ N || A ) -> ( ( A ^ B ) mod N ) = ( A mod N ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4130  (class class class)co 6085   0cc0 8180   < clt 8361   NNcn 9307   NN0cn0 9568   ZZcz 9649   QQcq 10029   RR+crp 10065   mod cmo 10774   ^cexp 10990   || cdvds 12573

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296  ax-pre-mulgt0 8297  ax-pre-mulext 8298  ax-arch 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-reap 8906  df-ap 8913  df-div 9006  df-inn 9308  df-n0 9569  df-z 9650  df-uz 9932  df-q 10030  df-rp 10066  df-fl 10716  df-mod 10775  df-seqfrec 10900  df-exp 10991  df-dvds 12574

This theorem is used by:  fermltl  13035