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Theorem divalglem2 13598
Description: Lemma for divalg 13606. (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 3436 . . . 4  |-  { r  e.  NN0  |  D  ||  ( N  -  r
) }  C_  NN0
31, 2eqsstri 3385 . . 3  |-  S  C_  NN0
4 nn0uz 10894 . . 3  |-  NN0  =  ( ZZ>= `  0 )
53, 4sseqtri 3387 . 2  |-  S  C_  ( ZZ>= `  0 )
6 divalglem0.1 . . . . . 6  |-  N  e.  ZZ
7 divalglem0.2 . . . . . . . . 9  |-  D  e.  ZZ
8 zmulcl 10692 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  ZZ )  ->  ( N  x.  D
)  e.  ZZ )
96, 7, 8mp2an 672 . . . . . . . 8  |-  ( N  x.  D )  e.  ZZ
10 nn0abscl 12800 . . . . . . . 8  |-  ( ( N  x.  D )  e.  ZZ  ->  ( abs `  ( N  x.  D ) )  e. 
NN0 )
119, 10ax-mp 5 . . . . . . 7  |-  ( abs `  ( N  x.  D
) )  e.  NN0
1211nn0zi 10670 . . . . . 6  |-  ( abs `  ( N  x.  D
) )  e.  ZZ
13 zaddcl 10684 . . . . . 6  |-  ( ( N  e.  ZZ  /\  ( abs `  ( N  x.  D ) )  e.  ZZ )  -> 
( N  +  ( abs `  ( N  x.  D ) ) )  e.  ZZ )
146, 12, 13mp2an 672 . . . . 5  |-  ( N  +  ( abs `  ( N  x.  D )
) )  e.  ZZ
15 divalglem1.3 . . . . . 6  |-  D  =/=  0
166, 7, 15divalglem1 13597 . . . . 5  |-  0  <_  ( N  +  ( abs `  ( N  x.  D ) ) )
17 elnn0z 10658 . . . . 5  |-  ( ( N  +  ( abs `  ( N  x.  D
) ) )  e. 
NN0 
<->  ( ( N  +  ( abs `  ( N  x.  D ) ) )  e.  ZZ  /\  0  <_  ( N  +  ( abs `  ( N  x.  D ) ) ) ) )
1814, 16, 17mpbir2an 911 . . . 4  |-  ( N  +  ( abs `  ( N  x.  D )
) )  e.  NN0
19 iddvds 13545 . . . . . . . 8  |-  ( D  e.  ZZ  ->  D  ||  D )
20 dvdsabsb 13551 . . . . . . . . 9  |-  ( ( D  e.  ZZ  /\  D  e.  ZZ )  ->  ( D  ||  D  <->  D 
||  ( abs `  D
) ) )
2120anidms 645 . . . . . . . 8  |-  ( D  e.  ZZ  ->  ( D  ||  D  <->  D  ||  ( abs `  D ) ) )
2219, 21mpbid 210 . . . . . . 7  |-  ( D  e.  ZZ  ->  D  ||  ( abs `  D
) )
237, 22ax-mp 5 . . . . . 6  |-  D  ||  ( abs `  D )
24 nn0abscl 12800 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  ( abs `  N )  e. 
NN0 )
256, 24ax-mp 5 . . . . . . . 8  |-  ( abs `  N )  e.  NN0
2625nn0negzi 10683 . . . . . . 7  |-  -u ( abs `  N )  e.  ZZ
27 nn0abscl 12800 . . . . . . . . 9  |-  ( D  e.  ZZ  ->  ( abs `  D )  e. 
NN0 )
287, 27ax-mp 5 . . . . . . . 8  |-  ( abs `  D )  e.  NN0
2928nn0zi 10670 . . . . . . 7  |-  ( abs `  D )  e.  ZZ
30 dvdsmultr2 13567 . . . . . . 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 1314 . . . . . 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 10650 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  CC )
346, 33ax-mp 5 . . . . . . . 8  |-  N  e.  CC
35 zcn 10650 . . . . . . . . 9  |-  ( D  e.  ZZ  ->  D  e.  CC )
367, 35ax-mp 5 . . . . . . . 8  |-  D  e.  CC
3734, 36absmuli 12890 . . . . . . 7  |-  ( abs `  ( N  x.  D
) )  =  ( ( abs `  N
)  x.  ( abs `  D ) )
3837negeqi 9602 . . . . . 6  |-  -u ( abs `  ( N  x.  D ) )  = 
-u ( ( abs `  N )  x.  ( abs `  D ) )
39 df-neg 9597 . . . . . . 7  |-  -u ( abs `  ( N  x.  D ) )  =  ( 0  -  ( abs `  ( N  x.  D ) ) )
4034subidi 9678 . . . . . . . 8  |-  ( N  -  N )  =  0
4140oveq1i 6100 . . . . . . 7  |-  ( ( N  -  N )  -  ( abs `  ( N  x.  D )
) )  =  ( 0  -  ( abs `  ( N  x.  D
) ) )
4211nn0cni 10590 . . . . . . . 8  |-  ( abs `  ( N  x.  D
) )  e.  CC
43 subsub4 9641 . . . . . . . 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 1314 . . . . . . 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 12881 . . . . . . . 8  |-  ( abs `  N )  e.  RR
4746recni 9397 . . . . . . 7  |-  ( abs `  N )  e.  CC
4836abscli 12881 . . . . . . . 8  |-  ( abs `  D )  e.  RR
4948recni 9397 . . . . . . 7  |-  ( abs `  D )  e.  CC
5047, 49mulneg1i 9789 . . . . . 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 4316 . . . 4  |-  D  ||  ( N  -  ( N  +  ( abs `  ( N  x.  D
) ) ) )
53 oveq2 6098 . . . . . 6  |-  ( r  =  ( N  +  ( abs `  ( N  x.  D ) ) )  ->  ( N  -  r )  =  ( N  -  ( N  +  ( abs `  ( N  x.  D
) ) ) ) )
5453breq2d 4303 . . . . 5  |-  ( r  =  ( N  +  ( abs `  ( N  x.  D ) ) )  ->  ( D  ||  ( N  -  r
)  <->  D  ||  ( N  -  ( N  +  ( abs `  ( N  x.  D ) ) ) ) ) )
5554, 1elrab2 3118 . . . 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 911 . . 3  |-  ( N  +  ( abs `  ( N  x.  D )
) )  e.  S
57 ne0i 3642 . . 3  |-  ( ( N  +  ( abs `  ( N  x.  D
) ) )  e.  S  ->  S  =/=  (/) )
5856, 57ax-mp 5 . 2  |-  S  =/=  (/)
59 infmssuzcl 10937 . 2  |-  ( ( S  C_  ( ZZ>= ` 
0 )  /\  S  =/=  (/) )  ->  sup ( S ,  RR ,  `'  <  )  e.  S
)
605, 58, 59mp2an 672 1  |-  sup ( S ,  RR ,  `'  <  )  e.  S
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756    =/= wne 2605   {crab 2718    C_ wss 3327   (/)c0 3636   class class class wbr 4291   `'ccnv 4838   ` cfv 5417  (class class class)co 6090   supcsup 7689   CCcc 9279   RRcr 9280   0cc0 9281    + caddc 9284    x. cmul 9286    < clt 9417    <_ cle 9418    - cmin 9594   -ucneg 9595   NN0cn0 10578   ZZcz 10645   ZZ>=cuz 10860   abscabs 12722    || cdivides 13534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  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-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-2nd 6577  df-recs 6831  df-rdg 6865  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-sup 7690  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-n0 10579  df-z 10646  df-uz 10861  df-rp 10991  df-seq 11806  df-exp 11865  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-dvds 13535
This theorem is referenced by:  divalglem5  13600  divalglem9  13604
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