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Theorem divalglem5 12596
Description: Lemma for divalg 12602. (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 12594 . . . . . 6  |-  sup ( S ,  RR ,  `'  <  )  e.  S
71, 6eqeltri 2353 . . . . 5  |-  R  e.  S
8 oveq2 5866 . . . . . . 7  |-  ( x  =  R  ->  ( N  -  x )  =  ( N  -  R ) )
98breq2d 4035 . . . . . 6  |-  ( x  =  R  ->  ( D  ||  ( N  -  x )  <->  D  ||  ( N  -  R )
) )
10 oveq2 5866 . . . . . . . . 9  |-  ( r  =  x  ->  ( N  -  r )  =  ( N  -  x ) )
1110breq2d 4035 . . . . . . . 8  |-  ( r  =  x  ->  ( D  ||  ( N  -  r )  <->  D  ||  ( N  -  x )
) )
1211cbvrabv 2787 . . . . . . 7  |-  { r  e.  NN0  |  D  ||  ( N  -  r
) }  =  {
x  e.  NN0  |  D  ||  ( N  -  x ) }
135, 12eqtri 2303 . . . . . 6  |-  S  =  { x  e.  NN0  |  D  ||  ( N  -  x ) }
149, 13elrab2 2925 . . . . 5  |-  ( R  e.  S  <->  ( R  e.  NN0  /\  D  ||  ( N  -  R
) ) )
157, 14mpbi 199 . . . 4  |-  ( R  e.  NN0  /\  D  ||  ( N  -  R
) )
1615simpli 444 . . 3  |-  R  e. 
NN0
1716nn0ge0i 9993 . 2  |-  0  <_  R
18 nnabscl 11809 . . . . . . 7  |-  ( ( D  e.  ZZ  /\  D  =/=  0 )  -> 
( abs `  D
)  e.  NN )
193, 4, 18mp2an 653 . . . . . 6  |-  ( abs `  D )  e.  NN
2019nngt0i 9779 . . . . 5  |-  0  <  ( abs `  D
)
21 0re 8838 . . . . . 6  |-  0  e.  RR
22 zcn 10029 . . . . . . . 8  |-  ( D  e.  ZZ  ->  D  e.  CC )
233, 22ax-mp 8 . . . . . . 7  |-  D  e.  CC
2423abscli 11878 . . . . . 6  |-  ( abs `  D )  e.  RR
2521, 24ltnlei 8939 . . . . 5  |-  ( 0  <  ( abs `  D
)  <->  -.  ( abs `  D )  <_  0
)
2620, 25mpbi 199 . . . 4  |-  -.  ( abs `  D )  <_ 
0
27 ssrab2 3258 . . . . . . . . 9  |-  { r  e.  NN0  |  D  ||  ( N  -  r
) }  C_  NN0
285, 27eqsstri 3208 . . . . . . . 8  |-  S  C_  NN0
29 nn0uz 10262 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
3028, 29sseqtri 3210 . . . . . . 7  |-  S  C_  ( ZZ>= `  0 )
31 nn0abscl 11797 . . . . . . . . . 10  |-  ( D  e.  ZZ  ->  ( abs `  D )  e. 
NN0 )
323, 31ax-mp 8 . . . . . . . . 9  |-  ( abs `  D )  e.  NN0
33 nn0sub2 10077 . . . . . . . . 9  |-  ( ( ( abs `  D
)  e.  NN0  /\  R  e.  NN0  /\  ( abs `  D )  <_  R )  ->  ( R  -  ( abs `  D ) )  e. 
NN0 )
3432, 16, 33mp3an12 1267 . . . . . . . 8  |-  ( ( abs `  D )  <_  R  ->  ( R  -  ( abs `  D ) )  e. 
NN0 )
3515a1i 10 . . . . . . . . 9  |-  ( ( abs `  D )  <_  R  ->  ( R  e.  NN0  /\  D  ||  ( N  -  R
) ) )
36 nn0z 10046 . . . . . . . . . . 11  |-  ( R  e.  NN0  ->  R  e.  ZZ )
37 1z 10053 . . . . . . . . . . . . 13  |-  1  e.  ZZ
382, 3divalglem0 12592 . . . . . . . . . . . . 13  |-  ( ( R  e.  ZZ  /\  1  e.  ZZ )  ->  ( D  ||  ( N  -  R )  ->  D  ||  ( N  -  ( R  -  ( 1  x.  ( abs `  D ) ) ) ) ) )
3937, 38mpan2 652 . . . . . . . . . . . 12  |-  ( R  e.  ZZ  ->  ( D  ||  ( N  -  R )  ->  D  ||  ( N  -  ( R  -  ( 1  x.  ( abs `  D
) ) ) ) ) )
4024recni 8849 . . . . . . . . . . . . . . . 16  |-  ( abs `  D )  e.  CC
4140mulid2i 8840 . . . . . . . . . . . . . . 15  |-  ( 1  x.  ( abs `  D
) )  =  ( abs `  D )
4241oveq2i 5869 . . . . . . . . . . . . . 14  |-  ( R  -  ( 1  x.  ( abs `  D
) ) )  =  ( R  -  ( abs `  D ) )
4342oveq2i 5869 . . . . . . . . . . . . 13  |-  ( N  -  ( R  -  ( 1  x.  ( abs `  D ) ) ) )  =  ( N  -  ( R  -  ( abs `  D
) ) )
4443breq2i 4031 . . . . . . . . . . . 12  |-  ( D 
||  ( N  -  ( R  -  (
1  x.  ( abs `  D ) ) ) )  <->  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) )
4539, 44syl6ib 217 . . . . . . . . . . 11  |-  ( R  e.  ZZ  ->  ( D  ||  ( N  -  R )  ->  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) ) )
4636, 45syl 15 . . . . . . . . . 10  |-  ( R  e.  NN0  ->  ( D 
||  ( N  -  R )  ->  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) ) )
4746imp 418 . . . . . . . . 9  |-  ( ( R  e.  NN0  /\  D  ||  ( N  -  R ) )  ->  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) )
4835, 47syl 15 . . . . . . . 8  |-  ( ( abs `  D )  <_  R  ->  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) )
49 oveq2 5866 . . . . . . . . . 10  |-  ( x  =  ( R  -  ( abs `  D ) )  ->  ( N  -  x )  =  ( N  -  ( R  -  ( abs `  D
) ) ) )
5049breq2d 4035 . . . . . . . . 9  |-  ( x  =  ( R  -  ( abs `  D ) )  ->  ( D  ||  ( N  -  x
)  <->  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) ) )
5150, 13elrab2 2925 . . . . . . . 8  |-  ( ( R  -  ( abs `  D ) )  e.  S  <->  ( ( R  -  ( abs `  D
) )  e.  NN0  /\  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) ) )
5234, 48, 51sylanbrc 645 . . . . . . 7  |-  ( ( abs `  D )  <_  R  ->  ( R  -  ( abs `  D ) )  e.  S )
53 infmssuzle 10300 . . . . . . 7  |-  ( ( S  C_  ( ZZ>= ` 
0 )  /\  ( R  -  ( abs `  D ) )  e.  S )  ->  sup ( S ,  RR ,  `'  <  )  <_  ( R  -  ( abs `  D ) ) )
5430, 52, 53sylancr 644 . . . . . 6  |-  ( ( abs `  D )  <_  R  ->  sup ( S ,  RR ,  `'  <  )  <_  ( R  -  ( abs `  D ) ) )
551, 54syl5eqbr 4056 . . . . 5  |-  ( ( abs `  D )  <_  R  ->  R  <_  ( R  -  ( abs `  D ) ) )
5635simpld 445 . . . . . . . 8  |-  ( ( abs `  D )  <_  R  ->  R  e.  NN0 )
57 nn0re 9974 . . . . . . . 8  |-  ( R  e.  NN0  ->  R  e.  RR )
5856, 57syl 15 . . . . . . 7  |-  ( ( abs `  D )  <_  R  ->  R  e.  RR )
59 lesub 9253 . . . . . . . 8  |-  ( ( R  e.  RR  /\  R  e.  RR  /\  ( abs `  D )  e.  RR )  ->  ( R  <_  ( R  -  ( abs `  D ) )  <->  ( abs `  D
)  <_  ( R  -  R ) ) )
6024, 59mp3an3 1266 . . . . . . 7  |-  ( ( R  e.  RR  /\  R  e.  RR )  ->  ( R  <_  ( R  -  ( abs `  D ) )  <->  ( abs `  D )  <_  ( R  -  R )
) )
6158, 58, 60syl2anc 642 . . . . . 6  |-  ( ( abs `  D )  <_  R  ->  ( R  <_  ( R  -  ( abs `  D ) )  <->  ( abs `  D
)  <_  ( R  -  R ) ) )
6258recnd 8861 . . . . . . . 8  |-  ( ( abs `  D )  <_  R  ->  R  e.  CC )
6362subidd 9145 . . . . . . 7  |-  ( ( abs `  D )  <_  R  ->  ( R  -  R )  =  0 )
6463breq2d 4035 . . . . . 6  |-  ( ( abs `  D )  <_  R  ->  (
( abs `  D
)  <_  ( R  -  R )  <->  ( abs `  D )  <_  0
) )
6561, 64bitrd 244 . . . . 5  |-  ( ( abs `  D )  <_  R  ->  ( R  <_  ( R  -  ( abs `  D ) )  <->  ( abs `  D
)  <_  0 ) )
6655, 65mpbid 201 . . . 4  |-  ( ( abs `  D )  <_  R  ->  ( abs `  D )  <_ 
0 )
6726, 66mto 167 . . 3  |-  -.  ( abs `  D )  <_  R
6816, 57ax-mp 8 . . . 4  |-  R  e.  RR
6968, 24ltnlei 8939 . . 3  |-  ( R  <  ( abs `  D
)  <->  -.  ( abs `  D )  <_  R
)
7067, 69mpbir 200 . 2  |-  R  < 
( abs `  D
)
7117, 70pm3.2i 441 1  |-  ( 0  <_  R  /\  R  <  ( abs `  D
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547    C_ wss 3152   class class class wbr 4023   `'ccnv 4688   ` cfv 5255  (class class class)co 5858   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037   NNcn 9746   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   abscabs 11719    || cdivides 12531
This theorem is referenced by:  divalglem9  12600
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532
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