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Theorem divalglem5 15120
Description: Lemma for divalg 15126. (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 ) }
divalglem5.5 |- R = inf ( S , RR , < )
Assertion
Ref Expression
divalglem5 |- ( 0 <_ R /\ R < ( abs ` D ) )
Distinct variable groups:   D, r   N, r
Allowed substitution hints:   R( r)   S( r)
Proof of Theorem divalglem5
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 divalglem5.5 . . . . . 6 |- R = inf ( S , RR , < )
2 divalglem0.1 . . . . . . 7 |- N e. ZZ
3 divalglem0.2 . . . . . . 7 |- D e. ZZ
4 divalglem1.3 . . . . . . 7 |- D =/= 0
5 divalglem2.4 . . . . . . 7 |- S = { r e. NN0 | D || ( N - r ) }
62, 3, 4, 5divalglem2 15118 . . . . . 6 |- inf ( S , RR , < ) e. S
71, 6eqeltri 2697 . . . . 5 |- R e. S
8 oveq2 6658 . . . . . . 7 |- ( x = R -> ( N - x ) = ( N - R ) )
98breq2d 4665 . . . . . 6 |- ( x = R -> ( D || ( N - x ) <-> D || ( N - R ) ) )
10 oveq2 6658 . . . . . . . . 9 |- ( r = x -> ( N - r ) = ( N - x ) )
1110breq2d 4665 . . . . . . . 8 |- ( r = x -> ( D || ( N - r ) <-> D || ( N - x ) ) )
1211cbvrabv 3199 . . . . . . 7 |- { r e. NN0 | D || ( N - r ) } = { x e. NN0 | D || ( N - x ) }
135, 12eqtri 2644 . . . . . 6 |- S = { x e. NN0 | D || ( N - x ) }
149, 13elrab2 3366 . . . . 5 |- ( R e. S <-> ( R e. NN0 /\ D || ( N - R ) ) )
157, 14mpbi 220 . . . 4 |- ( R e. NN0 /\ D || ( N - R ) )
1615simpli 474 . . 3 |- R e. NN0
1716nn0ge0i 11320 . 2 |- 0 <_ R
18 nnabscl 14065 . . . . . . 7 |- ( ( D e. ZZ /\ D =/= 0 ) -> ( abs ` D ) e. NN )
193, 4, 18mp2an 708 . . . . . 6 |- ( abs ` D ) e. NN
2019nngt0i 11054 . . . . 5 |- 0 < ( abs ` D )
21 0re 10040 . . . . . 6 |- 0 e. RR
22 zcn 11382 . . . . . . . 8 |- ( D e. ZZ -> D e. CC )
233, 22ax-mp 5 . . . . . . 7 |- D e. CC
2423abscli 14134 . . . . . 6 |- ( abs ` D ) e. RR
2521, 24ltnlei 10158 . . . . 5 |- ( 0 < ( abs ` D ) <-> -. ( abs ` D ) <_ 0 )
2620, 25mpbi 220 . . . 4 |- -. ( abs ` D ) <_ 0
27 ssrab2 3687 . . . . . . . . 9 |- { r e. NN0 | D || ( N - r ) } C_ NN0
285, 27eqsstri 3635 . . . . . . . 8 |- S C_ NN0
29 nn0uz 11722 . . . . . . . 8 |- NN0 = ( ZZ>= ` 0 )
3028, 29sseqtri 3637 . . . . . . 7 |- S C_ ( ZZ>= ` 0 )
31 nn0abscl 14052 . . . . . . . . . 10 |- ( D e. ZZ -> ( abs ` D ) e. NN0 )
323, 31ax-mp 5 . . . . . . . . 9 |- ( abs ` D ) e. NN0
33 nn0sub2 11438 . . . . . . . . 9 |- ( ( ( abs ` D ) e. NN0 /\ R e. NN0 /\ ( abs ` D ) <_ R ) -> ( R - ( abs ` D ) ) e. NN0 )
3432, 16, 33mp3an12 1414 . . . . . . . 8 |- ( ( abs ` D ) <_ R -> ( R - ( abs ` D ) ) e. NN0 )
3515a1i 11 . . . . . . . . 9 |- ( ( abs ` D ) <_ R -> ( R e. NN0 /\ D || ( N - R ) ) )
36 nn0z 11400 . . . . . . . . . . 11 |- ( R e. NN0 -> R e. ZZ )
37 1z 11407 . . . . . . . . . . . . 13 |- 1 e. ZZ
382, 3divalglem0 15116 . . . . . . . . . . . . 13 |- ( ( R e. ZZ /\ 1 e. ZZ ) -> ( D || ( N - R ) -> D || ( N - ( R - ( 1 x. ( abs ` D ) ) ) ) ) )
3937, 38mpan2 707 . . . . . . . . . . . 12 |- ( R e. ZZ -> ( D || ( N - R ) -> D || ( N - ( R - ( 1 x. ( abs ` D ) ) ) ) ) )
4024recni 10052 . . . . . . . . . . . . . . . 16 |- ( abs ` D ) e. CC
4140mulid2i 10043 . . . . . . . . . . . . . . 15 |- ( 1 x. ( abs ` D ) ) = ( abs ` D )
4241oveq2i 6661 . . . . . . . . . . . . . 14 |- ( R - ( 1 x. ( abs ` D ) ) ) = ( R - ( abs ` D ) )
4342oveq2i 6661 . . . . . . . . . . . . 13 |- ( N - ( R - ( 1 x. ( abs ` D ) ) ) ) = ( N - ( R - ( abs ` D ) ) )
4443breq2i 4661 . . . . . . . . . . . 12 |- ( D || ( N - ( R - ( 1 x. ( abs ` D ) ) ) ) <-> D || ( N - ( R - ( abs ` D ) ) ) )
4539, 44syl6ib 241 . . . . . . . . . . 11 |- ( R e. ZZ -> ( D || ( N - R ) -> D || ( N - ( R - ( abs ` D ) ) ) ) )
4636, 45syl 17 . . . . . . . . . 10 |- ( R e. NN0 -> ( D || ( N - R ) -> D || ( N - ( R - ( abs ` D ) ) ) ) )
4746imp 445 . . . . . . . . 9 |- ( ( R e. NN0 /\ D || ( N - R ) ) -> D || ( N - ( R - ( abs ` D ) ) ) )
4835, 47syl 17 . . . . . . . 8 |- ( ( abs ` D ) <_ R -> D || ( N - ( R - ( abs ` D ) ) ) )
49 oveq2 6658 . . . . . . . . . 10 |- ( x = ( R - ( abs ` D ) ) -> ( N - x ) = ( N - ( R - ( abs ` D ) ) ) )
5049breq2d 4665 . . . . . . . . 9 |- ( x = ( R - ( abs ` D ) ) -> ( D || ( N - x ) <-> D || ( N - ( R - ( abs ` D ) ) ) ) )
5150, 13elrab2 3366 . . . . . . . 8 |- ( ( R - ( abs ` D ) ) e. S <-> ( ( R - ( abs ` D ) ) e. NN0 /\ D || ( N - ( R - ( abs ` D ) ) ) ) )
5234, 48, 51sylanbrc 698 . . . . . . 7 |- ( ( abs ` D ) <_ R -> ( R - ( abs ` D ) ) e. S )
53 infssuzle 11771 . . . . . . 7 |- ( ( S C_ ( ZZ>= ` 0 ) /\ ( R - ( abs ` D ) ) e. S ) -> inf ( S , RR , < ) <_ ( R - ( abs ` D ) ) )
5430, 52, 53sylancr 695 . . . . . 6 |- ( ( abs ` D ) <_ R -> inf ( S , RR , < ) <_ ( R - ( abs ` D ) ) )
551, 54syl5eqbr 4688 . . . . 5 |- ( ( abs ` D ) <_ R -> R <_ ( R - ( abs ` D ) ) )
5635simpld 475 . . . . . . . 8 |- ( ( abs ` D ) <_ R -> R e. NN0 )
57 nn0re 11301 . . . . . . . 8 |- ( R e. NN0 -> R e. RR )
5856, 57syl 17 . . . . . . 7 |- ( ( abs ` D ) <_ R -> R e. RR )
59 lesub 10507 . . . . . . . 8 |- ( ( R e. RR /\ R e. RR /\ ( abs ` D ) e. RR ) -> ( R <_ ( R - ( abs ` D ) ) <-> ( abs ` D ) <_ ( R - R ) ) )
6024, 59mp3an3 1413 . . . . . . 7 |- ( ( R e. RR /\ R e. RR ) -> ( R <_ ( R - ( abs ` D ) ) <-> ( abs ` D ) <_ ( R - R ) ) )
6158, 58, 60syl2anc 693 . . . . . 6 |- ( ( abs ` D ) <_ R -> ( R <_ ( R - ( abs ` D ) ) <-> ( abs ` D ) <_ ( R - R ) ) )
6258recnd 10068 . . . . . . . 8 |- ( ( abs ` D ) <_ R -> R e. CC )
6362subidd 10380 . . . . . . 7 |- ( ( abs ` D ) <_ R -> ( R - R ) = 0 )
6463breq2d 4665 . . . . . 6 |- ( ( abs ` D ) <_ R -> ( ( abs ` D ) <_ ( R - R ) <-> ( abs ` D ) <_ 0 ) )
6561, 64bitrd 268 . . . . 5 |- ( ( abs ` D ) <_ R -> ( R <_ ( R - ( abs ` D ) ) <-> ( abs ` D ) <_ 0 ) )
6655, 65mpbid 222 . . . 4 |- ( ( abs ` D ) <_ R -> ( abs ` D ) <_ 0 )
6726, 66mto 188 . . 3 |- -. ( abs ` D ) <_ R
6816, 57ax-mp 5 . . . 4 |- R e. RR
6968, 24ltnlei 10158 . . 3 |- ( R < ( abs ` D ) <-> -. ( abs ` D ) <_ R )
7067, 69mpbir 221 . 2 |- R < ( abs ` D )
7117, 70pm3.2i 471 1 |- ( 0 <_ R /\ R < ( abs ` D ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650  infcinf 8347   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   abscabs 13974   || cdvds 14983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984
This theorem is referenced by:  divalglem9  15124
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