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Theorem congr 11708
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 11429 . . 3  |-  ( ( M  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( A  mod  M
)  =  ( B  mod  M )  <->  M  ||  ( A  -  B )
) )
213coml 1173 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN )  ->  (
( A  mod  M
)  =  ( B  mod  M )  <->  M  ||  ( A  -  B )
) )
3 simp3 968 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN )  ->  M  e.  NN )
43nnzd 9140 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN )  ->  M  e.  ZZ )
5 zsubcl 9063 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B
)  e.  ZZ )
653adant3 986 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN )  ->  ( A  -  B )  e.  ZZ )
7 divides 11422 . . 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 947    = wceq 1316    e. wcel 1465   E.wrex 2394   class class class wbr 3899  (class class class)co 5742    x. cmul 7593    - cmin 7901   NNcn 8688   ZZcz 9022    mod cmo 10063    || cdvds 11420
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-po 4188  df-iso 4189  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-n0 8946  df-z 9023  df-q 9380  df-rp 9410  df-fl 10011  df-mod 10064  df-dvds 11421
This theorem is referenced by:  cncongr1  11711
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