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Theorem divalglem10 14062
Description: Lemma for divalg 14063. (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 2382 . . . 4  |-  sup ( S ,  RR ,  `'  <  )  =  sup ( S ,  RR ,  `'  <  )
61, 2, 3, 4, 5divalglem9 14061 . . 3  |-  E! x  e.  S  x  <  ( abs `  D )
7 elnn0z 10794 . . . . . . . . . 10  |-  ( x  e.  NN0  <->  ( x  e.  ZZ  /\  0  <_  x ) )
87anbi2i 692 . . . . . . . . 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 448 . . . . . . . . . . 11  |-  ( ( x  <  ( abs `  D )  /\  0  <_  x )  <->  ( 0  <_  x  /\  x  <  ( abs `  D
) ) )
1110anbi2i 692 . . . . . . . . . 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 693 . . . . . . 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 647 . . . . . . 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 6204 . . . . . . . . . . 11  |-  ( r  =  x  ->  (
( q  x.  D
)  +  r )  =  ( ( q  x.  D )  +  x ) )
1817eqeq2d 2396 . . . . . . . . . 10  |-  ( r  =  x  ->  ( N  =  ( (
q  x.  D )  +  r )  <->  N  =  ( ( q  x.  D )  +  x
) ) )
1918rexbidv 2893 . . . . . . . . 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 14056 . . . . . . . . 9  |-  S  =  { r  e.  NN0  |  E. q  e.  ZZ  N  =  ( (
q  x.  D )  +  r ) }
2119, 20elrab2 3184 . . . . . . . 8  |-  ( x  e.  S  <->  ( x  e.  NN0  /\  E. q  e.  ZZ  N  =  ( ( q  x.  D
)  +  x ) ) )
2221anbi2i 692 . . . . . . 7  |-  ( ( x  <  ( abs `  D )  /\  x  e.  S )  <->  ( x  <  ( abs `  D
)  /\  ( x  e.  NN0  /\  E. q  e.  ZZ  N  =  ( ( q  x.  D
)  +  x ) ) ) )
23 ancom 448 . . . . . . 7  |-  ( ( x  e.  S  /\  x  <  ( abs `  D
) )  <->  ( x  <  ( abs `  D
)  /\  x  e.  S ) )
24 anass 647 . . . . . . 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 973 . . . . . . . . 9  |-  ( ( 0  <_  x  /\  x  <  ( abs `  D
)  /\  N  =  ( ( q  x.  D )  +  x
) )  <->  ( (
0  <_  x  /\  x  <  ( abs `  D
) )  /\  N  =  ( ( q  x.  D )  +  x ) ) )
2726rexbii 2884 . . . . . . . 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 2937 . . . . . . . 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 692 . . . . . 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 2242 . . . 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 2739 . . . 4  |-  ( E! x  e.  S  x  <  ( abs `  D
)  <->  E! x ( x  e.  S  /\  x  <  ( abs `  D
) ) )
34 df-reu 2739 . . . 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 4371 . . . . 5  |-  ( x  =  r  ->  (
0  <_  x  <->  0  <_  r ) )
38 breq1 4370 . . . . 5  |-  ( x  =  r  ->  (
x  <  ( abs `  D )  <->  r  <  ( abs `  D ) ) )
39 oveq2 6204 . . . . . 6  |-  ( x  =  r  ->  (
( q  x.  D
)  +  x )  =  ( ( q  x.  D )  +  r ) )
4039eqeq2d 2396 . . . . 5  |-  ( x  =  r  ->  ( N  =  ( (
q  x.  D )  +  x )  <->  N  =  ( ( q  x.  D )  +  r ) ) )
4137, 38, 403anbi123d 1297 . . . 4  |-  ( x  =  r  ->  (
( 0  <_  x  /\  x  <  ( abs `  D )  /\  N  =  ( ( q  x.  D )  +  x ) )  <->  ( 0  <_  r  /\  r  <  ( abs `  D
)  /\  N  =  ( ( q  x.  D )  +  r ) ) ) )
4241rexbidv 2893 . . 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 3011 . 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826   E!weu 2218    =/= wne 2577   E.wrex 2733   E!wreu 2734   {crab 2736   class class class wbr 4367   `'ccnv 4912   ` cfv 5496  (class class class)co 6196   supcsup 7815   RRcr 9402   0cc0 9403    + caddc 9406    x. cmul 9408    < clt 9539    <_ cle 9540    - cmin 9718   NN0cn0 10712   ZZcz 10781   abscabs 13069    || cdvds 13988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-rp 11140  df-fz 11594  df-seq 12011  df-exp 12070  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-dvds 13989
This theorem is referenced by:  divalg  14063
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