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Theorem divalglem4 12595
Description: Lemma for divalg 12602. (Contributed by Paul Chapman, 21-Mar-2011.)
Hypotheses
Ref Expression
divalglem0.1  |-  N  e.  ZZ
divalglem0.2  |-  D  e.  ZZ
divalglem1.3  |-  D  =/=  0
divalglem2.4  |-  S  =  { r  e.  NN0  |  D  ||  ( N  -  r ) }
Assertion
Ref Expression
divalglem4  |-  S  =  { r  e.  NN0  |  E. q  e.  ZZ  N  =  ( (
q  x.  D )  +  r ) }
Distinct variable groups:    D, r    N, r    D, q, r    N, q
Allowed substitution hints:    S( r, q)

Proof of Theorem divalglem4
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 divalglem0.2 . . . . . 6  |-  D  e.  ZZ
2 divalglem0.1 . . . . . . 7  |-  N  e.  ZZ
3 nn0z 10046 . . . . . . 7  |-  ( z  e.  NN0  ->  z  e.  ZZ )
4 zsubcl 10061 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  z  e.  ZZ )  ->  ( N  -  z
)  e.  ZZ )
52, 3, 4sylancr 644 . . . . . 6  |-  ( z  e.  NN0  ->  ( N  -  z )  e.  ZZ )
6 divides 12533 . . . . . 6  |-  ( ( D  e.  ZZ  /\  ( N  -  z
)  e.  ZZ )  ->  ( D  ||  ( N  -  z
)  <->  E. q  e.  ZZ  ( q  x.  D
)  =  ( N  -  z ) ) )
71, 5, 6sylancr 644 . . . . 5  |-  ( z  e.  NN0  ->  ( D 
||  ( N  -  z )  <->  E. q  e.  ZZ  ( q  x.  D )  =  ( N  -  z ) ) )
8 nn0cn 9975 . . . . . . . 8  |-  ( z  e.  NN0  ->  z  e.  CC )
9 zmulcl 10066 . . . . . . . . . 10  |-  ( ( q  e.  ZZ  /\  D  e.  ZZ )  ->  ( q  x.  D
)  e.  ZZ )
101, 9mpan2 652 . . . . . . . . 9  |-  ( q  e.  ZZ  ->  (
q  x.  D )  e.  ZZ )
1110zcnd 10118 . . . . . . . 8  |-  ( q  e.  ZZ  ->  (
q  x.  D )  e.  CC )
12 zcn 10029 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  CC )
132, 12ax-mp 8 . . . . . . . . . 10  |-  N  e.  CC
14 subadd 9054 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  z  e.  CC  /\  (
q  x.  D )  e.  CC )  -> 
( ( N  -  z )  =  ( q  x.  D )  <-> 
( z  +  ( q  x.  D ) )  =  N ) )
1513, 14mp3an1 1264 . . . . . . . . 9  |-  ( ( z  e.  CC  /\  ( q  x.  D
)  e.  CC )  ->  ( ( N  -  z )  =  ( q  x.  D
)  <->  ( z  +  ( q  x.  D
) )  =  N ) )
16 addcom 8998 . . . . . . . . . 10  |-  ( ( z  e.  CC  /\  ( q  x.  D
)  e.  CC )  ->  ( z  +  ( q  x.  D
) )  =  ( ( q  x.  D
)  +  z ) )
1716eqeq1d 2291 . . . . . . . . 9  |-  ( ( z  e.  CC  /\  ( q  x.  D
)  e.  CC )  ->  ( ( z  +  ( q  x.  D ) )  =  N  <->  ( ( q  x.  D )  +  z )  =  N ) )
1815, 17bitrd 244 . . . . . . . 8  |-  ( ( z  e.  CC  /\  ( q  x.  D
)  e.  CC )  ->  ( ( N  -  z )  =  ( q  x.  D
)  <->  ( ( q  x.  D )  +  z )  =  N ) )
198, 11, 18syl2an 463 . . . . . . 7  |-  ( ( z  e.  NN0  /\  q  e.  ZZ )  ->  ( ( N  -  z )  =  ( q  x.  D )  <-> 
( ( q  x.  D )  +  z )  =  N ) )
20 eqcom 2285 . . . . . . 7  |-  ( ( N  -  z )  =  ( q  x.  D )  <->  ( q  x.  D )  =  ( N  -  z ) )
21 eqcom 2285 . . . . . . 7  |-  ( ( ( q  x.  D
)  +  z )  =  N  <->  N  =  ( ( q  x.  D )  +  z ) )
2219, 20, 213bitr3g 278 . . . . . 6  |-  ( ( z  e.  NN0  /\  q  e.  ZZ )  ->  ( ( q  x.  D )  =  ( N  -  z )  <-> 
N  =  ( ( q  x.  D )  +  z ) ) )
2322rexbidva 2560 . . . . 5  |-  ( z  e.  NN0  ->  ( E. q  e.  ZZ  (
q  x.  D )  =  ( N  -  z )  <->  E. q  e.  ZZ  N  =  ( ( q  x.  D
)  +  z ) ) )
247, 23bitrd 244 . . . 4  |-  ( z  e.  NN0  ->  ( D 
||  ( N  -  z )  <->  E. q  e.  ZZ  N  =  ( ( q  x.  D
)  +  z ) ) )
2524pm5.32i 618 . . 3  |-  ( ( z  e.  NN0  /\  D  ||  ( N  -  z ) )  <->  ( z  e.  NN0  /\  E. q  e.  ZZ  N  =  ( ( q  x.  D
)  +  z ) ) )
26 oveq2 5866 . . . . 5  |-  ( r  =  z  ->  ( N  -  r )  =  ( N  -  z ) )
2726breq2d 4035 . . . 4  |-  ( r  =  z  ->  ( D  ||  ( N  -  r )  <->  D  ||  ( N  -  z )
) )
28 divalglem2.4 . . . 4  |-  S  =  { r  e.  NN0  |  D  ||  ( N  -  r ) }
2927, 28elrab2 2925 . . 3  |-  ( z  e.  S  <->  ( z  e.  NN0  /\  D  ||  ( N  -  z
) ) )
30 oveq2 5866 . . . . . 6  |-  ( r  =  z  ->  (
( q  x.  D
)  +  r )  =  ( ( q  x.  D )  +  z ) )
3130eqeq2d 2294 . . . . 5  |-  ( r  =  z  ->  ( N  =  ( (
q  x.  D )  +  r )  <->  N  =  ( ( q  x.  D )  +  z ) ) )
3231rexbidv 2564 . . . 4  |-  ( r  =  z  ->  ( E. q  e.  ZZ  N  =  ( (
q  x.  D )  +  r )  <->  E. q  e.  ZZ  N  =  ( ( q  x.  D
)  +  z ) ) )
3332elrab 2923 . . 3  |-  ( z  e.  { r  e. 
NN0  |  E. q  e.  ZZ  N  =  ( ( q  x.  D
)  +  r ) }  <->  ( z  e. 
NN0  /\  E. q  e.  ZZ  N  =  ( ( q  x.  D
)  +  z ) ) )
3425, 29, 333bitr4i 268 . 2  |-  ( z  e.  S  <->  z  e.  { r  e.  NN0  |  E. q  e.  ZZ  N  =  ( (
q  x.  D )  +  r ) } )
3534eqriv 2280 1  |-  S  =  { r  e.  NN0  |  E. q  e.  ZZ  N  =  ( (
q  x.  D )  +  r ) }
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   {crab 2547   class class class wbr 4023  (class class class)co 5858   CCcc 8735   0cc0 8737    + caddc 8740    x. cmul 8742    - cmin 9037   NN0cn0 9965   ZZcz 10024    || cdivides 12531
This theorem is referenced by:  divalglem10  12601
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-dvds 12532
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