MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  divalglem10 Unicode version

Theorem divalglem10 12648
Description: Lemma for divalg 12649. (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 2316 . . . 4  |-  sup ( S ,  RR ,  `'  <  )  =  sup ( S ,  RR ,  `'  <  )
61, 2, 3, 4, 5divalglem9 12647 . . 3  |-  E! x  e.  S  x  <  ( abs `  D )
7 elnn0z 10083 . . . . . . . . . 10  |-  ( x  e.  NN0  <->  ( x  e.  ZZ  /\  0  <_  x ) )
87anbi2i 675 . . . . . . . . 9  |-  ( ( x  <  ( abs `  D )  /\  x  e.  NN0 )  <->  ( x  <  ( abs `  D
)  /\  ( x  e.  ZZ  /\  0  <_  x ) ) )
9 an12 772 . . . . . . . . . 10  |-  ( ( x  <  ( abs `  D )  /\  (
x  e.  ZZ  /\  0  <_  x ) )  <-> 
( x  e.  ZZ  /\  ( x  <  ( abs `  D )  /\  0  <_  x ) ) )
10 ancom 437 . . . . . . . . . . 11  |-  ( ( x  <  ( abs `  D )  /\  0  <_  x )  <->  ( 0  <_  x  /\  x  <  ( abs `  D
) ) )
1110anbi2i 675 . . . . . . . . . 10  |-  ( ( x  e.  ZZ  /\  ( x  <  ( abs `  D )  /\  0  <_  x ) )  <->  ( x  e.  ZZ  /\  ( 0  <_  x  /\  x  <  ( abs `  D
) ) ) )
129, 11bitri 240 . . . . . . . . 9  |-  ( ( x  <  ( abs `  D )  /\  (
x  e.  ZZ  /\  0  <_  x ) )  <-> 
( x  e.  ZZ  /\  ( 0  <_  x  /\  x  <  ( abs `  D ) ) ) )
138, 12bitri 240 . . . . . . . 8  |-  ( ( x  <  ( abs `  D )  /\  x  e.  NN0 )  <->  ( x  e.  ZZ  /\  ( 0  <_  x  /\  x  <  ( abs `  D
) ) ) )
1413anbi1i 676 . . . . . . 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 630 . . . . . . 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 240 . . . . . 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 5908 . . . . . . . . . . 11  |-  ( r  =  x  ->  (
( q  x.  D
)  +  r )  =  ( ( q  x.  D )  +  x ) )
1817eqeq2d 2327 . . . . . . . . . 10  |-  ( r  =  x  ->  ( N  =  ( (
q  x.  D )  +  r )  <->  N  =  ( ( q  x.  D )  +  x
) ) )
1918rexbidv 2598 . . . . . . . . 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 12642 . . . . . . . . 9  |-  S  =  { r  e.  NN0  |  E. q  e.  ZZ  N  =  ( (
q  x.  D )  +  r ) }
2119, 20elrab2 2959 . . . . . . . 8  |-  ( x  e.  S  <->  ( x  e.  NN0  /\  E. q  e.  ZZ  N  =  ( ( q  x.  D
)  +  x ) ) )
2221anbi2i 675 . . . . . . 7  |-  ( ( x  <  ( abs `  D )  /\  x  e.  S )  <->  ( x  <  ( abs `  D
)  /\  ( x  e.  NN0  /\  E. q  e.  ZZ  N  =  ( ( q  x.  D
)  +  x ) ) ) )
23 ancom 437 . . . . . . 7  |-  ( ( x  e.  S  /\  x  <  ( abs `  D
) )  <->  ( x  <  ( abs `  D
)  /\  x  e.  S ) )
24 anass 630 . . . . . . 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 268 . . . . . 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 936 . . . . . . . . 9  |-  ( ( 0  <_  x  /\  x  <  ( abs `  D
)  /\  N  =  ( ( q  x.  D )  +  x
) )  <->  ( (
0  <_  x  /\  x  <  ( abs `  D
) )  /\  N  =  ( ( q  x.  D )  +  x ) ) )
2726rexbii 2602 . . . . . . . 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 2728 . . . . . . . 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 240 . . . . . . 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 675 . . . . . 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 268 . . . . 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 2185 . . . 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 2584 . . . 4  |-  ( E! x  e.  S  x  <  ( abs `  D
)  <->  E! x ( x  e.  S  /\  x  <  ( abs `  D
) ) )
34 df-reu 2584 . . . 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 268 . . 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 199 . 2  |-  E! x  e.  ZZ  E. q  e.  ZZ  ( 0  <_  x  /\  x  <  ( abs `  D )  /\  N  =  ( (
q  x.  D )  +  x ) )
37 breq2 4064 . . . . 5  |-  ( x  =  r  ->  (
0  <_  x  <->  0  <_  r ) )
38 breq1 4063 . . . . 5  |-  ( x  =  r  ->  (
x  <  ( abs `  D )  <->  r  <  ( abs `  D ) ) )
39 oveq2 5908 . . . . . 6  |-  ( x  =  r  ->  (
( q  x.  D
)  +  x )  =  ( ( q  x.  D )  +  r ) )
4039eqeq2d 2327 . . . . 5  |-  ( x  =  r  ->  ( N  =  ( (
q  x.  D )  +  x )  <->  N  =  ( ( q  x.  D )  +  r ) ) )
4137, 38, 403anbi123d 1252 . . . 4  |-  ( x  =  r  ->  (
( 0  <_  x  /\  x  <  ( abs `  D )  /\  N  =  ( ( q  x.  D )  +  x ) )  <->  ( 0  <_  r  /\  r  <  ( abs `  D
)  /\  N  =  ( ( q  x.  D )  +  r ) ) ) )
4241rexbidv 2598 . . 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 2800 . 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 199 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 358    /\ w3a 934    = wceq 1633    e. wcel 1701   E!weu 2176    =/= wne 2479   E.wrex 2578   E!wreu 2579   {crab 2581   class class class wbr 4060   `'ccnv 4725   ` cfv 5292  (class class class)co 5900   supcsup 7238   RRcr 8781   0cc0 8782    + caddc 8785    x. cmul 8787    < clt 8912    <_ cle 8913    - cmin 9082   NN0cn0 10012   ZZcz 10071   abscabs 11766    || cdivides 12578
This theorem is referenced by:  divalg  12649
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-n0 10013  df-z 10072  df-uz 10278  df-rp 10402  df-fz 10830  df-seq 11094  df-exp 11152  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-dvds 12579
  Copyright terms: Public domain W3C validator