Theorem congr 12422

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 12110	. . 3 |- ( ( M e. NN /\ A e. ZZ /\ B e. ZZ ) -> ( ( A mod M ) = ( B mod M ) <-> M || ( A - B ) ) )
2	1	3coml 1213	. 2 |- ( ( A e. ZZ /\ B e. ZZ /\ M e. NN ) -> ( ( A mod M ) = ( B mod M ) <-> M || ( A - B ) ) )
3		simp3 1002	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ /\ M e. NN ) -> M e. NN )
4	3	nnzd 9494	. . 3 |- ( ( A e. ZZ /\ B e. ZZ /\ M e. NN ) -> M e. ZZ )
5		zsubcl 9413	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A - B ) e. ZZ )
6	5	3adant3 1020	. . 3 |- ( ( A e. ZZ /\ B e. ZZ /\ M e. NN ) -> ( A - B ) e. ZZ )
7		divides 12100	. . 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 981   = wceq 1373   e. wcel 2176   E.wrex 2485   class class class wbr 4044  (class class class)co 5944   x. cmul 7930   - cmin 8243   NNcn 9036   ZZcz 9372   mod cmo 10467   || cdvds 12098

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-po 4343  df-iso 4344  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-n0 9296  df-z 9373  df-q 9741  df-rp 9776  df-fl 10413  df-mod 10468  df-dvds 12099

This theorem is referenced by:  cncongr1  12425