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Theorem divalglem5OLD 14455
Description: Lemma for divalg 14463. (Contributed by Paul Chapman, 21-Mar-2011.) Obsolete version of divalglem5 14456 as of 2-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
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 ) }
divalglem5OLD.5  |-  R  =  sup ( S ,  RR ,  `'  <  )
Assertion
Ref Expression
divalglem5OLD  |-  ( 0  <_  R  /\  R  <  ( abs `  D
) )
Distinct variable groups:    D, r    N, r
Allowed substitution hints:    R( r)    S( r)

Proof of Theorem divalglem5OLD
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 divalglem5OLD.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, 5divalglem2OLD 14453 . . . . . 6  |-  sup ( S ,  RR ,  `'  <  )  e.  S
71, 6eqeltri 2545 . . . . 5  |-  R  e.  S
8 oveq2 6316 . . . . . . 7  |-  ( x  =  R  ->  ( N  -  x )  =  ( N  -  R ) )
98breq2d 4407 . . . . . 6  |-  ( x  =  R  ->  ( D  ||  ( N  -  x )  <->  D  ||  ( N  -  R )
) )
10 oveq2 6316 . . . . . . . . 9  |-  ( r  =  x  ->  ( N  -  r )  =  ( N  -  x ) )
1110breq2d 4407 . . . . . . . 8  |-  ( r  =  x  ->  ( D  ||  ( N  -  r )  <->  D  ||  ( N  -  x )
) )
1211cbvrabv 3030 . . . . . . 7  |-  { r  e.  NN0  |  D  ||  ( N  -  r
) }  =  {
x  e.  NN0  |  D  ||  ( N  -  x ) }
135, 12eqtri 2493 . . . . . 6  |-  S  =  { x  e.  NN0  |  D  ||  ( N  -  x ) }
149, 13elrab2 3186 . . . . 5  |-  ( R  e.  S  <->  ( R  e.  NN0  /\  D  ||  ( N  -  R
) ) )
157, 14mpbi 213 . . . 4  |-  ( R  e.  NN0  /\  D  ||  ( N  -  R
) )
1615simpli 465 . . 3  |-  R  e. 
NN0
1716nn0ge0i 10921 . 2  |-  0  <_  R
18 nnabscl 13465 . . . . . . 7  |-  ( ( D  e.  ZZ  /\  D  =/=  0 )  -> 
( abs `  D
)  e.  NN )
193, 4, 18mp2an 686 . . . . . 6  |-  ( abs `  D )  e.  NN
2019nngt0i 10665 . . . . 5  |-  0  <  ( abs `  D
)
21 0re 9661 . . . . . 6  |-  0  e.  RR
22 zcn 10966 . . . . . . . 8  |-  ( D  e.  ZZ  ->  D  e.  CC )
233, 22ax-mp 5 . . . . . . 7  |-  D  e.  CC
2423abscli 13534 . . . . . 6  |-  ( abs `  D )  e.  RR
2521, 24ltnlei 9773 . . . . 5  |-  ( 0  <  ( abs `  D
)  <->  -.  ( abs `  D )  <_  0
)
2620, 25mpbi 213 . . . 4  |-  -.  ( abs `  D )  <_ 
0
27 ssrab2 3500 . . . . . . . . 9  |-  { r  e.  NN0  |  D  ||  ( N  -  r
) }  C_  NN0
285, 27eqsstri 3448 . . . . . . . 8  |-  S  C_  NN0
29 nn0uz 11217 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
3028, 29sseqtri 3450 . . . . . . 7  |-  S  C_  ( ZZ>= `  0 )
31 nn0abscl 13452 . . . . . . . . . 10  |-  ( D  e.  ZZ  ->  ( abs `  D )  e. 
NN0 )
323, 31ax-mp 5 . . . . . . . . 9  |-  ( abs `  D )  e.  NN0
33 nn0sub2 11021 . . . . . . . . 9  |-  ( ( ( abs `  D
)  e.  NN0  /\  R  e.  NN0  /\  ( abs `  D )  <_  R )  ->  ( R  -  ( abs `  D ) )  e. 
NN0 )
3432, 16, 33mp3an12 1380 . . . . . . . 8  |-  ( ( abs `  D )  <_  R  ->  ( R  -  ( abs `  D ) )  e. 
NN0 )
3515a1i 11 . . . . . . . . 9  |-  ( ( abs `  D )  <_  R  ->  ( R  e.  NN0  /\  D  ||  ( N  -  R
) ) )
36 nn0z 10984 . . . . . . . . . . 11  |-  ( R  e.  NN0  ->  R  e.  ZZ )
37 1z 10991 . . . . . . . . . . . . 13  |-  1  e.  ZZ
382, 3divalglem0 14450 . . . . . . . . . . . . 13  |-  ( ( R  e.  ZZ  /\  1  e.  ZZ )  ->  ( D  ||  ( N  -  R )  ->  D  ||  ( N  -  ( R  -  ( 1  x.  ( abs `  D ) ) ) ) ) )
3937, 38mpan2 685 . . . . . . . . . . . 12  |-  ( R  e.  ZZ  ->  ( D  ||  ( N  -  R )  ->  D  ||  ( N  -  ( R  -  ( 1  x.  ( abs `  D
) ) ) ) ) )
4024recni 9673 . . . . . . . . . . . . . . . 16  |-  ( abs `  D )  e.  CC
4140mulid2i 9664 . . . . . . . . . . . . . . 15  |-  ( 1  x.  ( abs `  D
) )  =  ( abs `  D )
4241oveq2i 6319 . . . . . . . . . . . . . 14  |-  ( R  -  ( 1  x.  ( abs `  D
) ) )  =  ( R  -  ( abs `  D ) )
4342oveq2i 6319 . . . . . . . . . . . . 13  |-  ( N  -  ( R  -  ( 1  x.  ( abs `  D ) ) ) )  =  ( N  -  ( R  -  ( abs `  D
) ) )
4443breq2i 4403 . . . . . . . . . . . 12  |-  ( D 
||  ( N  -  ( R  -  (
1  x.  ( abs `  D ) ) ) )  <->  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) )
4539, 44syl6ib 234 . . . . . . . . . . 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 436 . . . . . . . . 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 6316 . . . . . . . . . 10  |-  ( x  =  ( R  -  ( abs `  D ) )  ->  ( N  -  x )  =  ( N  -  ( R  -  ( abs `  D
) ) ) )
5049breq2d 4407 . . . . . . . . 9  |-  ( x  =  ( R  -  ( abs `  D ) )  ->  ( D  ||  ( N  -  x
)  <->  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) ) )
5150, 13elrab2 3186 . . . . . . . 8  |-  ( ( R  -  ( abs `  D ) )  e.  S  <->  ( ( R  -  ( abs `  D
) )  e.  NN0  /\  D  ||  ( N  -  ( R  -  ( abs `  D ) ) ) ) )
5234, 48, 51sylanbrc 677 . . . . . . 7  |-  ( ( abs `  D )  <_  R  ->  ( R  -  ( abs `  D ) )  e.  S )
53 infmssuzleOLD 11269 . . . . . . 7  |-  ( ( S  C_  ( ZZ>= ` 
0 )  /\  ( R  -  ( abs `  D ) )  e.  S )  ->  sup ( S ,  RR ,  `'  <  )  <_  ( R  -  ( abs `  D ) ) )
5430, 52, 53sylancr 676 . . . . . 6  |-  ( ( abs `  D )  <_  R  ->  sup ( S ,  RR ,  `'  <  )  <_  ( R  -  ( abs `  D ) ) )
551, 54syl5eqbr 4429 . . . . 5  |-  ( ( abs `  D )  <_  R  ->  R  <_  ( R  -  ( abs `  D ) ) )
5635simpld 466 . . . . . . . 8  |-  ( ( abs `  D )  <_  R  ->  R  e.  NN0 )
57 nn0re 10902 . . . . . . . 8  |-  ( R  e.  NN0  ->  R  e.  RR )
5856, 57syl 17 . . . . . . 7  |-  ( ( abs `  D )  <_  R  ->  R  e.  RR )
59 lesub 10114 . . . . . . . 8  |-  ( ( R  e.  RR  /\  R  e.  RR  /\  ( abs `  D )  e.  RR )  ->  ( R  <_  ( R  -  ( abs `  D ) )  <->  ( abs `  D
)  <_  ( R  -  R ) ) )
6024, 59mp3an3 1379 . . . . . . 7  |-  ( ( R  e.  RR  /\  R  e.  RR )  ->  ( R  <_  ( R  -  ( abs `  D ) )  <->  ( abs `  D )  <_  ( R  -  R )
) )
6158, 58, 60syl2anc 673 . . . . . 6  |-  ( ( abs `  D )  <_  R  ->  ( R  <_  ( R  -  ( abs `  D ) )  <->  ( abs `  D
)  <_  ( R  -  R ) ) )
6258recnd 9687 . . . . . . . 8  |-  ( ( abs `  D )  <_  R  ->  R  e.  CC )
6362subidd 9993 . . . . . . 7  |-  ( ( abs `  D )  <_  R  ->  ( R  -  R )  =  0 )
6463breq2d 4407 . . . . . 6  |-  ( ( abs `  D )  <_  R  ->  (
( abs `  D
)  <_  ( R  -  R )  <->  ( abs `  D )  <_  0
) )
6561, 64bitrd 261 . . . . 5  |-  ( ( abs `  D )  <_  R  ->  ( R  <_  ( R  -  ( abs `  D ) )  <->  ( abs `  D
)  <_  0 ) )
6655, 65mpbid 215 . . . 4  |-  ( ( abs `  D )  <_  R  ->  ( abs `  D )  <_ 
0 )
6726, 66mto 181 . . 3  |-  -.  ( abs `  D )  <_  R
6816, 57ax-mp 5 . . . 4  |-  R  e.  RR
6968, 24ltnlei 9773 . . 3  |-  ( R  <  ( abs `  D
)  <->  -.  ( abs `  D )  <_  R
)
7067, 69mpbir 214 . 2  |-  R  < 
( abs `  D
)
7117, 70pm3.2i 462 1  |-  ( 0  <_  R  /\  R  <  ( abs `  D
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   {crab 2760    C_ wss 3390   class class class wbr 4395   `'ccnv 4838   ` cfv 5589  (class class class)co 6308   supcsup 7972   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    x. cmul 9562    < clt 9693    <_ cle 9694    - cmin 9880   NNcn 10631   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   abscabs 13374    || cdvds 14382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  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-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-dvds 14383
This theorem is referenced by:  divalglem9OLD  14461
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