Theorem congr 12102

Description: Definition of congruence by integer multiple (see ProofWiki "Congruence (Number Theory)", 11-Jul-2021, https://proofwiki.org/wiki/Definition:Congruence_(Number_Theory)): An integer A is congruent to an integer B modulo M if their difference is a multiple of M. See also the definition in [ApostolNT] p. 104: "... a is congruent to b modulo m, and we write a == b (mod m) if m divides the difference a - b", or Wikipedia "Modular arithmetic - Congruence", https://en.wikipedia.org/wiki/Modular_arithmetic#Congruence, 11-Jul-2021,: "Given an integer n > 1, called a modulus, two integers are said to be congruent modulo n, if n is a divisor of their difference (i.e., if there is an integer k such that a-b = kn)". (Contributed by AV, 11-Jul-2021.)

Assertion

Ref	Expression
congr	|- ( ( A e. ZZ /\ B e. ZZ /\ M e. NN ) -> ( ( A mod M ) = ( B mod M ) <-> E. n e. ZZ ( n x. M ) = ( A - B ) ) )

Distinct variable groups:   A, n   B, n   n, M

Proof of Theorem congr

Step	Hyp	Ref	Expression
1		moddvds 11808	. . 3 |- ( ( M e. NN /\ A e. ZZ /\ B e. ZZ ) -> ( ( A mod M ) = ( B mod M ) <-> M || ( A - B ) ) )
2	1	3coml 1210	. 2 |- ( ( A e. ZZ /\ B e. ZZ /\ M e. NN ) -> ( ( A mod M ) = ( B mod M ) <-> M || ( A - B ) ) )
3		simp3 999	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ /\ M e. NN ) -> M e. NN )
4	3	nnzd 9376	. . 3 |- ( ( A e. ZZ /\ B e. ZZ /\ M e. NN ) -> M e. ZZ )
5		zsubcl 9296	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A - B ) e. ZZ )
6	5	3adant3 1017	. . 3 |- ( ( A e. ZZ /\ B e. ZZ /\ M e. NN ) -> ( A - B ) e. ZZ )
7		divides 11798	. . 3 |- ( ( M e. ZZ /\ ( A - B ) e. ZZ ) -> ( M || ( A - B ) <-> E. n e. ZZ ( n x. M ) = ( A - B ) ) )
8	4, 6, 7	syl2anc 411	. 2 |- ( ( A e. ZZ /\ B e. ZZ /\ M e. NN ) -> ( M || ( A - B ) <-> E. n e. ZZ ( n x. M ) = ( A - B ) ) )
9	2, 8	bitrd 188	1 |- ( ( A e. ZZ /\ B e. ZZ /\ M e. NN ) -> ( ( A mod M ) = ( B mod M ) <-> E. n e. ZZ ( n x. M ) = ( A - B ) ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   E.wrex 2456   class class class wbr 4005  (class class class)co 5877   x. cmul 7818   - cmin 8130   NNcn 8921   ZZcz 9255   mod cmo 10324   || 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-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  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-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-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-id 4295  df-po 4298  df-iso 4299  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-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  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-q 9622  df-rp 9656  df-fl 10272  df-mod 10325  df-dvds 11797

This theorem is referenced by:  cncongr1  12105