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Theorem divalglem10 12849
Description: Lemma for divalg 12850. (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 2387 . . . 4  |-  sup ( S ,  RR ,  `'  <  )  =  sup ( S ,  RR ,  `'  <  )
61, 2, 3, 4, 5divalglem9 12848 . . 3  |-  E! x  e.  S  x  <  ( abs `  D )
7 elnn0z 10226 . . . . . . . . . 10  |-  ( x  e.  NN0  <->  ( x  e.  ZZ  /\  0  <_  x ) )
87anbi2i 676 . . . . . . . . 9  |-  ( ( x  <  ( abs `  D )  /\  x  e.  NN0 )  <->  ( x  <  ( abs `  D
)  /\  ( x  e.  ZZ  /\  0  <_  x ) ) )
9 an12 773 . . . . . . . . . 10  |-  ( ( x  <  ( abs `  D )  /\  (
x  e.  ZZ  /\  0  <_  x ) )  <-> 
( x  e.  ZZ  /\  ( x  <  ( abs `  D )  /\  0  <_  x ) ) )
10 ancom 438 . . . . . . . . . . 11  |-  ( ( x  <  ( abs `  D )  /\  0  <_  x )  <->  ( 0  <_  x  /\  x  <  ( abs `  D
) ) )
1110anbi2i 676 . . . . . . . . . 10  |-  ( ( x  e.  ZZ  /\  ( x  <  ( abs `  D )  /\  0  <_  x ) )  <->  ( x  e.  ZZ  /\  ( 0  <_  x  /\  x  <  ( abs `  D
) ) ) )
129, 11bitri 241 . . . . . . . . 9  |-  ( ( x  <  ( abs `  D )  /\  (
x  e.  ZZ  /\  0  <_  x ) )  <-> 
( x  e.  ZZ  /\  ( 0  <_  x  /\  x  <  ( abs `  D ) ) ) )
138, 12bitri 241 . . . . . . . 8  |-  ( ( x  <  ( abs `  D )  /\  x  e.  NN0 )  <->  ( x  e.  ZZ  /\  ( 0  <_  x  /\  x  <  ( abs `  D
) ) ) )
1413anbi1i 677 . . . . . . 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 631 . . . . . . 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 241 . . . . . 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 6028 . . . . . . . . . . 11  |-  ( r  =  x  ->  (
( q  x.  D
)  +  r )  =  ( ( q  x.  D )  +  x ) )
1817eqeq2d 2398 . . . . . . . . . 10  |-  ( r  =  x  ->  ( N  =  ( (
q  x.  D )  +  r )  <->  N  =  ( ( q  x.  D )  +  x
) ) )
1918rexbidv 2670 . . . . . . . . 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 12843 . . . . . . . . 9  |-  S  =  { r  e.  NN0  |  E. q  e.  ZZ  N  =  ( (
q  x.  D )  +  r ) }
2119, 20elrab2 3037 . . . . . . . 8  |-  ( x  e.  S  <->  ( x  e.  NN0  /\  E. q  e.  ZZ  N  =  ( ( q  x.  D
)  +  x ) ) )
2221anbi2i 676 . . . . . . 7  |-  ( ( x  <  ( abs `  D )  /\  x  e.  S )  <->  ( x  <  ( abs `  D
)  /\  ( x  e.  NN0  /\  E. q  e.  ZZ  N  =  ( ( q  x.  D
)  +  x ) ) ) )
23 ancom 438 . . . . . . 7  |-  ( ( x  e.  S  /\  x  <  ( abs `  D
) )  <->  ( x  <  ( abs `  D
)  /\  x  e.  S ) )
24 anass 631 . . . . . . 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 269 . . . . . 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 938 . . . . . . . . 9  |-  ( ( 0  <_  x  /\  x  <  ( abs `  D
)  /\  N  =  ( ( q  x.  D )  +  x
) )  <->  ( (
0  <_  x  /\  x  <  ( abs `  D
) )  /\  N  =  ( ( q  x.  D )  +  x ) ) )
2726rexbii 2674 . . . . . . . 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 2805 . . . . . . . 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 241 . . . . . . 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 676 . . . . . 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 269 . . . . 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 2247 . . . 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 2656 . . . 4  |-  ( E! x  e.  S  x  <  ( abs `  D
)  <->  E! x ( x  e.  S  /\  x  <  ( abs `  D
) ) )
34 df-reu 2656 . . . 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 269 . . 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 200 . 2  |-  E! x  e.  ZZ  E. q  e.  ZZ  ( 0  <_  x  /\  x  <  ( abs `  D )  /\  N  =  ( (
q  x.  D )  +  x ) )
37 breq2 4157 . . . . 5  |-  ( x  =  r  ->  (
0  <_  x  <->  0  <_  r ) )
38 breq1 4156 . . . . 5  |-  ( x  =  r  ->  (
x  <  ( abs `  D )  <->  r  <  ( abs `  D ) ) )
39 oveq2 6028 . . . . . 6  |-  ( x  =  r  ->  (
( q  x.  D
)  +  x )  =  ( ( q  x.  D )  +  r ) )
4039eqeq2d 2398 . . . . 5  |-  ( x  =  r  ->  ( N  =  ( (
q  x.  D )  +  x )  <->  N  =  ( ( q  x.  D )  +  r ) ) )
4137, 38, 403anbi123d 1254 . . . 4  |-  ( x  =  r  ->  (
( 0  <_  x  /\  x  <  ( abs `  D )  /\  N  =  ( ( q  x.  D )  +  x ) )  <->  ( 0  <_  r  /\  r  <  ( abs `  D
)  /\  N  =  ( ( q  x.  D )  +  r ) ) ) )
4241rexbidv 2670 . . 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 2877 . 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 200 1  |-  E! r  e.  ZZ  E. q  e.  ZZ  ( 0  <_ 
r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D
)  +  r ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   E!weu 2238    =/= wne 2550   E.wrex 2650   E!wreu 2651   {crab 2653   class class class wbr 4153   `'ccnv 4817   ` cfv 5394  (class class class)co 6020   supcsup 7380   RRcr 8922   0cc0 8923    + caddc 8926    x. cmul 8928    < clt 9053    <_ cle 9054    - cmin 9223   NN0cn0 10153   ZZcz 10214   abscabs 11966    || cdivides 12779
This theorem is referenced by:  divalg  12850
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-fz 10976  df-seq 11251  df-exp 11310  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-dvds 12780
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