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Theorem congr 12041
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 11748 . . 3  |-  ( ( M  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( A  mod  M
)  =  ( B  mod  M )  <->  M  ||  ( A  -  B )
) )
213coml 1205 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN )  ->  (
( A  mod  M
)  =  ( B  mod  M )  <->  M  ||  ( A  -  B )
) )
3 simp3 994 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN )  ->  M  e.  NN )
43nnzd 9320 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN )  ->  M  e.  ZZ )
5 zsubcl 9240 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B
)  e.  ZZ )
653adant3 1012 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN )  ->  ( A  -  B )  e.  ZZ )
7 divides 11738 . . 3  |-  ( ( M  e.  ZZ  /\  ( A  -  B
)  e.  ZZ )  ->  ( M  ||  ( A  -  B
)  <->  E. n  e.  ZZ  ( n  x.  M
)  =  ( A  -  B ) ) )
84, 6, 7syl2anc 409 . 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 973    = wceq 1348    e. wcel 2141   E.wrex 2449   class class class wbr 3987  (class class class)co 5850    x. cmul 7766    - cmin 8077   NNcn 8865   ZZcz 9199    mod cmo 10265    || cdvds 11736
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879  ax-arch 7880
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-po 4279  df-iso 4280  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577  df-inn 8866  df-n0 9123  df-z 9200  df-q 9566  df-rp 9598  df-fl 10213  df-mod 10266  df-dvds 11737
This theorem is referenced by:  cncongr1  12044
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