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Theorem congr 11770
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
StepHypRef Expression
1 moddvds 11491 . . 3  |-  ( ( M  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( A  mod  M
)  =  ( B  mod  M )  <->  M  ||  ( A  -  B )
) )
213coml 1188 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN )  ->  (
( A  mod  M
)  =  ( B  mod  M )  <->  M  ||  ( A  -  B )
) )
3 simp3 983 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN )  ->  M  e.  NN )
43nnzd 9165 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN )  ->  M  e.  ZZ )
5 zsubcl 9088 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B
)  e.  ZZ )
653adant3 1001 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN )  ->  ( A  -  B )  e.  ZZ )
7 divides 11484 . . 3  |-  ( ( M  e.  ZZ  /\  ( A  -  B
)  e.  ZZ )  ->  ( M  ||  ( A  -  B
)  <->  E. n  e.  ZZ  ( n  x.  M
)  =  ( A  -  B ) ) )
84, 6, 7syl2anc 408 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN )  ->  ( M  ||  ( A  -  B )  <->  E. n  e.  ZZ  ( n  x.  M )  =  ( A  -  B ) ) )
92, 8bitrd 187 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 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   E.wrex 2415   class class class wbr 3924  (class class class)co 5767    x. cmul 7618    - cmin 7926   NNcn 8713   ZZcz 9047    mod cmo 10088    || cdvds 11482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-po 4213  df-iso 4214  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-n0 8971  df-z 9048  df-q 9405  df-rp 9435  df-fl 10036  df-mod 10089  df-dvds 11483
This theorem is referenced by:  cncongr1  11773
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