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Theorem divalglem10 13606
Description: Lemma for divalg 13607. (Contributed by Paul Chapman, 21-Mar-2011.)
Hypotheses
Ref Expression
divalglem8.1  |-  N  e.  ZZ
divalglem8.2  |-  D  e.  ZZ
divalglem8.3  |-  D  =/=  0
divalglem8.4  |-  S  =  { r  e.  NN0  |  D  ||  ( N  -  r ) }
Assertion
Ref Expression
divalglem10  |-  E! r  e.  ZZ  E. q  e.  ZZ  ( 0  <_ 
r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D
)  +  r ) )
Distinct variable groups:    D, q,
r    N, q, r
Allowed substitution hints:    S( r, q)

Proof of Theorem divalglem10
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 divalglem8.1 . . . 4  |-  N  e.  ZZ
2 divalglem8.2 . . . 4  |-  D  e.  ZZ
3 divalglem8.3 . . . 4  |-  D  =/=  0
4 divalglem8.4 . . . 4  |-  S  =  { r  e.  NN0  |  D  ||  ( N  -  r ) }
5 eqid 2443 . . . 4  |-  sup ( S ,  RR ,  `'  <  )  =  sup ( S ,  RR ,  `'  <  )
61, 2, 3, 4, 5divalglem9 13605 . . 3  |-  E! x  e.  S  x  <  ( abs `  D )
7 elnn0z 10659 . . . . . . . . . 10  |-  ( x  e.  NN0  <->  ( x  e.  ZZ  /\  0  <_  x ) )
87anbi2i 694 . . . . . . . . 9  |-  ( ( x  <  ( abs `  D )  /\  x  e.  NN0 )  <->  ( x  <  ( abs `  D
)  /\  ( x  e.  ZZ  /\  0  <_  x ) ) )
9 an12 795 . . . . . . . . . 10  |-  ( ( x  <  ( abs `  D )  /\  (
x  e.  ZZ  /\  0  <_  x ) )  <-> 
( x  e.  ZZ  /\  ( x  <  ( abs `  D )  /\  0  <_  x ) ) )
10 ancom 450 . . . . . . . . . . 11  |-  ( ( x  <  ( abs `  D )  /\  0  <_  x )  <->  ( 0  <_  x  /\  x  <  ( abs `  D
) ) )
1110anbi2i 694 . . . . . . . . . 10  |-  ( ( x  e.  ZZ  /\  ( x  <  ( abs `  D )  /\  0  <_  x ) )  <->  ( x  e.  ZZ  /\  ( 0  <_  x  /\  x  <  ( abs `  D
) ) ) )
129, 11bitri 249 . . . . . . . . 9  |-  ( ( x  <  ( abs `  D )  /\  (
x  e.  ZZ  /\  0  <_  x ) )  <-> 
( x  e.  ZZ  /\  ( 0  <_  x  /\  x  <  ( abs `  D ) ) ) )
138, 12bitri 249 . . . . . . . 8  |-  ( ( x  <  ( abs `  D )  /\  x  e.  NN0 )  <->  ( x  e.  ZZ  /\  ( 0  <_  x  /\  x  <  ( abs `  D
) ) ) )
1413anbi1i 695 . . . . . . 7  |-  ( ( ( x  <  ( abs `  D )  /\  x  e.  NN0 )  /\  E. q  e.  ZZ  N  =  ( ( q  x.  D )  +  x ) )  <->  ( (
x  e.  ZZ  /\  ( 0  <_  x  /\  x  <  ( abs `  D ) ) )  /\  E. q  e.  ZZ  N  =  ( ( q  x.  D
)  +  x ) ) )
15 anass 649 . . . . . . 7  |-  ( ( ( x  e.  ZZ  /\  ( 0  <_  x  /\  x  <  ( abs `  D ) ) )  /\  E. q  e.  ZZ  N  =  ( ( q  x.  D
)  +  x ) )  <->  ( x  e.  ZZ  /\  ( ( 0  <_  x  /\  x  <  ( abs `  D
) )  /\  E. q  e.  ZZ  N  =  ( ( q  x.  D )  +  x ) ) ) )
1614, 15bitri 249 . . . . . 6  |-  ( ( ( x  <  ( abs `  D )  /\  x  e.  NN0 )  /\  E. q  e.  ZZ  N  =  ( ( q  x.  D )  +  x ) )  <->  ( x  e.  ZZ  /\  ( ( 0  <_  x  /\  x  <  ( abs `  D
) )  /\  E. q  e.  ZZ  N  =  ( ( q  x.  D )  +  x ) ) ) )
17 oveq2 6099 . . . . . . . . . . 11  |-  ( r  =  x  ->  (
( q  x.  D
)  +  r )  =  ( ( q  x.  D )  +  x ) )
1817eqeq2d 2454 . . . . . . . . . 10  |-  ( r  =  x  ->  ( N  =  ( (
q  x.  D )  +  r )  <->  N  =  ( ( q  x.  D )  +  x
) ) )
1918rexbidv 2736 . . . . . . . . 9  |-  ( r  =  x  ->  ( E. q  e.  ZZ  N  =  ( (
q  x.  D )  +  r )  <->  E. q  e.  ZZ  N  =  ( ( q  x.  D
)  +  x ) ) )
201, 2, 3, 4divalglem4 13600 . . . . . . . . 9  |-  S  =  { r  e.  NN0  |  E. q  e.  ZZ  N  =  ( (
q  x.  D )  +  r ) }
2119, 20elrab2 3119 . . . . . . . 8  |-  ( x  e.  S  <->  ( x  e.  NN0  /\  E. q  e.  ZZ  N  =  ( ( q  x.  D
)  +  x ) ) )
2221anbi2i 694 . . . . . . 7  |-  ( ( x  <  ( abs `  D )  /\  x  e.  S )  <->  ( x  <  ( abs `  D
)  /\  ( x  e.  NN0  /\  E. q  e.  ZZ  N  =  ( ( q  x.  D
)  +  x ) ) ) )
23 ancom 450 . . . . . . 7  |-  ( ( x  e.  S  /\  x  <  ( abs `  D
) )  <->  ( x  <  ( abs `  D
)  /\  x  e.  S ) )
24 anass 649 . . . . . . 7  |-  ( ( ( x  <  ( abs `  D )  /\  x  e.  NN0 )  /\  E. q  e.  ZZ  N  =  ( ( q  x.  D )  +  x ) )  <->  ( x  <  ( abs `  D
)  /\  ( x  e.  NN0  /\  E. q  e.  ZZ  N  =  ( ( q  x.  D
)  +  x ) ) ) )
2522, 23, 243bitr4i 277 . . . . . 6  |-  ( ( x  e.  S  /\  x  <  ( abs `  D
) )  <->  ( (
x  <  ( abs `  D )  /\  x  e.  NN0 )  /\  E. q  e.  ZZ  N  =  ( ( q  x.  D )  +  x ) ) )
26 df-3an 967 . . . . . . . . 9  |-  ( ( 0  <_  x  /\  x  <  ( abs `  D
)  /\  N  =  ( ( q  x.  D )  +  x
) )  <->  ( (
0  <_  x  /\  x  <  ( abs `  D
) )  /\  N  =  ( ( q  x.  D )  +  x ) ) )
2726rexbii 2740 . . . . . . . 8  |-  ( E. q  e.  ZZ  (
0  <_  x  /\  x  <  ( abs `  D
)  /\  N  =  ( ( q  x.  D )  +  x
) )  <->  E. q  e.  ZZ  ( ( 0  <_  x  /\  x  <  ( abs `  D
) )  /\  N  =  ( ( q  x.  D )  +  x ) ) )
28 r19.42v 2875 . . . . . . . 8  |-  ( E. q  e.  ZZ  (
( 0  <_  x  /\  x  <  ( abs `  D ) )  /\  N  =  ( (
q  x.  D )  +  x ) )  <-> 
( ( 0  <_  x  /\  x  <  ( abs `  D ) )  /\  E. q  e.  ZZ  N  =  ( ( q  x.  D
)  +  x ) ) )
2927, 28bitri 249 . . . . . . 7  |-  ( E. q  e.  ZZ  (
0  <_  x  /\  x  <  ( abs `  D
)  /\  N  =  ( ( q  x.  D )  +  x
) )  <->  ( (
0  <_  x  /\  x  <  ( abs `  D
) )  /\  E. q  e.  ZZ  N  =  ( ( q  x.  D )  +  x ) ) )
3029anbi2i 694 . . . . . 6  |-  ( ( x  e.  ZZ  /\  E. q  e.  ZZ  (
0  <_  x  /\  x  <  ( abs `  D
)  /\  N  =  ( ( q  x.  D )  +  x
) ) )  <->  ( x  e.  ZZ  /\  ( ( 0  <_  x  /\  x  <  ( abs `  D
) )  /\  E. q  e.  ZZ  N  =  ( ( q  x.  D )  +  x ) ) ) )
3116, 25, 303bitr4i 277 . . . . 5  |-  ( ( x  e.  S  /\  x  <  ( abs `  D
) )  <->  ( x  e.  ZZ  /\  E. q  e.  ZZ  ( 0  <_  x  /\  x  <  ( abs `  D )  /\  N  =  ( (
q  x.  D )  +  x ) ) ) )
3231eubii 2278 . . . 4  |-  ( E! x ( x  e.  S  /\  x  < 
( abs `  D
) )  <->  E! x
( x  e.  ZZ  /\ 
E. q  e.  ZZ  ( 0  <_  x  /\  x  <  ( abs `  D )  /\  N  =  ( ( q  x.  D )  +  x ) ) ) )
33 df-reu 2722 . . . 4  |-  ( E! x  e.  S  x  <  ( abs `  D
)  <->  E! x ( x  e.  S  /\  x  <  ( abs `  D
) ) )
34 df-reu 2722 . . . 4  |-  ( E! x  e.  ZZ  E. q  e.  ZZ  (
0  <_  x  /\  x  <  ( abs `  D
)  /\  N  =  ( ( q  x.  D )  +  x
) )  <->  E! x
( x  e.  ZZ  /\ 
E. q  e.  ZZ  ( 0  <_  x  /\  x  <  ( abs `  D )  /\  N  =  ( ( q  x.  D )  +  x ) ) ) )
3532, 33, 343bitr4i 277 . . 3  |-  ( E! x  e.  S  x  <  ( abs `  D
)  <->  E! x  e.  ZZ  E. q  e.  ZZ  (
0  <_  x  /\  x  <  ( abs `  D
)  /\  N  =  ( ( q  x.  D )  +  x
) ) )
366, 35mpbi 208 . 2  |-  E! x  e.  ZZ  E. q  e.  ZZ  ( 0  <_  x  /\  x  <  ( abs `  D )  /\  N  =  ( (
q  x.  D )  +  x ) )
37 breq2 4296 . . . . 5  |-  ( x  =  r  ->  (
0  <_  x  <->  0  <_  r ) )
38 breq1 4295 . . . . 5  |-  ( x  =  r  ->  (
x  <  ( abs `  D )  <->  r  <  ( abs `  D ) ) )
39 oveq2 6099 . . . . . 6  |-  ( x  =  r  ->  (
( q  x.  D
)  +  x )  =  ( ( q  x.  D )  +  r ) )
4039eqeq2d 2454 . . . . 5  |-  ( x  =  r  ->  ( N  =  ( (
q  x.  D )  +  x )  <->  N  =  ( ( q  x.  D )  +  r ) ) )
4137, 38, 403anbi123d 1289 . . . 4  |-  ( x  =  r  ->  (
( 0  <_  x  /\  x  <  ( abs `  D )  /\  N  =  ( ( q  x.  D )  +  x ) )  <->  ( 0  <_  r  /\  r  <  ( abs `  D
)  /\  N  =  ( ( q  x.  D )  +  r ) ) ) )
4241rexbidv 2736 . . 3  |-  ( x  =  r  ->  ( E. q  e.  ZZ  ( 0  <_  x  /\  x  <  ( abs `  D )  /\  N  =  ( ( q  x.  D )  +  x ) )  <->  E. q  e.  ZZ  ( 0  <_ 
r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D
)  +  r ) ) ) )
4342cbvreuv 2949 . 2  |-  ( E! x  e.  ZZ  E. q  e.  ZZ  (
0  <_  x  /\  x  <  ( abs `  D
)  /\  N  =  ( ( q  x.  D )  +  x
) )  <->  E! r  e.  ZZ  E. q  e.  ZZ  ( 0  <_ 
r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D
)  +  r ) ) )
4436, 43mpbi 208 1  |-  E! r  e.  ZZ  E. q  e.  ZZ  ( 0  <_ 
r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D
)  +  r ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E!weu 2253    =/= wne 2606   E.wrex 2716   E!wreu 2717   {crab 2719   class class class wbr 4292   `'ccnv 4839   ` cfv 5418  (class class class)co 6091   supcsup 7690   RRcr 9281   0cc0 9282    + caddc 9285    x. cmul 9287    < clt 9418    <_ cle 9419    - cmin 9595   NN0cn0 10579   ZZcz 10646   abscabs 12723    || 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-cnex 9338  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  ax-pre-sup 9360
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-rmo 2723  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-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-dvds 13536
This theorem is referenced by:  divalg  13607
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