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Theorem divalglem2 12946
Description: Lemma for divalg 12954. (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 3414 . . . 4  |-  { r  e.  NN0  |  D  ||  ( N  -  r
) }  C_  NN0
31, 2eqsstri 3364 . . 3  |-  S  C_  NN0
4 nn0uz 10551 . . 3  |-  NN0  =  ( ZZ>= `  0 )
53, 4sseqtri 3366 . 2  |-  S  C_  ( ZZ>= `  0 )
6 divalglem0.1 . . . . . 6  |-  N  e.  ZZ
7 divalglem0.2 . . . . . . . . 9  |-  D  e.  ZZ
8 zmulcl 10355 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  ZZ )  ->  ( N  x.  D
)  e.  ZZ )
96, 7, 8mp2an 655 . . . . . . . 8  |-  ( N  x.  D )  e.  ZZ
10 nn0abscl 12148 . . . . . . . 8  |-  ( ( N  x.  D )  e.  ZZ  ->  ( abs `  ( N  x.  D ) )  e. 
NN0 )
119, 10ax-mp 5 . . . . . . 7  |-  ( abs `  ( N  x.  D
) )  e.  NN0
1211nn0zi 10337 . . . . . 6  |-  ( abs `  ( N  x.  D
) )  e.  ZZ
13 zaddcl 10348 . . . . . 6  |-  ( ( N  e.  ZZ  /\  ( abs `  ( N  x.  D ) )  e.  ZZ )  -> 
( N  +  ( abs `  ( N  x.  D ) ) )  e.  ZZ )
146, 12, 13mp2an 655 . . . . 5  |-  ( N  +  ( abs `  ( N  x.  D )
) )  e.  ZZ
15 divalglem1.3 . . . . . 6  |-  D  =/=  0
166, 7, 15divalglem1 12945 . . . . 5  |-  0  <_  ( N  +  ( abs `  ( N  x.  D ) ) )
17 elnn0z 10325 . . . . 5  |-  ( ( N  +  ( abs `  ( N  x.  D
) ) )  e. 
NN0 
<->  ( ( N  +  ( abs `  ( N  x.  D ) ) )  e.  ZZ  /\  0  <_  ( N  +  ( abs `  ( N  x.  D ) ) ) ) )
1814, 16, 17mpbir2an 888 . . . 4  |-  ( N  +  ( abs `  ( N  x.  D )
) )  e.  NN0
19 iddvds 12894 . . . . . . . 8  |-  ( D  e.  ZZ  ->  D  ||  D )
20 dvdsabsb 12900 . . . . . . . . 9  |-  ( ( D  e.  ZZ  /\  D  e.  ZZ )  ->  ( D  ||  D  <->  D 
||  ( abs `  D
) ) )
2120anidms 628 . . . . . . . 8  |-  ( D  e.  ZZ  ->  ( D  ||  D  <->  D  ||  ( abs `  D ) ) )
2219, 21mpbid 203 . . . . . . 7  |-  ( D  e.  ZZ  ->  D  ||  ( abs `  D
) )
237, 22ax-mp 5 . . . . . 6  |-  D  ||  ( abs `  D )
24 nn0abscl 12148 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  ( abs `  N )  e. 
NN0 )
256, 24ax-mp 5 . . . . . . . 8  |-  ( abs `  N )  e.  NN0
2625nn0negzi 10347 . . . . . . 7  |-  -u ( abs `  N )  e.  ZZ
27 nn0abscl 12148 . . . . . . . . 9  |-  ( D  e.  ZZ  ->  ( abs `  D )  e. 
NN0 )
287, 27ax-mp 5 . . . . . . . 8  |-  ( abs `  D )  e.  NN0
2928nn0zi 10337 . . . . . . 7  |-  ( abs `  D )  e.  ZZ
30 dvdsmultr2 12916 . . . . . . 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 1280 . . . . . 6  |-  ( D 
||  ( abs `  D
)  ->  D  ||  ( -u ( abs `  N
)  x.  ( abs `  D ) ) )
3223, 31ax-mp 5 . . . . 5  |-  D  ||  ( -u ( abs `  N
)  x.  ( abs `  D ) )
33 zcn 10318 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  CC )
346, 33ax-mp 5 . . . . . . . 8  |-  N  e.  CC
35 zcn 10318 . . . . . . . . 9  |-  ( D  e.  ZZ  ->  D  e.  CC )
367, 35ax-mp 5 . . . . . . . 8  |-  D  e.  CC
3734, 36absmuli 12238 . . . . . . 7  |-  ( abs `  ( N  x.  D
) )  =  ( ( abs `  N
)  x.  ( abs `  D ) )
3837negeqi 9330 . . . . . 6  |-  -u ( abs `  ( N  x.  D ) )  = 
-u ( ( abs `  N )  x.  ( abs `  D ) )
39 df-neg 9325 . . . . . . 7  |-  -u ( abs `  ( N  x.  D ) )  =  ( 0  -  ( abs `  ( N  x.  D ) ) )
4034subidi 9402 . . . . . . . 8  |-  ( N  -  N )  =  0
4140oveq1i 6120 . . . . . . 7  |-  ( ( N  -  N )  -  ( abs `  ( N  x.  D )
) )  =  ( 0  -  ( abs `  ( N  x.  D
) ) )
4211nn0cni 10264 . . . . . . . 8  |-  ( abs `  ( N  x.  D
) )  e.  CC
43 subsub4 9365 . . . . . . . 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 1280 . . . . . . 7  |-  ( ( N  -  N )  -  ( abs `  ( N  x.  D )
) )  =  ( N  -  ( N  +  ( abs `  ( N  x.  D )
) ) )
4539, 41, 443eqtr2ri 2469 . . . . . 6  |-  ( N  -  ( N  +  ( abs `  ( N  x.  D ) ) ) )  =  -u ( abs `  ( N  x.  D ) )
4634abscli 12229 . . . . . . . 8  |-  ( abs `  N )  e.  RR
4746recni 9133 . . . . . . 7  |-  ( abs `  N )  e.  CC
4836abscli 12229 . . . . . . . 8  |-  ( abs `  D )  e.  RR
4948recni 9133 . . . . . . 7  |-  ( abs `  D )  e.  CC
5047, 49mulneg1i 9510 . . . . . 6  |-  ( -u ( abs `  N )  x.  ( abs `  D
) )  =  -u ( ( abs `  N
)  x.  ( abs `  D ) )
5138, 45, 503eqtr4i 2472 . . . . 5  |-  ( N  -  ( N  +  ( abs `  ( N  x.  D ) ) ) )  =  (
-u ( abs `  N
)  x.  ( abs `  D ) )
5232, 51breqtrri 4262 . . . 4  |-  D  ||  ( N  -  ( N  +  ( abs `  ( N  x.  D
) ) ) )
53 oveq2 6118 . . . . . 6  |-  ( r  =  ( N  +  ( abs `  ( N  x.  D ) ) )  ->  ( N  -  r )  =  ( N  -  ( N  +  ( abs `  ( N  x.  D
) ) ) ) )
5453breq2d 4249 . . . . 5  |-  ( r  =  ( N  +  ( abs `  ( N  x.  D ) ) )  ->  ( D  ||  ( N  -  r
)  <->  D  ||  ( N  -  ( N  +  ( abs `  ( N  x.  D ) ) ) ) ) )
5554, 1elrab2 3100 . . . 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 888 . . 3  |-  ( N  +  ( abs `  ( N  x.  D )
) )  e.  S
57 ne0i 3619 . . 3  |-  ( ( N  +  ( abs `  ( N  x.  D
) ) )  e.  S  ->  S  =/=  (/) )
5856, 57ax-mp 5 . 2  |-  S  =/=  (/)
59 infmssuzcl 10590 . 2  |-  ( ( S  C_  ( ZZ>= ` 
0 )  /\  S  =/=  (/) )  ->  sup ( S ,  RR ,  `'  <  )  e.  S
)
605, 58, 59mp2an 655 1  |-  sup ( S ,  RR ,  `'  <  )  e.  S
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1727    =/= wne 2605   {crab 2715    C_ wss 3306   (/)c0 3613   class class class wbr 4237   `'ccnv 4906   ` cfv 5483  (class class class)co 6110   supcsup 7474   CCcc 9019   RRcr 9020   0cc0 9021    + caddc 9024    x. cmul 9026    < clt 9151    <_ cle 9152    - cmin 9322   -ucneg 9323   NN0cn0 10252   ZZcz 10313   ZZ>=cuz 10519   abscabs 12070    || cdivides 12883
This theorem is referenced by:  divalglem5  12948  divalglem9  12952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098  ax-pre-sup 9099
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-sup 7475  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709  df-nn 10032  df-2 10089  df-3 10090  df-n0 10253  df-z 10314  df-uz 10520  df-rp 10644  df-seq 11355  df-exp 11414  df-cj 11935  df-re 11936  df-im 11937  df-sqr 12071  df-abs 12072  df-dvds 12884
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