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Theorem divalglem5 14353
Description: Lemma for divalg 14359. (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 14351 . . . . . 6  |-  sup ( S ,  RR ,  `'  <  )  e.  S
71, 6eqeltri 2513 . . . . 5  |-  R  e.  S
8 oveq2 6313 . . . . . . 7  |-  ( x  =  R  ->  ( N  -  x )  =  ( N  -  R ) )
98breq2d 4438 . . . . . 6  |-  ( x  =  R  ->  ( D  ||  ( N  -  x )  <->  D  ||  ( N  -  R )
) )
10 oveq2 6313 . . . . . . . . 9  |-  ( r  =  x  ->  ( N  -  r )  =  ( N  -  x ) )
1110breq2d 4438 . . . . . . . 8  |-  ( r  =  x  ->  ( D  ||  ( N  -  r )  <->  D  ||  ( N  -  x )
) )
1211cbvrabv 3086 . . . . . . 7  |-  { r  e.  NN0  |  D  ||  ( N  -  r
) }  =  {
x  e.  NN0  |  D  ||  ( N  -  x ) }
135, 12eqtri 2458 . . . . . 6  |-  S  =  { x  e.  NN0  |  D  ||  ( N  -  x ) }
149, 13elrab2 3237 . . . . 5  |-  ( R  e.  S  <->  ( R  e.  NN0  /\  D  ||  ( N  -  R
) ) )
157, 14mpbi 211 . . . 4  |-  ( R  e.  NN0  /\  D  ||  ( N  -  R
) )
1615simpli 459 . . 3  |-  R  e. 
NN0
1716nn0ge0i 10897 . 2  |-  0  <_  R
18 nnabscl 13367 . . . . . . 7  |-  ( ( D  e.  ZZ  /\  D  =/=  0 )  -> 
( abs `  D
)  e.  NN )
193, 4, 18mp2an 676 . . . . . 6  |-  ( abs `  D )  e.  NN
2019nngt0i 10643 . . . . 5  |-  0  <  ( abs `  D
)
21 0re 9642 . . . . . 6  |-  0  e.  RR
22 zcn 10942 . . . . . . . 8  |-  ( D  e.  ZZ  ->  D  e.  CC )
233, 22ax-mp 5 . . . . . . 7  |-  D  e.  CC
2423abscli 13436 . . . . . 6  |-  ( abs `  D )  e.  RR
2521, 24ltnlei 9754 . . . . 5  |-  ( 0  <  ( abs `  D
)  <->  -.  ( abs `  D )  <_  0
)
2620, 25mpbi 211 . . . 4  |-  -.  ( abs `  D )  <_ 
0
27 ssrab2 3552 . . . . . . . . 9  |-  { r  e.  NN0  |  D  ||  ( N  -  r
) }  C_  NN0
285, 27eqsstri 3500 . . . . . . . 8  |-  S  C_  NN0
29 nn0uz 11193 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
3028, 29sseqtri 3502 . . . . . . 7  |-  S  C_  ( ZZ>= `  0 )
31 nn0abscl 13354 . . . . . . . . . 10  |-  ( D  e.  ZZ  ->  ( abs `  D )  e. 
NN0 )
323, 31ax-mp 5 . . . . . . . . 9  |-  ( abs `  D )  e.  NN0
33 nn0sub2 10997 . . . . . . . . 9  |-  ( ( ( abs `  D
)  e.  NN0  /\  R  e.  NN0  /\  ( abs `  D )  <_  R )  ->  ( R  -  ( abs `  D ) )  e. 
NN0 )
3432, 16, 33mp3an12 1350 . . . . . . . 8  |-  ( ( abs `  D )  <_  R  ->  ( R  -  ( abs `  D ) )  e. 
NN0 )
3515a1i 11 . . . . . . . . 9  |-  ( ( abs `  D )  <_  R  ->  ( R  e.  NN0  /\  D  ||  ( N  -  R
) ) )
36 nn0z 10960 . . . . . . . . . . 11  |-  ( R  e.  NN0  ->  R  e.  ZZ )
37 1z 10967 . . . . . . . . . . . . 13  |-  1  e.  ZZ
382, 3divalglem0 14349 . . . . . . . . . . . . 13  |-  ( ( R  e.  ZZ  /\  1  e.  ZZ )  ->  ( D  ||  ( N  -  R )  ->  D  ||  ( N  -  ( R  -  ( 1  x.  ( abs `  D ) ) ) ) ) )
3937, 38mpan2 675 . . . . . . . . . . . 12  |-  ( R  e.  ZZ  ->  ( D  ||  ( N  -  R )  ->  D  ||  ( N  -  ( R  -  ( 1  x.  ( abs `  D
) ) ) ) ) )
4024recni 9654 . . . . . . . . . . . . . . . 16  |-  ( abs `  D )  e.  CC
4140mulid2i 9645 . . . . . . . . . . . . . . 15  |-  ( 1  x.  ( abs `  D
) )  =  ( abs `  D )
4241oveq2i 6316 . . . . . . . . . . . . . 14  |-  ( R  -  ( 1  x.  ( abs `  D
) ) )  =  ( R  -  ( abs `  D ) )
4342oveq2i 6316 . . . . . . . . . . . . 13  |-  ( N  -  ( R  -  ( 1  x.  ( abs `  D ) ) ) )  =  ( N  -  ( R  -  ( abs `  D
) ) )
4443breq2i 4434 . . . . . . . . . . . 12  |-  ( D 
||  ( N  -  ( R  -  (
1  x.  ( abs `  D ) ) ) )  <->  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) )
4539, 44syl6ib 229 . . . . . . . . . . 11  |-  ( R  e.  ZZ  ->  ( D  ||  ( N  -  R )  ->  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) ) )
4636, 45syl 17 . . . . . . . . . 10  |-  ( R  e.  NN0  ->  ( D 
||  ( N  -  R )  ->  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) ) )
4746imp 430 . . . . . . . . 9  |-  ( ( R  e.  NN0  /\  D  ||  ( N  -  R ) )  ->  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) )
4835, 47syl 17 . . . . . . . 8  |-  ( ( abs `  D )  <_  R  ->  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) )
49 oveq2 6313 . . . . . . . . . 10  |-  ( x  =  ( R  -  ( abs `  D ) )  ->  ( N  -  x )  =  ( N  -  ( R  -  ( abs `  D
) ) ) )
5049breq2d 4438 . . . . . . . . 9  |-  ( x  =  ( R  -  ( abs `  D ) )  ->  ( D  ||  ( N  -  x
)  <->  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) ) )
5150, 13elrab2 3237 . . . . . . . 8  |-  ( ( R  -  ( abs `  D ) )  e.  S  <->  ( ( R  -  ( abs `  D
) )  e.  NN0  /\  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) ) )
5234, 48, 51sylanbrc 668 . . . . . . 7  |-  ( ( abs `  D )  <_  R  ->  ( R  -  ( abs `  D ) )  e.  S )
53 infmssuzleOLD 11246 . . . . . . 7  |-  ( ( S  C_  ( ZZ>= ` 
0 )  /\  ( R  -  ( abs `  D ) )  e.  S )  ->  sup ( S ,  RR ,  `'  <  )  <_  ( R  -  ( abs `  D ) ) )
5430, 52, 53sylancr 667 . . . . . 6  |-  ( ( abs `  D )  <_  R  ->  sup ( S ,  RR ,  `'  <  )  <_  ( R  -  ( abs `  D ) ) )
551, 54syl5eqbr 4459 . . . . 5  |-  ( ( abs `  D )  <_  R  ->  R  <_  ( R  -  ( abs `  D ) ) )
5635simpld 460 . . . . . . . 8  |-  ( ( abs `  D )  <_  R  ->  R  e.  NN0 )
57 nn0re 10878 . . . . . . . 8  |-  ( R  e.  NN0  ->  R  e.  RR )
5856, 57syl 17 . . . . . . 7  |-  ( ( abs `  D )  <_  R  ->  R  e.  RR )
59 lesub 10092 . . . . . . . 8  |-  ( ( R  e.  RR  /\  R  e.  RR  /\  ( abs `  D )  e.  RR )  ->  ( R  <_  ( R  -  ( abs `  D ) )  <->  ( abs `  D
)  <_  ( R  -  R ) ) )
6024, 59mp3an3 1349 . . . . . . 7  |-  ( ( R  e.  RR  /\  R  e.  RR )  ->  ( R  <_  ( R  -  ( abs `  D ) )  <->  ( abs `  D )  <_  ( R  -  R )
) )
6158, 58, 60syl2anc 665 . . . . . 6  |-  ( ( abs `  D )  <_  R  ->  ( R  <_  ( R  -  ( abs `  D ) )  <->  ( abs `  D
)  <_  ( R  -  R ) ) )
6258recnd 9668 . . . . . . . 8  |-  ( ( abs `  D )  <_  R  ->  R  e.  CC )
6362subidd 9973 . . . . . . 7  |-  ( ( abs `  D )  <_  R  ->  ( R  -  R )  =  0 )
6463breq2d 4438 . . . . . 6  |-  ( ( abs `  D )  <_  R  ->  (
( abs `  D
)  <_  ( R  -  R )  <->  ( abs `  D )  <_  0
) )
6561, 64bitrd 256 . . . . 5  |-  ( ( abs `  D )  <_  R  ->  ( R  <_  ( R  -  ( abs `  D ) )  <->  ( abs `  D
)  <_  0 ) )
6655, 65mpbid 213 . . . 4  |-  ( ( abs `  D )  <_  R  ->  ( abs `  D )  <_ 
0 )
6726, 66mto 179 . . 3  |-  -.  ( abs `  D )  <_  R
6816, 57ax-mp 5 . . . 4  |-  R  e.  RR
6968, 24ltnlei 9754 . . 3  |-  ( R  <  ( abs `  D
)  <->  -.  ( abs `  D )  <_  R
)
7067, 69mpbir 212 . 2  |-  R  < 
( abs `  D
)
7117, 70pm3.2i 456 1  |-  ( 0  <_  R  /\  R  <  ( abs `  D
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   {crab 2786    C_ wss 3442   class class class wbr 4426   `'ccnv 4853   ` cfv 5601  (class class class)co 6305   supcsup 7960   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539    x. cmul 9543    < clt 9674    <_ cle 9675    - cmin 9859   NNcn 10609   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   abscabs 13276    || cdvds 14283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-seq 12211  df-exp 12270  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-dvds 14284
This theorem is referenced by:  divalglem9  14357
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