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Theorem divalglem10 12914
Description: Lemma for divalg 12915. (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 2435 . . . 4  |-  sup ( S ,  RR ,  `'  <  )  =  sup ( S ,  RR ,  `'  <  )
61, 2, 3, 4, 5divalglem9 12913 . . 3  |-  E! x  e.  S  x  <  ( abs `  D )
7 elnn0z 10286 . . . . . . . . . 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 6081 . . . . . . . . . . 11  |-  ( r  =  x  ->  (
( q  x.  D
)  +  r )  =  ( ( q  x.  D )  +  x ) )
1817eqeq2d 2446 . . . . . . . . . 10  |-  ( r  =  x  ->  ( N  =  ( (
q  x.  D )  +  r )  <->  N  =  ( ( q  x.  D )  +  x
) ) )
1918rexbidv 2718 . . . . . . . . 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 12908 . . . . . . . . 9  |-  S  =  { r  e.  NN0  |  E. q  e.  ZZ  N  =  ( (
q  x.  D )  +  r ) }
2119, 20elrab2 3086 . . . . . . . 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 2722 . . . . . . . 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 2854 . . . . . . . 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 2289 . . . 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 2704 . . . 4  |-  ( E! x  e.  S  x  <  ( abs `  D
)  <->  E! x ( x  e.  S  /\  x  <  ( abs `  D
) ) )
34 df-reu 2704 . . . 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 4208 . . . . 5  |-  ( x  =  r  ->  (
0  <_  x  <->  0  <_  r ) )
38 breq1 4207 . . . . 5  |-  ( x  =  r  ->  (
x  <  ( abs `  D )  <->  r  <  ( abs `  D ) ) )
39 oveq2 6081 . . . . . 6  |-  ( x  =  r  ->  (
( q  x.  D
)  +  x )  =  ( ( q  x.  D )  +  r ) )
4039eqeq2d 2446 . . . . 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 2718 . . 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 2926 . 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 1652    e. wcel 1725   E!weu 2280    =/= wne 2598   E.wrex 2698   E!wreu 2699   {crab 2701   class class class wbr 4204   `'ccnv 4869   ` cfv 5446  (class class class)co 6073   supcsup 7437   RRcr 8981   0cc0 8982    + caddc 8985    x. cmul 8987    < clt 9112    <_ cle 9113    - cmin 9283   NN0cn0 10213   ZZcz 10274   abscabs 12031    || cdivides 12844
This theorem is referenced by:  divalg  12915
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845
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