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Theorem divalgmodcl 27096
Description: divalgmod 12957 using a class variable. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Assertion
Ref Expression
divalgmodcl  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  R  e.  NN0 )  ->  ( R  =  ( N  mod  D )  <->  ( R  <  D  /\  D  ||  ( N  -  R
) ) ) )

Proof of Theorem divalgmodcl
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2448 . . . . . . 7  |-  ( a  =  R  ->  (
a  =  ( N  mod  D )  <->  R  =  ( N  mod  D ) ) )
2 eleq1 2502 . . . . . . . 8  |-  ( a  =  R  ->  (
a  e.  NN0  <->  R  e.  NN0 ) )
3 breq1 4240 . . . . . . . . 9  |-  ( a  =  R  ->  (
a  <  D  <->  R  <  D ) )
4 oveq2 6118 . . . . . . . . . 10  |-  ( a  =  R  ->  ( N  -  a )  =  ( N  -  R ) )
54breq2d 4249 . . . . . . . . 9  |-  ( a  =  R  ->  ( D  ||  ( N  -  a )  <->  D  ||  ( N  -  R )
) )
63, 5anbi12d 693 . . . . . . . 8  |-  ( a  =  R  ->  (
( a  <  D  /\  D  ||  ( N  -  a ) )  <-> 
( R  <  D  /\  D  ||  ( N  -  R ) ) ) )
72, 6anbi12d 693 . . . . . . 7  |-  ( a  =  R  ->  (
( a  e.  NN0  /\  ( a  <  D  /\  D  ||  ( N  -  a ) ) )  <->  ( R  e. 
NN0  /\  ( R  <  D  /\  D  ||  ( N  -  R
) ) ) ) )
81, 7bibi12d 314 . . . . . 6  |-  ( a  =  R  ->  (
( a  =  ( N  mod  D )  <-> 
( a  e.  NN0  /\  ( a  <  D  /\  D  ||  ( N  -  a ) ) ) )  <->  ( R  =  ( N  mod  D )  <->  ( R  e. 
NN0  /\  ( R  <  D  /\  D  ||  ( N  -  R
) ) ) ) ) )
98imbi2d 309 . . . . 5  |-  ( a  =  R  ->  (
( ( N  e.  ZZ  /\  D  e.  NN )  ->  (
a  =  ( N  mod  D )  <->  ( a  e.  NN0  /\  ( a  <  D  /\  D  ||  ( N  -  a
) ) ) ) )  <->  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( R  =  ( N  mod  D )  <->  ( R  e.  NN0  /\  ( R  <  D  /\  D  ||  ( N  -  R
) ) ) ) ) ) )
10 divalgmod 12957 . . . . 5  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( a  =  ( N  mod  D )  <-> 
( a  e.  NN0  /\  ( a  <  D  /\  D  ||  ( N  -  a ) ) ) ) )
119, 10vtoclg 3017 . . . 4  |-  ( R  e.  NN0  ->  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( R  =  ( N  mod  D )  <-> 
( R  e.  NN0  /\  ( R  <  D  /\  D  ||  ( N  -  R ) ) ) ) ) )
1211impcom 421 . . 3  |-  ( ( ( N  e.  ZZ  /\  D  e.  NN )  /\  R  e.  NN0 )  ->  ( R  =  ( N  mod  D
)  <->  ( R  e. 
NN0  /\  ( R  <  D  /\  D  ||  ( N  -  R
) ) ) ) )
13 ibar 492 . . . 4  |-  ( R  e.  NN0  ->  ( ( R  <  D  /\  D  ||  ( N  -  R ) )  <->  ( R  e.  NN0  /\  ( R  <  D  /\  D  ||  ( N  -  R
) ) ) ) )
1413adantl 454 . . 3  |-  ( ( ( N  e.  ZZ  /\  D  e.  NN )  /\  R  e.  NN0 )  ->  ( ( R  <  D  /\  D  ||  ( N  -  R
) )  <->  ( R  e.  NN0  /\  ( R  <  D  /\  D  ||  ( N  -  R
) ) ) ) )
1512, 14bitr4d 249 . 2  |-  ( ( ( N  e.  ZZ  /\  D  e.  NN )  /\  R  e.  NN0 )  ->  ( R  =  ( N  mod  D
)  <->  ( R  < 
D  /\  D  ||  ( N  -  R )
) ) )
16153impa 1149 1  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  R  e.  NN0 )  ->  ( R  =  ( N  mod  D )  <->  ( R  <  D  /\  D  ||  ( N  -  R
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727   class class class wbr 4237  (class class class)co 6110    < clt 9151    - cmin 9322   NNcn 10031   NN0cn0 10252   ZZcz 10313    mod cmo 11281    || cdivides 12883
This theorem is referenced by:  jm3.1  27129  expdiophlem1  27130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098  ax-pre-sup 9099
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-sup 7475  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709  df-nn 10032  df-2 10089  df-3 10090  df-n0 10253  df-z 10314  df-uz 10520  df-rp 10644  df-fz 11075  df-fl 11233  df-mod 11282  df-seq 11355  df-exp 11414  df-cj 11935  df-re 11936  df-im 11937  df-sqr 12071  df-abs 12072  df-dvds 12884
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