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Theorem congr 10862
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 10585 . . 3  |-  ( ( M  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( A  mod  M
)  =  ( B  mod  M )  <->  M  ||  ( A  -  B )
) )
213coml 1146 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN )  ->  (
( A  mod  M
)  =  ( B  mod  M )  <->  M  ||  ( A  -  B )
) )
3 simp3 941 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN )  ->  M  e.  NN )
43nnzd 8763 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN )  ->  M  e.  ZZ )
5 zsubcl 8687 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B
)  e.  ZZ )
653adant3 959 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN )  ->  ( A  -  B )  e.  ZZ )
7 divides 10578 . . 3  |-  ( ( M  e.  ZZ  /\  ( A  -  B
)  e.  ZZ )  ->  ( M  ||  ( A  -  B
)  <->  E. n  e.  ZZ  ( n  x.  M
)  =  ( A  -  B ) ) )
84, 6, 7syl2anc 403 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN )  ->  ( M  ||  ( A  -  B )  <->  E. n  e.  ZZ  ( n  x.  M )  =  ( A  -  B ) ) )
92, 8bitrd 186 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 103    /\ w3a 920    = wceq 1285    e. wcel 1434   E.wrex 2354   class class class wbr 3811  (class class class)co 5591    x. cmul 7258    - cmin 7556   NNcn 8316   ZZcz 8646    mod cmo 9618    || cdvds 10576
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-cnex 7339  ax-resscn 7340  ax-1cn 7341  ax-1re 7342  ax-icn 7343  ax-addcl 7344  ax-addrcl 7345  ax-mulcl 7346  ax-mulrcl 7347  ax-addcom 7348  ax-mulcom 7349  ax-addass 7350  ax-mulass 7351  ax-distr 7352  ax-i2m1 7353  ax-0lt1 7354  ax-1rid 7355  ax-0id 7356  ax-rnegex 7357  ax-precex 7358  ax-cnre 7359  ax-pre-ltirr 7360  ax-pre-ltwlin 7361  ax-pre-lttrn 7362  ax-pre-apti 7363  ax-pre-ltadd 7364  ax-pre-mulgt0 7365  ax-pre-mulext 7366  ax-arch 7367
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4084  df-po 4087  df-iso 4088  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-fv 4977  df-riota 5547  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-pnf 7427  df-mnf 7428  df-xr 7429  df-ltxr 7430  df-le 7431  df-sub 7558  df-neg 7559  df-reap 7952  df-ap 7959  df-div 8038  df-inn 8317  df-n0 8566  df-z 8647  df-q 9000  df-rp 9030  df-fl 9566  df-mod 9619  df-dvds 10577
This theorem is referenced by:  cncongr1  10865
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