MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  divalglem5 Structured version   Unicode version

Theorem divalglem5 13600
Description: Lemma for divalg 13606. (Contributed by Paul Chapman, 21-Mar-2011.)
Hypotheses
Ref Expression
divalglem0.1  |-  N  e.  ZZ
divalglem0.2  |-  D  e.  ZZ
divalglem1.3  |-  D  =/=  0
divalglem2.4  |-  S  =  { r  e.  NN0  |  D  ||  ( N  -  r ) }
divalglem5.5  |-  R  =  sup ( S ,  RR ,  `'  <  )
Assertion
Ref Expression
divalglem5  |-  ( 0  <_  R  /\  R  <  ( abs `  D
) )
Distinct variable groups:    D, r    N, r
Allowed substitution hints:    R( r)    S( r)

Proof of Theorem divalglem5
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 divalglem5.5 . . . . . 6  |-  R  =  sup ( S ,  RR ,  `'  <  )
2 divalglem0.1 . . . . . . 7  |-  N  e.  ZZ
3 divalglem0.2 . . . . . . 7  |-  D  e.  ZZ
4 divalglem1.3 . . . . . . 7  |-  D  =/=  0
5 divalglem2.4 . . . . . . 7  |-  S  =  { r  e.  NN0  |  D  ||  ( N  -  r ) }
62, 3, 4, 5divalglem2 13598 . . . . . 6  |-  sup ( S ,  RR ,  `'  <  )  e.  S
71, 6eqeltri 2512 . . . . 5  |-  R  e.  S
8 oveq2 6098 . . . . . . 7  |-  ( x  =  R  ->  ( N  -  x )  =  ( N  -  R ) )
98breq2d 4303 . . . . . 6  |-  ( x  =  R  ->  ( D  ||  ( N  -  x )  <->  D  ||  ( N  -  R )
) )
10 oveq2 6098 . . . . . . . . 9  |-  ( r  =  x  ->  ( N  -  r )  =  ( N  -  x ) )
1110breq2d 4303 . . . . . . . 8  |-  ( r  =  x  ->  ( D  ||  ( N  -  r )  <->  D  ||  ( N  -  x )
) )
1211cbvrabv 2970 . . . . . . 7  |-  { r  e.  NN0  |  D  ||  ( N  -  r
) }  =  {
x  e.  NN0  |  D  ||  ( N  -  x ) }
135, 12eqtri 2462 . . . . . 6  |-  S  =  { x  e.  NN0  |  D  ||  ( N  -  x ) }
149, 13elrab2 3118 . . . . 5  |-  ( R  e.  S  <->  ( R  e.  NN0  /\  D  ||  ( N  -  R
) ) )
157, 14mpbi 208 . . . 4  |-  ( R  e.  NN0  /\  D  ||  ( N  -  R
) )
1615simpli 458 . . 3  |-  R  e. 
NN0
1716nn0ge0i 10606 . 2  |-  0  <_  R
18 nnabscl 12812 . . . . . . 7  |-  ( ( D  e.  ZZ  /\  D  =/=  0 )  -> 
( abs `  D
)  e.  NN )
193, 4, 18mp2an 672 . . . . . 6  |-  ( abs `  D )  e.  NN
2019nngt0i 10354 . . . . 5  |-  0  <  ( abs `  D
)
21 0re 9385 . . . . . 6  |-  0  e.  RR
22 zcn 10650 . . . . . . . 8  |-  ( D  e.  ZZ  ->  D  e.  CC )
233, 22ax-mp 5 . . . . . . 7  |-  D  e.  CC
2423abscli 12881 . . . . . 6  |-  ( abs `  D )  e.  RR
2521, 24ltnlei 9494 . . . . 5  |-  ( 0  <  ( abs `  D
)  <->  -.  ( abs `  D )  <_  0
)
2620, 25mpbi 208 . . . 4  |-  -.  ( abs `  D )  <_ 
0
27 ssrab2 3436 . . . . . . . . 9  |-  { r  e.  NN0  |  D  ||  ( N  -  r
) }  C_  NN0
285, 27eqsstri 3385 . . . . . . . 8  |-  S  C_  NN0
29 nn0uz 10894 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
3028, 29sseqtri 3387 . . . . . . 7  |-  S  C_  ( ZZ>= `  0 )
31 nn0abscl 12800 . . . . . . . . . 10  |-  ( D  e.  ZZ  ->  ( abs `  D )  e. 
NN0 )
323, 31ax-mp 5 . . . . . . . . 9  |-  ( abs `  D )  e.  NN0
33 nn0sub2 10704 . . . . . . . . 9  |-  ( ( ( abs `  D
)  e.  NN0  /\  R  e.  NN0  /\  ( abs `  D )  <_  R )  ->  ( R  -  ( abs `  D ) )  e. 
NN0 )
3432, 16, 33mp3an12 1304 . . . . . . . 8  |-  ( ( abs `  D )  <_  R  ->  ( R  -  ( abs `  D ) )  e. 
NN0 )
3515a1i 11 . . . . . . . . 9  |-  ( ( abs `  D )  <_  R  ->  ( R  e.  NN0  /\  D  ||  ( N  -  R
) ) )
36 nn0z 10668 . . . . . . . . . . 11  |-  ( R  e.  NN0  ->  R  e.  ZZ )
37 1z 10675 . . . . . . . . . . . . 13  |-  1  e.  ZZ
382, 3divalglem0 13596 . . . . . . . . . . . . 13  |-  ( ( R  e.  ZZ  /\  1  e.  ZZ )  ->  ( D  ||  ( N  -  R )  ->  D  ||  ( N  -  ( R  -  ( 1  x.  ( abs `  D ) ) ) ) ) )
3937, 38mpan2 671 . . . . . . . . . . . 12  |-  ( R  e.  ZZ  ->  ( D  ||  ( N  -  R )  ->  D  ||  ( N  -  ( R  -  ( 1  x.  ( abs `  D
) ) ) ) ) )
4024recni 9397 . . . . . . . . . . . . . . . 16  |-  ( abs `  D )  e.  CC
4140mulid2i 9388 . . . . . . . . . . . . . . 15  |-  ( 1  x.  ( abs `  D
) )  =  ( abs `  D )
4241oveq2i 6101 . . . . . . . . . . . . . 14  |-  ( R  -  ( 1  x.  ( abs `  D
) ) )  =  ( R  -  ( abs `  D ) )
4342oveq2i 6101 . . . . . . . . . . . . 13  |-  ( N  -  ( R  -  ( 1  x.  ( abs `  D ) ) ) )  =  ( N  -  ( R  -  ( abs `  D
) ) )
4443breq2i 4299 . . . . . . . . . . . 12  |-  ( D 
||  ( N  -  ( R  -  (
1  x.  ( abs `  D ) ) ) )  <->  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) )
4539, 44syl6ib 226 . . . . . . . . . . 11  |-  ( R  e.  ZZ  ->  ( D  ||  ( N  -  R )  ->  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) ) )
4636, 45syl 16 . . . . . . . . . 10  |-  ( R  e.  NN0  ->  ( D 
||  ( N  -  R )  ->  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) ) )
4746imp 429 . . . . . . . . 9  |-  ( ( R  e.  NN0  /\  D  ||  ( N  -  R ) )  ->  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) )
4835, 47syl 16 . . . . . . . 8  |-  ( ( abs `  D )  <_  R  ->  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) )
49 oveq2 6098 . . . . . . . . . 10  |-  ( x  =  ( R  -  ( abs `  D ) )  ->  ( N  -  x )  =  ( N  -  ( R  -  ( abs `  D
) ) ) )
5049breq2d 4303 . . . . . . . . 9  |-  ( x  =  ( R  -  ( abs `  D ) )  ->  ( D  ||  ( N  -  x
)  <->  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) ) )
5150, 13elrab2 3118 . . . . . . . 8  |-  ( ( R  -  ( abs `  D ) )  e.  S  <->  ( ( R  -  ( abs `  D
) )  e.  NN0  /\  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) ) )
5234, 48, 51sylanbrc 664 . . . . . . 7  |-  ( ( abs `  D )  <_  R  ->  ( R  -  ( abs `  D ) )  e.  S )
53 infmssuzle 10936 . . . . . . 7  |-  ( ( S  C_  ( ZZ>= ` 
0 )  /\  ( R  -  ( abs `  D ) )  e.  S )  ->  sup ( S ,  RR ,  `'  <  )  <_  ( R  -  ( abs `  D ) ) )
5430, 52, 53sylancr 663 . . . . . 6  |-  ( ( abs `  D )  <_  R  ->  sup ( S ,  RR ,  `'  <  )  <_  ( R  -  ( abs `  D ) ) )
551, 54syl5eqbr 4324 . . . . 5  |-  ( ( abs `  D )  <_  R  ->  R  <_  ( R  -  ( abs `  D ) ) )
5635simpld 459 . . . . . . . 8  |-  ( ( abs `  D )  <_  R  ->  R  e.  NN0 )
57 nn0re 10587 . . . . . . . 8  |-  ( R  e.  NN0  ->  R  e.  RR )
5856, 57syl 16 . . . . . . 7  |-  ( ( abs `  D )  <_  R  ->  R  e.  RR )
59 lesub 9817 . . . . . . . 8  |-  ( ( R  e.  RR  /\  R  e.  RR  /\  ( abs `  D )  e.  RR )  ->  ( R  <_  ( R  -  ( abs `  D ) )  <->  ( abs `  D
)  <_  ( R  -  R ) ) )
6024, 59mp3an3 1303 . . . . . . 7  |-  ( ( R  e.  RR  /\  R  e.  RR )  ->  ( R  <_  ( R  -  ( abs `  D ) )  <->  ( abs `  D )  <_  ( R  -  R )
) )
6158, 58, 60syl2anc 661 . . . . . 6  |-  ( ( abs `  D )  <_  R  ->  ( R  <_  ( R  -  ( abs `  D ) )  <->  ( abs `  D
)  <_  ( R  -  R ) ) )
6258recnd 9411 . . . . . . . 8  |-  ( ( abs `  D )  <_  R  ->  R  e.  CC )
6362subidd 9706 . . . . . . 7  |-  ( ( abs `  D )  <_  R  ->  ( R  -  R )  =  0 )
6463breq2d 4303 . . . . . 6  |-  ( ( abs `  D )  <_  R  ->  (
( abs `  D
)  <_  ( R  -  R )  <->  ( abs `  D )  <_  0
) )
6561, 64bitrd 253 . . . . 5  |-  ( ( abs `  D )  <_  R  ->  ( R  <_  ( R  -  ( abs `  D ) )  <->  ( abs `  D
)  <_  0 ) )
6655, 65mpbid 210 . . . 4  |-  ( ( abs `  D )  <_  R  ->  ( abs `  D )  <_ 
0 )
6726, 66mto 176 . . 3  |-  -.  ( abs `  D )  <_  R
6816, 57ax-mp 5 . . . 4  |-  R  e.  RR
6968, 24ltnlei 9494 . . 3  |-  ( R  <  ( abs `  D
)  <->  -.  ( abs `  D )  <_  R
)
7067, 69mpbir 209 . 2  |-  R  < 
( abs `  D
)
7117, 70pm3.2i 455 1  |-  ( 0  <_  R  /\  R  <  ( abs `  D
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2605   {crab 2718    C_ wss 3327   class class class wbr 4291   `'ccnv 4838   ` cfv 5417  (class class class)co 6090   supcsup 7689   CCcc 9279   RRcr 9280   0cc0 9281   1c1 9282    x. cmul 9286    < clt 9417    <_ cle 9418    - cmin 9594   NNcn 10321   NN0cn0 10578   ZZcz 10645   ZZ>=cuz 10860   abscabs 12722    || cdivides 13534
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 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-2nd 6577  df-recs 6831  df-rdg 6865  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-sup 7690  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-n0 10579  df-z 10646  df-uz 10861  df-rp 10991  df-seq 11806  df-exp 11865  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-dvds 13535
This theorem is referenced by:  divalglem9  13604
  Copyright terms: Public domain W3C validator