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Theorem divalglem2 14385
Description: Lemma for divalg 14396. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by AV, 2-Oct-2020.)
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  |- inf ( 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 3516 . . . 4  |-  { r  e.  NN0  |  D  ||  ( N  -  r
) }  C_  NN0
31, 2eqsstri 3464 . . 3  |-  S  C_  NN0
4 nn0uz 11200 . . 3  |-  NN0  =  ( ZZ>= `  0 )
53, 4sseqtri 3466 . 2  |-  S  C_  ( ZZ>= `  0 )
6 divalglem0.1 . . . . . 6  |-  N  e.  ZZ
7 divalglem0.2 . . . . . . . . 9  |-  D  e.  ZZ
8 zmulcl 10992 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  ZZ )  ->  ( N  x.  D
)  e.  ZZ )
96, 7, 8mp2an 679 . . . . . . . 8  |-  ( N  x.  D )  e.  ZZ
10 nn0abscl 13387 . . . . . . . 8  |-  ( ( N  x.  D )  e.  ZZ  ->  ( abs `  ( N  x.  D ) )  e. 
NN0 )
119, 10ax-mp 5 . . . . . . 7  |-  ( abs `  ( N  x.  D
) )  e.  NN0
1211nn0zi 10969 . . . . . 6  |-  ( abs `  ( N  x.  D
) )  e.  ZZ
13 zaddcl 10984 . . . . . 6  |-  ( ( N  e.  ZZ  /\  ( abs `  ( N  x.  D ) )  e.  ZZ )  -> 
( N  +  ( abs `  ( N  x.  D ) ) )  e.  ZZ )
146, 12, 13mp2an 679 . . . . 5  |-  ( N  +  ( abs `  ( N  x.  D )
) )  e.  ZZ
15 divalglem1.3 . . . . . 6  |-  D  =/=  0
166, 7, 15divalglem1 14384 . . . . 5  |-  0  <_  ( N  +  ( abs `  ( N  x.  D ) ) )
17 elnn0z 10957 . . . . 5  |-  ( ( N  +  ( abs `  ( N  x.  D
) ) )  e. 
NN0 
<->  ( ( N  +  ( abs `  ( N  x.  D ) ) )  e.  ZZ  /\  0  <_  ( N  +  ( abs `  ( N  x.  D ) ) ) ) )
1814, 16, 17mpbir2an 932 . . . 4  |-  ( N  +  ( abs `  ( N  x.  D )
) )  e.  NN0
19 iddvds 14328 . . . . . . . 8  |-  ( D  e.  ZZ  ->  D  ||  D )
20 dvdsabsb 14334 . . . . . . . . 9  |-  ( ( D  e.  ZZ  /\  D  e.  ZZ )  ->  ( D  ||  D  <->  D 
||  ( abs `  D
) ) )
2120anidms 651 . . . . . . . 8  |-  ( D  e.  ZZ  ->  ( D  ||  D  <->  D  ||  ( abs `  D ) ) )
2219, 21mpbid 214 . . . . . . 7  |-  ( D  e.  ZZ  ->  D  ||  ( abs `  D
) )
237, 22ax-mp 5 . . . . . 6  |-  D  ||  ( abs `  D )
24 nn0abscl 13387 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  ( abs `  N )  e. 
NN0 )
256, 24ax-mp 5 . . . . . . . 8  |-  ( abs `  N )  e.  NN0
2625nn0negzi 10983 . . . . . . 7  |-  -u ( abs `  N )  e.  ZZ
27 nn0abscl 13387 . . . . . . . . 9  |-  ( D  e.  ZZ  ->  ( abs `  D )  e. 
NN0 )
287, 27ax-mp 5 . . . . . . . 8  |-  ( abs `  D )  e.  NN0
2928nn0zi 10969 . . . . . . 7  |-  ( abs `  D )  e.  ZZ
30 dvdsmultr2 14352 . . . . . . 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 1366 . . . . . 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 10949 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  CC )
346, 33ax-mp 5 . . . . . . . 8  |-  N  e.  CC
35 zcn 10949 . . . . . . . . 9  |-  ( D  e.  ZZ  ->  D  e.  CC )
367, 35ax-mp 5 . . . . . . . 8  |-  D  e.  CC
3734, 36absmuli 13478 . . . . . . 7  |-  ( abs `  ( N  x.  D
) )  =  ( ( abs `  N
)  x.  ( abs `  D ) )
3837negeqi 9873 . . . . . 6  |-  -u ( abs `  ( N  x.  D ) )  = 
-u ( ( abs `  N )  x.  ( abs `  D ) )
39 df-neg 9868 . . . . . . 7  |-  -u ( abs `  ( N  x.  D ) )  =  ( 0  -  ( abs `  ( N  x.  D ) ) )
4034subidi 9950 . . . . . . . 8  |-  ( N  -  N )  =  0
4140oveq1i 6305 . . . . . . 7  |-  ( ( N  -  N )  -  ( abs `  ( N  x.  D )
) )  =  ( 0  -  ( abs `  ( N  x.  D
) ) )
4211nn0cni 10888 . . . . . . . 8  |-  ( abs `  ( N  x.  D
) )  e.  CC
43 subsub4 9912 . . . . . . . 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 1366 . . . . . . 7  |-  ( ( N  -  N )  -  ( abs `  ( N  x.  D )
) )  =  ( N  -  ( N  +  ( abs `  ( N  x.  D )
) ) )
4539, 41, 443eqtr2ri 2482 . . . . . 6  |-  ( N  -  ( N  +  ( abs `  ( N  x.  D ) ) ) )  =  -u ( abs `  ( N  x.  D ) )
4634abscli 13469 . . . . . . . 8  |-  ( abs `  N )  e.  RR
4746recni 9660 . . . . . . 7  |-  ( abs `  N )  e.  CC
4836abscli 13469 . . . . . . . 8  |-  ( abs `  D )  e.  RR
4948recni 9660 . . . . . . 7  |-  ( abs `  D )  e.  CC
5047, 49mulneg1i 10071 . . . . . 6  |-  ( -u ( abs `  N )  x.  ( abs `  D
) )  =  -u ( ( abs `  N
)  x.  ( abs `  D ) )
5138, 45, 503eqtr4i 2485 . . . . 5  |-  ( N  -  ( N  +  ( abs `  ( N  x.  D ) ) ) )  =  (
-u ( abs `  N
)  x.  ( abs `  D ) )
5232, 51breqtrri 4431 . . . 4  |-  D  ||  ( N  -  ( N  +  ( abs `  ( N  x.  D
) ) ) )
53 oveq2 6303 . . . . . 6  |-  ( r  =  ( N  +  ( abs `  ( N  x.  D ) ) )  ->  ( N  -  r )  =  ( N  -  ( N  +  ( abs `  ( N  x.  D
) ) ) ) )
5453breq2d 4417 . . . . 5  |-  ( r  =  ( N  +  ( abs `  ( N  x.  D ) ) )  ->  ( D  ||  ( N  -  r
)  <->  D  ||  ( N  -  ( N  +  ( abs `  ( N  x.  D ) ) ) ) ) )
5554, 1elrab2 3200 . . . 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 932 . . 3  |-  ( N  +  ( abs `  ( N  x.  D )
) )  e.  S
5756ne0ii 3740 . 2  |-  S  =/=  (/)
58 infssuzcl 11252 . 2  |-  ( ( S  C_  ( ZZ>= ` 
0 )  /\  S  =/=  (/) )  -> inf ( S ,  RR ,  <  )  e.  S )
595, 57, 58mp2an 679 1  |- inf ( S ,  RR ,  <  )  e.  S
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    = wceq 1446    e. wcel 1889    =/= wne 2624   {crab 2743    C_ wss 3406   (/)c0 3733   class class class wbr 4405   ` cfv 5585  (class class class)co 6295  infcinf 7960   CCcc 9542   RRcr 9543   0cc0 9544    + caddc 9547    x. cmul 9549    < clt 9680    <_ cle 9681    - cmin 9865   -ucneg 9866   NN0cn0 10876   ZZcz 10944   ZZ>=cuz 11166   abscabs 13309    || cdvds 14317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-sup 7961  df-inf 7962  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-seq 12221  df-exp 12280  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-dvds 14318
This theorem is referenced by:  divalglem5  14389  divalglem9  14393
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