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Theorem divalglem4 13600
Description: Lemma for divalg 13607. (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 10669 . . . . . . 7  |-  ( z  e.  NN0  ->  z  e.  ZZ )
4 zsubcl 10687 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  z  e.  ZZ )  ->  ( N  -  z
)  e.  ZZ )
52, 3, 4sylancr 663 . . . . . 6  |-  ( z  e.  NN0  ->  ( N  -  z )  e.  ZZ )
6 divides 13537 . . . . . 6  |-  ( ( D  e.  ZZ  /\  ( N  -  z
)  e.  ZZ )  ->  ( D  ||  ( N  -  z
)  <->  E. q  e.  ZZ  ( q  x.  D
)  =  ( N  -  z ) ) )
71, 5, 6sylancr 663 . . . . 5  |-  ( z  e.  NN0  ->  ( D 
||  ( N  -  z )  <->  E. q  e.  ZZ  ( q  x.  D )  =  ( N  -  z ) ) )
8 nn0cn 10589 . . . . . . . 8  |-  ( z  e.  NN0  ->  z  e.  CC )
9 zmulcl 10693 . . . . . . . . . 10  |-  ( ( q  e.  ZZ  /\  D  e.  ZZ )  ->  ( q  x.  D
)  e.  ZZ )
101, 9mpan2 671 . . . . . . . . 9  |-  ( q  e.  ZZ  ->  (
q  x.  D )  e.  ZZ )
1110zcnd 10748 . . . . . . . 8  |-  ( q  e.  ZZ  ->  (
q  x.  D )  e.  CC )
12 zcn 10651 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  CC )
132, 12ax-mp 5 . . . . . . . . . 10  |-  N  e.  CC
14 subadd 9613 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  z  e.  CC  /\  (
q  x.  D )  e.  CC )  -> 
( ( N  -  z )  =  ( q  x.  D )  <-> 
( z  +  ( q  x.  D ) )  =  N ) )
1513, 14mp3an1 1301 . . . . . . . . 9  |-  ( ( z  e.  CC  /\  ( q  x.  D
)  e.  CC )  ->  ( ( N  -  z )  =  ( q  x.  D
)  <->  ( z  +  ( q  x.  D
) )  =  N ) )
16 addcom 9555 . . . . . . . . . 10  |-  ( ( z  e.  CC  /\  ( q  x.  D
)  e.  CC )  ->  ( z  +  ( q  x.  D
) )  =  ( ( q  x.  D
)  +  z ) )
1716eqeq1d 2451 . . . . . . . . 9  |-  ( ( z  e.  CC  /\  ( q  x.  D
)  e.  CC )  ->  ( ( z  +  ( q  x.  D ) )  =  N  <->  ( ( q  x.  D )  +  z )  =  N ) )
1815, 17bitrd 253 . . . . . . . 8  |-  ( ( z  e.  CC  /\  ( q  x.  D
)  e.  CC )  ->  ( ( N  -  z )  =  ( q  x.  D
)  <->  ( ( q  x.  D )  +  z )  =  N ) )
198, 11, 18syl2an 477 . . . . . . 7  |-  ( ( z  e.  NN0  /\  q  e.  ZZ )  ->  ( ( N  -  z )  =  ( q  x.  D )  <-> 
( ( q  x.  D )  +  z )  =  N ) )
20 eqcom 2445 . . . . . . 7  |-  ( ( N  -  z )  =  ( q  x.  D )  <->  ( q  x.  D )  =  ( N  -  z ) )
21 eqcom 2445 . . . . . . 7  |-  ( ( ( q  x.  D
)  +  z )  =  N  <->  N  =  ( ( q  x.  D )  +  z ) )
2219, 20, 213bitr3g 287 . . . . . 6  |-  ( ( z  e.  NN0  /\  q  e.  ZZ )  ->  ( ( q  x.  D )  =  ( N  -  z )  <-> 
N  =  ( ( q  x.  D )  +  z ) ) )
2322rexbidva 2732 . . . . 5  |-  ( z  e.  NN0  ->  ( E. q  e.  ZZ  (
q  x.  D )  =  ( N  -  z )  <->  E. q  e.  ZZ  N  =  ( ( q  x.  D
)  +  z ) ) )
247, 23bitrd 253 . . . 4  |-  ( z  e.  NN0  ->  ( D 
||  ( N  -  z )  <->  E. q  e.  ZZ  N  =  ( ( q  x.  D
)  +  z ) ) )
2524pm5.32i 637 . . 3  |-  ( ( z  e.  NN0  /\  D  ||  ( N  -  z ) )  <->  ( z  e.  NN0  /\  E. q  e.  ZZ  N  =  ( ( q  x.  D
)  +  z ) ) )
26 oveq2 6099 . . . . 5  |-  ( r  =  z  ->  ( N  -  r )  =  ( N  -  z ) )
2726breq2d 4304 . . . 4  |-  ( r  =  z  ->  ( D  ||  ( N  -  r )  <->  D  ||  ( N  -  z )
) )
28 divalglem2.4 . . . 4  |-  S  =  { r  e.  NN0  |  D  ||  ( N  -  r ) }
2927, 28elrab2 3119 . . 3  |-  ( z  e.  S  <->  ( z  e.  NN0  /\  D  ||  ( N  -  z
) ) )
30 oveq2 6099 . . . . . 6  |-  ( r  =  z  ->  (
( q  x.  D
)  +  r )  =  ( ( q  x.  D )  +  z ) )
3130eqeq2d 2454 . . . . 5  |-  ( r  =  z  ->  ( N  =  ( (
q  x.  D )  +  r )  <->  N  =  ( ( q  x.  D )  +  z ) ) )
3231rexbidv 2736 . . . 4  |-  ( r  =  z  ->  ( E. q  e.  ZZ  N  =  ( (
q  x.  D )  +  r )  <->  E. q  e.  ZZ  N  =  ( ( q  x.  D
)  +  z ) ) )
3332elrab 3117 . . 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 277 . 2  |-  ( z  e.  S  <->  z  e.  { r  e.  NN0  |  E. q  e.  ZZ  N  =  ( (
q  x.  D )  +  r ) } )
3534eqriv 2440 1  |-  S  =  { r  e.  NN0  |  E. q  e.  ZZ  N  =  ( (
q  x.  D )  +  r ) }
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   E.wrex 2716   {crab 2719   class class class wbr 4292  (class class class)co 6091   CCcc 9280   0cc0 9282    + caddc 9285    x. cmul 9287    - cmin 9595   NN0cn0 10579   ZZcz 10646    || cdivides 13535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-n0 10580  df-z 10647  df-dvds 13536
This theorem is referenced by:  divalglem10  13606
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