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Theorem divalglem2 14065
Description: Lemma for divalg 14073. (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 3581 . . . 4  |-  { r  e.  NN0  |  D  ||  ( N  -  r
) }  C_  NN0
31, 2eqsstri 3529 . . 3  |-  S  C_  NN0
4 nn0uz 11140 . . 3  |-  NN0  =  ( ZZ>= `  0 )
53, 4sseqtri 3531 . 2  |-  S  C_  ( ZZ>= `  0 )
6 divalglem0.1 . . . . . 6  |-  N  e.  ZZ
7 divalglem0.2 . . . . . . . . 9  |-  D  e.  ZZ
8 zmulcl 10933 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  ZZ )  ->  ( N  x.  D
)  e.  ZZ )
96, 7, 8mp2an 672 . . . . . . . 8  |-  ( N  x.  D )  e.  ZZ
10 nn0abscl 13157 . . . . . . . 8  |-  ( ( N  x.  D )  e.  ZZ  ->  ( abs `  ( N  x.  D ) )  e. 
NN0 )
119, 10ax-mp 5 . . . . . . 7  |-  ( abs `  ( N  x.  D
) )  e.  NN0
1211nn0zi 10910 . . . . . 6  |-  ( abs `  ( N  x.  D
) )  e.  ZZ
13 zaddcl 10925 . . . . . 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 14064 . . . . 5  |-  0  <_  ( N  +  ( abs `  ( N  x.  D ) ) )
17 elnn0z 10898 . . . . 5  |-  ( ( N  +  ( abs `  ( N  x.  D
) ) )  e. 
NN0 
<->  ( ( N  +  ( abs `  ( N  x.  D ) ) )  e.  ZZ  /\  0  <_  ( N  +  ( abs `  ( N  x.  D ) ) ) ) )
1814, 16, 17mpbir2an 920 . . . 4  |-  ( N  +  ( abs `  ( N  x.  D )
) )  e.  NN0
19 iddvds 14009 . . . . . . . 8  |-  ( D  e.  ZZ  ->  D  ||  D )
20 dvdsabsb 14015 . . . . . . . . 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 13157 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  ( abs `  N )  e. 
NN0 )
256, 24ax-mp 5 . . . . . . . 8  |-  ( abs `  N )  e.  NN0
2625nn0negzi 10924 . . . . . . 7  |-  -u ( abs `  N )  e.  ZZ
27 nn0abscl 13157 . . . . . . . . 9  |-  ( D  e.  ZZ  ->  ( abs `  D )  e. 
NN0 )
287, 27ax-mp 5 . . . . . . . 8  |-  ( abs `  D )  e.  NN0
2928nn0zi 10910 . . . . . . 7  |-  ( abs `  D )  e.  ZZ
30 dvdsmultr2 14033 . . . . . . 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 1324 . . . . . 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 10890 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  CC )
346, 33ax-mp 5 . . . . . . . 8  |-  N  e.  CC
35 zcn 10890 . . . . . . . . 9  |-  ( D  e.  ZZ  ->  D  e.  CC )
367, 35ax-mp 5 . . . . . . . 8  |-  D  e.  CC
3734, 36absmuli 13248 . . . . . . 7  |-  ( abs `  ( N  x.  D
) )  =  ( ( abs `  N
)  x.  ( abs `  D ) )
3837negeqi 9832 . . . . . 6  |-  -u ( abs `  ( N  x.  D ) )  = 
-u ( ( abs `  N )  x.  ( abs `  D ) )
39 df-neg 9827 . . . . . . 7  |-  -u ( abs `  ( N  x.  D ) )  =  ( 0  -  ( abs `  ( N  x.  D ) ) )
4034subidi 9909 . . . . . . . 8  |-  ( N  -  N )  =  0
4140oveq1i 6306 . . . . . . 7  |-  ( ( N  -  N )  -  ( abs `  ( N  x.  D )
) )  =  ( 0  -  ( abs `  ( N  x.  D
) ) )
4211nn0cni 10828 . . . . . . . 8  |-  ( abs `  ( N  x.  D
) )  e.  CC
43 subsub4 9871 . . . . . . . 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 1324 . . . . . . 7  |-  ( ( N  -  N )  -  ( abs `  ( N  x.  D )
) )  =  ( N  -  ( N  +  ( abs `  ( N  x.  D )
) ) )
4539, 41, 443eqtr2ri 2493 . . . . . 6  |-  ( N  -  ( N  +  ( abs `  ( N  x.  D ) ) ) )  =  -u ( abs `  ( N  x.  D ) )
4634abscli 13239 . . . . . . . 8  |-  ( abs `  N )  e.  RR
4746recni 9625 . . . . . . 7  |-  ( abs `  N )  e.  CC
4836abscli 13239 . . . . . . . 8  |-  ( abs `  D )  e.  RR
4948recni 9625 . . . . . . 7  |-  ( abs `  D )  e.  CC
5047, 49mulneg1i 10023 . . . . . 6  |-  ( -u ( abs `  N )  x.  ( abs `  D
) )  =  -u ( ( abs `  N
)  x.  ( abs `  D ) )
5138, 45, 503eqtr4i 2496 . . . . 5  |-  ( N  -  ( N  +  ( abs `  ( N  x.  D ) ) ) )  =  (
-u ( abs `  N
)  x.  ( abs `  D ) )
5232, 51breqtrri 4481 . . . 4  |-  D  ||  ( N  -  ( N  +  ( abs `  ( N  x.  D
) ) ) )
53 oveq2 6304 . . . . . 6  |-  ( r  =  ( N  +  ( abs `  ( N  x.  D ) ) )  ->  ( N  -  r )  =  ( N  -  ( N  +  ( abs `  ( N  x.  D
) ) ) ) )
5453breq2d 4468 . . . . 5  |-  ( r  =  ( N  +  ( abs `  ( N  x.  D ) ) )  ->  ( D  ||  ( N  -  r
)  <->  D  ||  ( N  -  ( N  +  ( abs `  ( N  x.  D ) ) ) ) ) )
5554, 1elrab2 3259 . . . 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 920 . . 3  |-  ( N  +  ( abs `  ( N  x.  D )
) )  e.  S
5756ne0ii 3800 . 2  |-  S  =/=  (/)
58 infmssuzcl 11190 . 2  |-  ( ( S  C_  ( ZZ>= ` 
0 )  /\  S  =/=  (/) )  ->  sup ( S ,  RR ,  `'  <  )  e.  S
)
595, 57, 58mp2an 672 1  |-  sup ( S ,  RR ,  `'  <  )  e.  S
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1395    e. wcel 1819    =/= wne 2652   {crab 2811    C_ wss 3471   (/)c0 3793   class class class wbr 4456   `'ccnv 5007   ` cfv 5594  (class class class)co 6296   supcsup 7918   CCcc 9507   RRcr 9508   0cc0 9509    + caddc 9512    x. cmul 9514    < clt 9645    <_ cle 9646    - cmin 9824   -ucneg 9825   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106   abscabs 13079    || cdvds 13998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-seq 12111  df-exp 12170  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-dvds 13999
This theorem is referenced by:  divalglem5  14067  divalglem9  14071
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