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Theorem divalglem2 12594
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 ) }
Assertion
Ref Expression
divalglem2  |-  sup ( S ,  RR ,  `'  <  )  e.  S
Distinct variable groups:    D, r    N, r
Allowed substitution hint:    S( r)

Proof of Theorem divalglem2
StepHypRef Expression
1 divalglem2.4 . . . 4  |-  S  =  { r  e.  NN0  |  D  ||  ( N  -  r ) }
2 ssrab2 3258 . . . 4  |-  { r  e.  NN0  |  D  ||  ( N  -  r
) }  C_  NN0
31, 2eqsstri 3208 . . 3  |-  S  C_  NN0
4 nn0uz 10262 . . 3  |-  NN0  =  ( ZZ>= `  0 )
53, 4sseqtri 3210 . 2  |-  S  C_  ( ZZ>= `  0 )
6 divalglem0.1 . . . . . 6  |-  N  e.  ZZ
7 divalglem0.2 . . . . . . . . 9  |-  D  e.  ZZ
8 zmulcl 10066 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  ZZ )  ->  ( N  x.  D
)  e.  ZZ )
96, 7, 8mp2an 653 . . . . . . . 8  |-  ( N  x.  D )  e.  ZZ
10 nn0abscl 11797 . . . . . . . 8  |-  ( ( N  x.  D )  e.  ZZ  ->  ( abs `  ( N  x.  D ) )  e. 
NN0 )
119, 10ax-mp 8 . . . . . . 7  |-  ( abs `  ( N  x.  D
) )  e.  NN0
1211nn0zi 10048 . . . . . 6  |-  ( abs `  ( N  x.  D
) )  e.  ZZ
13 zaddcl 10059 . . . . . 6  |-  ( ( N  e.  ZZ  /\  ( abs `  ( N  x.  D ) )  e.  ZZ )  -> 
( N  +  ( abs `  ( N  x.  D ) ) )  e.  ZZ )
146, 12, 13mp2an 653 . . . . 5  |-  ( N  +  ( abs `  ( N  x.  D )
) )  e.  ZZ
15 divalglem1.3 . . . . . 6  |-  D  =/=  0
166, 7, 15divalglem1 12593 . . . . 5  |-  0  <_  ( N  +  ( abs `  ( N  x.  D ) ) )
17 elnn0z 10036 . . . . 5  |-  ( ( N  +  ( abs `  ( N  x.  D
) ) )  e. 
NN0 
<->  ( ( N  +  ( abs `  ( N  x.  D ) ) )  e.  ZZ  /\  0  <_  ( N  +  ( abs `  ( N  x.  D ) ) ) ) )
1814, 16, 17mpbir2an 886 . . . 4  |-  ( N  +  ( abs `  ( N  x.  D )
) )  e.  NN0
19 iddvds 12542 . . . . . . . 8  |-  ( D  e.  ZZ  ->  D  ||  D )
20 dvdsabsb 12548 . . . . . . . . 9  |-  ( ( D  e.  ZZ  /\  D  e.  ZZ )  ->  ( D  ||  D  <->  D 
||  ( abs `  D
) ) )
2120anidms 626 . . . . . . . 8  |-  ( D  e.  ZZ  ->  ( D  ||  D  <->  D  ||  ( abs `  D ) ) )
2219, 21mpbid 201 . . . . . . 7  |-  ( D  e.  ZZ  ->  D  ||  ( abs `  D
) )
237, 22ax-mp 8 . . . . . 6  |-  D  ||  ( abs `  D )
24 nn0abscl 11797 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  ( abs `  N )  e. 
NN0 )
256, 24ax-mp 8 . . . . . . . 8  |-  ( abs `  N )  e.  NN0
2625nn0negzi 10058 . . . . . . 7  |-  -u ( abs `  N )  e.  ZZ
27 nn0abscl 11797 . . . . . . . . 9  |-  ( D  e.  ZZ  ->  ( abs `  D )  e. 
NN0 )
287, 27ax-mp 8 . . . . . . . 8  |-  ( abs `  D )  e.  NN0
2928nn0zi 10048 . . . . . . 7  |-  ( abs `  D )  e.  ZZ
30 dvdsmultr2 12564 . . . . . . 7  |-  ( ( D  e.  ZZ  /\  -u ( abs `  N
)  e.  ZZ  /\  ( abs `  D )  e.  ZZ )  -> 
( D  ||  ( abs `  D )  ->  D  ||  ( -u ( abs `  N )  x.  ( abs `  D
) ) ) )
317, 26, 29, 30mp3an 1277 . . . . . 6  |-  ( D 
||  ( abs `  D
)  ->  D  ||  ( -u ( abs `  N
)  x.  ( abs `  D ) ) )
3223, 31ax-mp 8 . . . . 5  |-  D  ||  ( -u ( abs `  N
)  x.  ( abs `  D ) )
33 zcn 10029 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  CC )
346, 33ax-mp 8 . . . . . . . 8  |-  N  e.  CC
35 zcn 10029 . . . . . . . . 9  |-  ( D  e.  ZZ  ->  D  e.  CC )
367, 35ax-mp 8 . . . . . . . 8  |-  D  e.  CC
3734, 36absmuli 11887 . . . . . . 7  |-  ( abs `  ( N  x.  D
) )  =  ( ( abs `  N
)  x.  ( abs `  D ) )
3837negeqi 9045 . . . . . 6  |-  -u ( abs `  ( N  x.  D ) )  = 
-u ( ( abs `  N )  x.  ( abs `  D ) )
39 df-neg 9040 . . . . . . 7  |-  -u ( abs `  ( N  x.  D ) )  =  ( 0  -  ( abs `  ( N  x.  D ) ) )
4034subidi 9117 . . . . . . . 8  |-  ( N  -  N )  =  0
4140oveq1i 5868 . . . . . . 7  |-  ( ( N  -  N )  -  ( abs `  ( N  x.  D )
) )  =  ( 0  -  ( abs `  ( N  x.  D
) ) )
4211nn0cni 9977 . . . . . . . 8  |-  ( abs `  ( N  x.  D
) )  e.  CC
43 subsub4 9080 . . . . . . . 8  |-  ( ( N  e.  CC  /\  N  e.  CC  /\  ( abs `  ( N  x.  D ) )  e.  CC )  ->  (
( N  -  N
)  -  ( abs `  ( N  x.  D
) ) )  =  ( N  -  ( N  +  ( abs `  ( N  x.  D
) ) ) ) )
4434, 34, 42, 43mp3an 1277 . . . . . . 7  |-  ( ( N  -  N )  -  ( abs `  ( N  x.  D )
) )  =  ( N  -  ( N  +  ( abs `  ( N  x.  D )
) ) )
4539, 41, 443eqtr2ri 2310 . . . . . 6  |-  ( N  -  ( N  +  ( abs `  ( N  x.  D ) ) ) )  =  -u ( abs `  ( N  x.  D ) )
4634abscli 11878 . . . . . . . 8  |-  ( abs `  N )  e.  RR
4746recni 8849 . . . . . . 7  |-  ( abs `  N )  e.  CC
4836abscli 11878 . . . . . . . 8  |-  ( abs `  D )  e.  RR
4948recni 8849 . . . . . . 7  |-  ( abs `  D )  e.  CC
5047, 49mulneg1i 9225 . . . . . 6  |-  ( -u ( abs `  N )  x.  ( abs `  D
) )  =  -u ( ( abs `  N
)  x.  ( abs `  D ) )
5138, 45, 503eqtr4i 2313 . . . . 5  |-  ( N  -  ( N  +  ( abs `  ( N  x.  D ) ) ) )  =  (
-u ( abs `  N
)  x.  ( abs `  D ) )
5232, 51breqtrri 4048 . . . 4  |-  D  ||  ( N  -  ( N  +  ( abs `  ( N  x.  D
) ) ) )
53 oveq2 5866 . . . . . 6  |-  ( r  =  ( N  +  ( abs `  ( N  x.  D ) ) )  ->  ( N  -  r )  =  ( N  -  ( N  +  ( abs `  ( N  x.  D
) ) ) ) )
5453breq2d 4035 . . . . 5  |-  ( r  =  ( N  +  ( abs `  ( N  x.  D ) ) )  ->  ( D  ||  ( N  -  r
)  <->  D  ||  ( N  -  ( N  +  ( abs `  ( N  x.  D ) ) ) ) ) )
5554, 1elrab2 2925 . . . 4  |-  ( ( N  +  ( abs `  ( N  x.  D
) ) )  e.  S  <->  ( ( N  +  ( abs `  ( N  x.  D )
) )  e.  NN0  /\  D  ||  ( N  -  ( N  +  ( abs `  ( N  x.  D ) ) ) ) ) )
5618, 52, 55mpbir2an 886 . . 3  |-  ( N  +  ( abs `  ( N  x.  D )
) )  e.  S
57 ne0i 3461 . . 3  |-  ( ( N  +  ( abs `  ( N  x.  D
) ) )  e.  S  ->  S  =/=  (/) )
5856, 57ax-mp 8 . 2  |-  S  =/=  (/)
59 infmssuzcl 10301 . 2  |-  ( ( S  C_  ( ZZ>= ` 
0 )  /\  S  =/=  (/) )  ->  sup ( S ,  RR ,  `'  <  )  e.  S
)
605, 58, 59mp2an 653 1  |-  sup ( S ,  RR ,  `'  <  )  e.  S
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547    C_ wss 3152   (/)c0 3455   class class class wbr 4023   `'ccnv 4688   ` cfv 5255  (class class class)co 5858   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037   -ucneg 9038   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   abscabs 11719    || cdivides 12531
This theorem is referenced by:  divalglem5  12596  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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