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Theorem divalglem5 12917
Description: Lemma for divalg 12923. (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 12915 . . . . . 6  |-  sup ( S ,  RR ,  `'  <  )  e.  S
71, 6eqeltri 2506 . . . . 5  |-  R  e.  S
8 oveq2 6089 . . . . . . 7  |-  ( x  =  R  ->  ( N  -  x )  =  ( N  -  R ) )
98breq2d 4224 . . . . . 6  |-  ( x  =  R  ->  ( D  ||  ( N  -  x )  <->  D  ||  ( N  -  R )
) )
10 oveq2 6089 . . . . . . . . 9  |-  ( r  =  x  ->  ( N  -  r )  =  ( N  -  x ) )
1110breq2d 4224 . . . . . . . 8  |-  ( r  =  x  ->  ( D  ||  ( N  -  r )  <->  D  ||  ( N  -  x )
) )
1211cbvrabv 2955 . . . . . . 7  |-  { r  e.  NN0  |  D  ||  ( N  -  r
) }  =  {
x  e.  NN0  |  D  ||  ( N  -  x ) }
135, 12eqtri 2456 . . . . . 6  |-  S  =  { x  e.  NN0  |  D  ||  ( N  -  x ) }
149, 13elrab2 3094 . . . . 5  |-  ( R  e.  S  <->  ( R  e.  NN0  /\  D  ||  ( N  -  R
) ) )
157, 14mpbi 200 . . . 4  |-  ( R  e.  NN0  /\  D  ||  ( N  -  R
) )
1615simpli 445 . . 3  |-  R  e. 
NN0
1716nn0ge0i 10249 . 2  |-  0  <_  R
18 nnabscl 12129 . . . . . . 7  |-  ( ( D  e.  ZZ  /\  D  =/=  0 )  -> 
( abs `  D
)  e.  NN )
193, 4, 18mp2an 654 . . . . . 6  |-  ( abs `  D )  e.  NN
2019nngt0i 10033 . . . . 5  |-  0  <  ( abs `  D
)
21 0re 9091 . . . . . 6  |-  0  e.  RR
22 zcn 10287 . . . . . . . 8  |-  ( D  e.  ZZ  ->  D  e.  CC )
233, 22ax-mp 8 . . . . . . 7  |-  D  e.  CC
2423abscli 12198 . . . . . 6  |-  ( abs `  D )  e.  RR
2521, 24ltnlei 9194 . . . . 5  |-  ( 0  <  ( abs `  D
)  <->  -.  ( abs `  D )  <_  0
)
2620, 25mpbi 200 . . . 4  |-  -.  ( abs `  D )  <_ 
0
27 ssrab2 3428 . . . . . . . . 9  |-  { r  e.  NN0  |  D  ||  ( N  -  r
) }  C_  NN0
285, 27eqsstri 3378 . . . . . . . 8  |-  S  C_  NN0
29 nn0uz 10520 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
3028, 29sseqtri 3380 . . . . . . 7  |-  S  C_  ( ZZ>= `  0 )
31 nn0abscl 12117 . . . . . . . . . 10  |-  ( D  e.  ZZ  ->  ( abs `  D )  e. 
NN0 )
323, 31ax-mp 8 . . . . . . . . 9  |-  ( abs `  D )  e.  NN0
33 nn0sub2 10335 . . . . . . . . 9  |-  ( ( ( abs `  D
)  e.  NN0  /\  R  e.  NN0  /\  ( abs `  D )  <_  R )  ->  ( R  -  ( abs `  D ) )  e. 
NN0 )
3432, 16, 33mp3an12 1269 . . . . . . . 8  |-  ( ( abs `  D )  <_  R  ->  ( R  -  ( abs `  D ) )  e. 
NN0 )
3515a1i 11 . . . . . . . . 9  |-  ( ( abs `  D )  <_  R  ->  ( R  e.  NN0  /\  D  ||  ( N  -  R
) ) )
36 nn0z 10304 . . . . . . . . . . 11  |-  ( R  e.  NN0  ->  R  e.  ZZ )
37 1z 10311 . . . . . . . . . . . . 13  |-  1  e.  ZZ
382, 3divalglem0 12913 . . . . . . . . . . . . 13  |-  ( ( R  e.  ZZ  /\  1  e.  ZZ )  ->  ( D  ||  ( N  -  R )  ->  D  ||  ( N  -  ( R  -  ( 1  x.  ( abs `  D ) ) ) ) ) )
3937, 38mpan2 653 . . . . . . . . . . . 12  |-  ( R  e.  ZZ  ->  ( D  ||  ( N  -  R )  ->  D  ||  ( N  -  ( R  -  ( 1  x.  ( abs `  D
) ) ) ) ) )
4024recni 9102 . . . . . . . . . . . . . . . 16  |-  ( abs `  D )  e.  CC
4140mulid2i 9093 . . . . . . . . . . . . . . 15  |-  ( 1  x.  ( abs `  D
) )  =  ( abs `  D )
4241oveq2i 6092 . . . . . . . . . . . . . 14  |-  ( R  -  ( 1  x.  ( abs `  D
) ) )  =  ( R  -  ( abs `  D ) )
4342oveq2i 6092 . . . . . . . . . . . . 13  |-  ( N  -  ( R  -  ( 1  x.  ( abs `  D ) ) ) )  =  ( N  -  ( R  -  ( abs `  D
) ) )
4443breq2i 4220 . . . . . . . . . . . 12  |-  ( D 
||  ( N  -  ( R  -  (
1  x.  ( abs `  D ) ) ) )  <->  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) )
4539, 44syl6ib 218 . . . . . . . . . . 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 419 . . . . . . . . 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 6089 . . . . . . . . . 10  |-  ( x  =  ( R  -  ( abs `  D ) )  ->  ( N  -  x )  =  ( N  -  ( R  -  ( abs `  D
) ) ) )
5049breq2d 4224 . . . . . . . . 9  |-  ( x  =  ( R  -  ( abs `  D ) )  ->  ( D  ||  ( N  -  x
)  <->  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) ) )
5150, 13elrab2 3094 . . . . . . . 8  |-  ( ( R  -  ( abs `  D ) )  e.  S  <->  ( ( R  -  ( abs `  D
) )  e.  NN0  /\  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) ) )
5234, 48, 51sylanbrc 646 . . . . . . 7  |-  ( ( abs `  D )  <_  R  ->  ( R  -  ( abs `  D ) )  e.  S )
53 infmssuzle 10558 . . . . . . 7  |-  ( ( S  C_  ( ZZ>= ` 
0 )  /\  ( R  -  ( abs `  D ) )  e.  S )  ->  sup ( S ,  RR ,  `'  <  )  <_  ( R  -  ( abs `  D ) ) )
5430, 52, 53sylancr 645 . . . . . 6  |-  ( ( abs `  D )  <_  R  ->  sup ( S ,  RR ,  `'  <  )  <_  ( R  -  ( abs `  D ) ) )
551, 54syl5eqbr 4245 . . . . 5  |-  ( ( abs `  D )  <_  R  ->  R  <_  ( R  -  ( abs `  D ) ) )
5635simpld 446 . . . . . . . 8  |-  ( ( abs `  D )  <_  R  ->  R  e.  NN0 )
57 nn0re 10230 . . . . . . . 8  |-  ( R  e.  NN0  ->  R  e.  RR )
5856, 57syl 16 . . . . . . 7  |-  ( ( abs `  D )  <_  R  ->  R  e.  RR )
59 lesub 9507 . . . . . . . 8  |-  ( ( R  e.  RR  /\  R  e.  RR  /\  ( abs `  D )  e.  RR )  ->  ( R  <_  ( R  -  ( abs `  D ) )  <->  ( abs `  D
)  <_  ( R  -  R ) ) )
6024, 59mp3an3 1268 . . . . . . 7  |-  ( ( R  e.  RR  /\  R  e.  RR )  ->  ( R  <_  ( R  -  ( abs `  D ) )  <->  ( abs `  D )  <_  ( R  -  R )
) )
6158, 58, 60syl2anc 643 . . . . . 6  |-  ( ( abs `  D )  <_  R  ->  ( R  <_  ( R  -  ( abs `  D ) )  <->  ( abs `  D
)  <_  ( R  -  R ) ) )
6258recnd 9114 . . . . . . . 8  |-  ( ( abs `  D )  <_  R  ->  R  e.  CC )
6362subidd 9399 . . . . . . 7  |-  ( ( abs `  D )  <_  R  ->  ( R  -  R )  =  0 )
6463breq2d 4224 . . . . . 6  |-  ( ( abs `  D )  <_  R  ->  (
( abs `  D
)  <_  ( R  -  R )  <->  ( abs `  D )  <_  0
) )
6561, 64bitrd 245 . . . . 5  |-  ( ( abs `  D )  <_  R  ->  ( R  <_  ( R  -  ( abs `  D ) )  <->  ( abs `  D
)  <_  0 ) )
6655, 65mpbid 202 . . . 4  |-  ( ( abs `  D )  <_  R  ->  ( abs `  D )  <_ 
0 )
6726, 66mto 169 . . 3  |-  -.  ( abs `  D )  <_  R
6816, 57ax-mp 8 . . . 4  |-  R  e.  RR
6968, 24ltnlei 9194 . . 3  |-  ( R  <  ( abs `  D
)  <->  -.  ( abs `  D )  <_  R
)
7067, 69mpbir 201 . 2  |-  R  < 
( abs `  D
)
7117, 70pm3.2i 442 1  |-  ( 0  <_  R  /\  R  <  ( abs `  D
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   {crab 2709    C_ wss 3320   class class class wbr 4212   `'ccnv 4877   ` cfv 5454  (class class class)co 6081   supcsup 7445   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    x. cmul 8995    < clt 9120    <_ cle 9121    - cmin 9291   NNcn 10000   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488   abscabs 12039    || cdivides 12852
This theorem is referenced by:  divalglem9  12921
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-dvds 12853
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