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Theorem divalgmodcl 26956
Description: divalgmod 12889 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 2418 . . . . . . 7  |-  ( a  =  R  ->  (
a  =  ( N  mod  D )  <->  R  =  ( N  mod  D ) ) )
2 eleq1 2472 . . . . . . . 8  |-  ( a  =  R  ->  (
a  e.  NN0  <->  R  e.  NN0 ) )
3 breq1 4183 . . . . . . . . 9  |-  ( a  =  R  ->  (
a  <  D  <->  R  <  D ) )
4 oveq2 6056 . . . . . . . . . 10  |-  ( a  =  R  ->  ( N  -  a )  =  ( N  -  R ) )
54breq2d 4192 . . . . . . . . 9  |-  ( a  =  R  ->  ( D  ||  ( N  -  a )  <->  D  ||  ( N  -  R )
) )
63, 5anbi12d 692 . . . . . . . 8  |-  ( a  =  R  ->  (
( a  <  D  /\  D  ||  ( N  -  a ) )  <-> 
( R  <  D  /\  D  ||  ( N  -  R ) ) ) )
72, 6anbi12d 692 . . . . . . 7  |-  ( a  =  R  ->  (
( a  e.  NN0  /\  ( a  <  D  /\  D  ||  ( N  -  a ) ) )  <->  ( R  e. 
NN0  /\  ( R  <  D  /\  D  ||  ( N  -  R
) ) ) ) )
81, 7bibi12d 313 . . . . . 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 308 . . . . 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 12889 . . . . 5  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( a  =  ( N  mod  D )  <-> 
( a  e.  NN0  /\  ( a  <  D  /\  D  ||  ( N  -  a ) ) ) ) )
119, 10vtoclg 2979 . . . 4  |-  ( R  e.  NN0  ->  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( R  =  ( N  mod  D )  <-> 
( R  e.  NN0  /\  ( R  <  D  /\  D  ||  ( N  -  R ) ) ) ) ) )
1211impcom 420 . . 3  |-  ( ( ( N  e.  ZZ  /\  D  e.  NN )  /\  R  e.  NN0 )  ->  ( R  =  ( N  mod  D
)  <->  ( R  e. 
NN0  /\  ( R  <  D  /\  D  ||  ( N  -  R
) ) ) ) )
13 ibar 491 . . . 4  |-  ( R  e.  NN0  ->  ( ( R  <  D  /\  D  ||  ( N  -  R ) )  <->  ( R  e.  NN0  /\  ( R  <  D  /\  D  ||  ( N  -  R
) ) ) ) )
1413adantl 453 . . 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 248 . 2  |-  ( ( ( N  e.  ZZ  /\  D  e.  NN )  /\  R  e.  NN0 )  ->  ( R  =  ( N  mod  D
)  <->  ( R  < 
D  /\  D  ||  ( N  -  R )
) ) )
16153impa 1148 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4180  (class class class)co 6048    < clt 9084    - cmin 9255   NNcn 9964   NN0cn0 10185   ZZcz 10246    mod cmo 11213    || cdivides 12815
This theorem is referenced by:  jm3.1  26989  expdiophlem1  26990
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-fz 11008  df-fl 11165  df-mod 11214  df-seq 11287  df-exp 11346  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-dvds 12816
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