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Theorem divalgmodcl 26403
Description: divalgmod 12696 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 2364 . . . . . . 7  |-  ( a  =  R  ->  (
a  =  ( N  mod  D )  <->  R  =  ( N  mod  D ) ) )
2 eleq1 2418 . . . . . . . 8  |-  ( a  =  R  ->  (
a  e.  NN0  <->  R  e.  NN0 ) )
3 breq1 4105 . . . . . . . . 9  |-  ( a  =  R  ->  (
a  <  D  <->  R  <  D ) )
4 oveq2 5950 . . . . . . . . . 10  |-  ( a  =  R  ->  ( N  -  a )  =  ( N  -  R ) )
54breq2d 4114 . . . . . . . . 9  |-  ( a  =  R  ->  ( D  ||  ( N  -  a )  <->  D  ||  ( N  -  R )
) )
63, 5anbi12d 691 . . . . . . . 8  |-  ( a  =  R  ->  (
( a  <  D  /\  D  ||  ( N  -  a ) )  <-> 
( R  <  D  /\  D  ||  ( N  -  R ) ) ) )
72, 6anbi12d 691 . . . . . . 7  |-  ( a  =  R  ->  (
( a  e.  NN0  /\  ( a  <  D  /\  D  ||  ( N  -  a ) ) )  <->  ( R  e. 
NN0  /\  ( R  <  D  /\  D  ||  ( N  -  R
) ) ) ) )
81, 7bibi12d 312 . . . . . 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 307 . . . . 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 12696 . . . . 5  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( a  =  ( N  mod  D )  <-> 
( a  e.  NN0  /\  ( a  <  D  /\  D  ||  ( N  -  a ) ) ) ) )
119, 10vtoclg 2919 . . . 4  |-  ( R  e.  NN0  ->  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( R  =  ( N  mod  D )  <-> 
( R  e.  NN0  /\  ( R  <  D  /\  D  ||  ( N  -  R ) ) ) ) ) )
1211impcom 419 . . 3  |-  ( ( ( N  e.  ZZ  /\  D  e.  NN )  /\  R  e.  NN0 )  ->  ( R  =  ( N  mod  D
)  <->  ( R  e. 
NN0  /\  ( R  <  D  /\  D  ||  ( N  -  R
) ) ) ) )
13 ibar 490 . . . 4  |-  ( R  e.  NN0  ->  ( ( R  <  D  /\  D  ||  ( N  -  R ) )  <->  ( R  e.  NN0  /\  ( R  <  D  /\  D  ||  ( N  -  R
) ) ) ) )
1413adantl 452 . . 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 247 . 2  |-  ( ( ( N  e.  ZZ  /\  D  e.  NN )  /\  R  e.  NN0 )  ->  ( R  =  ( N  mod  D
)  <->  ( R  < 
D  /\  D  ||  ( N  -  R )
) ) )
16153impa 1146 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 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   class class class wbr 4102  (class class class)co 5942    < clt 8954    - cmin 9124   NNcn 9833   NN0cn0 10054   ZZcz 10113    mod cmo 11062    || cdivides 12622
This theorem is referenced by:  jm3.1  26436  expdiophlem1  26437
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-sup 7281  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-n0 10055  df-z 10114  df-uz 10320  df-rp 10444  df-fz 10872  df-fl 11014  df-mod 11063  df-seq 11136  df-exp 11195  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-dvds 12623
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