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Theorem moddvds 14991
Description: Two ways to say A == B (mod N), see also definition in [ApostolNT] p. 106. (Contributed by Mario Carneiro, 18-Feb-2014.)
Assertion
Ref Expression
moddvds |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> ( ( A mod N ) = ( B mod N ) <-> N || ( A - B ) ) )
Proof of Theorem moddvds
StepHypRef Expression
1 nnrp 11842 . . . . . 6 |- ( N e. NN -> N e. RR+ )
21adantr 481 . . . . 5 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> N e. RR+ )
3 0mod 12701 . . . . 5 |- ( N e. RR+ -> ( 0 mod N ) = 0 )
42, 3syl 17 . . . 4 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( 0 mod N ) = 0 )
54eqeq2d 2632 . . 3 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( ( ( A - B ) mod N ) = ( 0 mod N ) <-> ( ( A - B ) mod N ) = 0 ) )
6 zre 11381 . . . . . . 7 |- ( A e. ZZ -> A e. RR )
76ad2antrl 764 . . . . . 6 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> A e. RR )
8 zre 11381 . . . . . . 7 |- ( B e. ZZ -> B e. RR )
98ad2antll 765 . . . . . 6 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> B e. RR )
109renegcld 10457 . . . . . 6 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> -u B e. RR )
11 modadd1 12707 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -u B e. RR /\ N e. RR+ ) /\ ( A mod N ) = ( B mod N ) ) -> ( ( A + -u B ) mod N ) = ( ( B + -u B ) mod N ) )
12113expia 1267 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -u B e. RR /\ N e. RR+ ) ) -> ( ( A mod N ) = ( B mod N ) -> ( ( A + -u B ) mod N ) = ( ( B + -u B ) mod N ) ) )
137, 9, 10, 2, 12syl22anc 1327 . . . . 5 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( ( A mod N ) = ( B mod N ) -> ( ( A + -u B ) mod N ) = ( ( B + -u B ) mod N ) ) )
147recnd 10068 . . . . . . . 8 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> A e. CC )
159recnd 10068 . . . . . . . 8 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> B e. CC )
1614, 15negsubd 10398 . . . . . . 7 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( A + -u B ) = ( A - B ) )
1716oveq1d 6665 . . . . . 6 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( ( A + -u B ) mod N ) = ( ( A - B ) mod N ) )
1815negidd 10382 . . . . . . 7 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( B + -u B ) = 0 )
1918oveq1d 6665 . . . . . 6 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( ( B + -u B ) mod N ) = ( 0 mod N ) )
2017, 19eqeq12d 2637 . . . . 5 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( ( ( A + -u B ) mod N ) = ( ( B + -u B ) mod N ) <-> ( ( A - B ) mod N ) = ( 0 mod N ) ) )
2113, 20sylibd 229 . . . 4 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( ( A mod N ) = ( B mod N ) -> ( ( A - B ) mod N ) = ( 0 mod N ) ) )
227, 9resubcld 10458 . . . . . 6 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( A - B ) e. RR )
23 0red 10041 . . . . . 6 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> 0 e. RR )
24 modadd1 12707 . . . . . . 7 |- ( ( ( ( A - B ) e. RR /\ 0 e. RR ) /\ ( B e. RR /\ N e. RR+ ) /\ ( ( A - B ) mod N ) = ( 0 mod N ) ) -> ( ( ( A - B ) + B ) mod N ) = ( ( 0 + B ) mod N ) )
25243expia 1267 . . . . . 6 |- ( ( ( ( A - B ) e. RR /\ 0 e. RR ) /\ ( B e. RR /\ N e. RR+ ) ) -> ( ( ( A - B ) mod N ) = ( 0 mod N ) -> ( ( ( A - B ) + B ) mod N ) = ( ( 0 + B ) mod N ) ) )
2622, 23, 9, 2, 25syl22anc 1327 . . . . 5 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( ( ( A - B ) mod N ) = ( 0 mod N ) -> ( ( ( A - B ) + B ) mod N ) = ( ( 0 + B ) mod N ) ) )
2714, 15npcand 10396 . . . . . . 7 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( ( A - B ) + B ) = A )
2827oveq1d 6665 . . . . . 6 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( ( ( A - B ) + B ) mod N ) = ( A mod N ) )
2915addid2d 10237 . . . . . . 7 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( 0 + B ) = B )
3029oveq1d 6665 . . . . . 6 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( ( 0 + B ) mod N ) = ( B mod N ) )
3128, 30eqeq12d 2637 . . . . 5 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( ( ( ( A - B ) + B ) mod N ) = ( ( 0 + B ) mod N ) <-> ( A mod N ) = ( B mod N ) ) )
3226, 31sylibd 229 . . . 4 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( ( ( A - B ) mod N ) = ( 0 mod N ) -> ( A mod N ) = ( B mod N ) ) )
3321, 32impbid 202 . . 3 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( ( A mod N ) = ( B mod N ) <-> ( ( A - B ) mod N ) = ( 0 mod N ) ) )
34 zsubcl 11419 . . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A - B ) e. ZZ )
35 dvdsval3 14987 . . . 4 |- ( ( N e. NN /\ ( A - B ) e. ZZ ) -> ( N || ( A - B ) <-> ( ( A - B ) mod N ) = 0 ) )
3634, 35sylan2 491 . . 3 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( N || ( A - B ) <-> ( ( A - B ) mod N ) = 0 ) )
375, 33, 363bitr4d 300 . 2 |- ( ( N e. NN /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( ( A mod N ) = ( B mod N ) <-> N || ( A - B ) ) )
38373impb 1260 1 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> ( ( A mod N ) = ( B mod N ) <-> N || ( A - B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   RRcr 9935   0cc0 9936   + caddc 9939   - cmin 10266   -ucneg 10267   NNcn 11020   ZZcz 11377   RR+crp 11832   mod cmo 12668   || cdvds 14983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fl 12593  df-mod 12669  df-dvds 14984
This theorem is referenced by:  summodnegmod  15012  modmulconst  15013  addmodlteqALT  15047  dvdsmod  15050  sadadd3  15183  sadaddlem  15188  congr  15378  cncongr1  15381  cncongr2  15382  crth  15483  eulerthlem2  15487  prmdiv  15490  prmdiveq  15491  odzcllem  15497  odzdvds  15500  odzphi  15501  modprm1div  15502  pockthlem  15609  4sqlem11  15659  4sqlem12  15660  mndodcong  17961  dfod2  17981  sylow3lem6  18047  znf1o  19900  wilthlem1  24794  wilthlem2  24795  wilthlem3  24796  wilthimp  24798  ppiub  24929  lgslem1  25022  lgsmod  25048  lgsdirprm  25056  lgsqrlem1  25071  lgsqrlem2  25072  lgsqr  25076  lgsqrmod  25077  lgsqrmodndvds  25078  lgsdchrval  25079  lgseisenlem2  25101  lgseisenlem3  25102  lgseisenlem4  25103  m1lgs  25113  sfprmdvdsmersenne  41520
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