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Theorem divalglem10 15125
Description: Lemma for divalg 15126. (Contributed by Paul Chapman, 21-Mar-2011.) (Proof shortened by AV, 2-Oct-2020.)
Hypotheses
Ref Expression
divalglem8.1 |- N e. ZZ
divalglem8.2 |- D e. ZZ
divalglem8.3 |- D =/= 0
divalglem8.4 |- S = { r e. NN0 | D || ( N - r ) }
Assertion
Ref Expression
divalglem10 |- E! r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) )
Distinct variable groups:   D, q, r   N, q, r
Allowed substitution hints:   S( r, q)
Proof of Theorem divalglem10
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 divalglem8.1 . . . 4 |- N e. ZZ
2 divalglem8.2 . . . 4 |- D e. ZZ
3 divalglem8.3 . . . 4 |- D =/= 0
4 divalglem8.4 . . . 4 |- S = { r e. NN0 | D || ( N - r ) }
5 eqid 2622 . . . 4 |- inf ( S , RR , < ) = inf ( S , RR , < )
61, 2, 3, 4, 5divalglem9 15124 . . 3 |- E! x e. S x < ( abs ` D )
7 elnn0z 11390 . . . . . . . . . 10 |- ( x e. NN0 <-> ( x e. ZZ /\ 0 <_ x ) )
87anbi2i 730 . . . . . . . . 9 |- ( ( x < ( abs ` D ) /\ x e. NN0 ) <-> ( x < ( abs ` D ) /\ ( x e. ZZ /\ 0 <_ x ) ) )
9 an12 838 . . . . . . . . . 10 |- ( ( x < ( abs ` D ) /\ ( x e. ZZ /\ 0 <_ x ) ) <-> ( x e. ZZ /\ ( x < ( abs ` D ) /\ 0 <_ x ) ) )
10 ancom 466 . . . . . . . . . . 11 |- ( ( x < ( abs ` D ) /\ 0 <_ x ) <-> ( 0 <_ x /\ x < ( abs ` D ) ) )
1110anbi2i 730 . . . . . . . . . 10 |- ( ( x e. ZZ /\ ( x < ( abs ` D ) /\ 0 <_ x ) ) <-> ( x e. ZZ /\ ( 0 <_ x /\ x < ( abs ` D ) ) ) )
129, 11bitri 264 . . . . . . . . 9 |- ( ( x < ( abs ` D ) /\ ( x e. ZZ /\ 0 <_ x ) ) <-> ( x e. ZZ /\ ( 0 <_ x /\ x < ( abs ` D ) ) ) )
138, 12bitri 264 . . . . . . . 8 |- ( ( x < ( abs ` D ) /\ x e. NN0 ) <-> ( x e. ZZ /\ ( 0 <_ x /\ x < ( abs ` D ) ) ) )
1413anbi1i 731 . . . . . . 7 |- ( ( ( x < ( abs ` D ) /\ x e. NN0 ) /\ E. q e. ZZ N = ( ( q x. D ) + x ) ) <-> ( ( x e. ZZ /\ ( 0 <_ x /\ x < ( abs ` D ) ) ) /\ E. q e. ZZ N = ( ( q x. D ) + x ) ) )
15 anass 681 . . . . . . 7 |- ( ( ( x e. ZZ /\ ( 0 <_ x /\ x < ( abs ` D ) ) ) /\ E. q e. ZZ N = ( ( q x. D ) + x ) ) <-> ( x e. ZZ /\ ( ( 0 <_ x /\ x < ( abs ` D ) ) /\ E. q e. ZZ N = ( ( q x. D ) + x ) ) ) )
1614, 15bitri 264 . . . . . 6 |- ( ( ( x < ( abs ` D ) /\ x e. NN0 ) /\ E. q e. ZZ N = ( ( q x. D ) + x ) ) <-> ( x e. ZZ /\ ( ( 0 <_ x /\ x < ( abs ` D ) ) /\ E. q e. ZZ N = ( ( q x. D ) + x ) ) ) )
17 oveq2 6658 . . . . . . . . . . 11 |- ( r = x -> ( ( q x. D ) + r ) = ( ( q x. D ) + x ) )
1817eqeq2d 2632 . . . . . . . . . 10 |- ( r = x -> ( N = ( ( q x. D ) + r ) <-> N = ( ( q x. D ) + x ) ) )
1918rexbidv 3052 . . . . . . . . 9 |- ( r = x -> ( E. q e. ZZ N = ( ( q x. D ) + r ) <-> E. q e. ZZ N = ( ( q x. D ) + x ) ) )
201, 2, 3, 4divalglem4 15119 . . . . . . . . 9 |- S = { r e. NN0 | E. q e. ZZ N = ( ( q x. D ) + r ) }
2119, 20elrab2 3366 . . . . . . . 8 |- ( x e. S <-> ( x e. NN0 /\ E. q e. ZZ N = ( ( q x. D ) + x ) ) )
2221anbi2i 730 . . . . . . 7 |- ( ( x < ( abs ` D ) /\ x e. S ) <-> ( x < ( abs ` D ) /\ ( x e. NN0 /\ E. q e. ZZ N = ( ( q x. D ) + x ) ) ) )
23 ancom 466 . . . . . . 7 |- ( ( x e. S /\ x < ( abs ` D ) ) <-> ( x < ( abs ` D ) /\ x e. S ) )
24 anass 681 . . . . . . 7 |- ( ( ( x < ( abs ` D ) /\ x e. NN0 ) /\ E. q e. ZZ N = ( ( q x. D ) + x ) ) <-> ( x < ( abs ` D ) /\ ( x e. NN0 /\ E. q e. ZZ N = ( ( q x. D ) + x ) ) ) )
2522, 23, 243bitr4i 292 . . . . . 6 |- ( ( x e. S /\ x < ( abs ` D ) ) <-> ( ( x < ( abs ` D ) /\ x e. NN0 ) /\ E. q e. ZZ N = ( ( q x. D ) + x ) ) )
26 df-3an 1039 . . . . . . . . 9 |- ( ( 0 <_ x /\ x < ( abs ` D ) /\ N = ( ( q x. D ) + x ) ) <-> ( ( 0 <_ x /\ x < ( abs ` D ) ) /\ N = ( ( q x. D ) + x ) ) )
2726rexbii 3041 . . . . . . . 8 |- ( E. q e. ZZ ( 0 <_ x /\ x < ( abs ` D ) /\ N = ( ( q x. D ) + x ) ) <-> E. q e. ZZ ( ( 0 <_ x /\ x < ( abs ` D ) ) /\ N = ( ( q x. D ) + x ) ) )
28 r19.42v 3092 . . . . . . . 8 |- ( E. q e. ZZ ( ( 0 <_ x /\ x < ( abs ` D ) ) /\ N = ( ( q x. D ) + x ) ) <-> ( ( 0 <_ x /\ x < ( abs ` D ) ) /\ E. q e. ZZ N = ( ( q x. D ) + x ) ) )
2927, 28bitri 264 . . . . . . 7 |- ( E. q e. ZZ ( 0 <_ x /\ x < ( abs ` D ) /\ N = ( ( q x. D ) + x ) ) <-> ( ( 0 <_ x /\ x < ( abs ` D ) ) /\ E. q e. ZZ N = ( ( q x. D ) + x ) ) )
3029anbi2i 730 . . . . . 6 |- ( ( x e. ZZ /\ E. q e. ZZ ( 0 <_ x /\ x < ( abs ` D ) /\ N = ( ( q x. D ) + x ) ) ) <-> ( x e. ZZ /\ ( ( 0 <_ x /\ x < ( abs ` D ) ) /\ E. q e. ZZ N = ( ( q x. D ) + x ) ) ) )
3116, 25, 303bitr4i 292 . . . . 5 |- ( ( x e. S /\ x < ( abs ` D ) ) <-> ( x e. ZZ /\ E. q e. ZZ ( 0 <_ x /\ x < ( abs ` D ) /\ N = ( ( q x. D ) + x ) ) ) )
3231eubii 2492 . . . 4 |- ( E! x ( x e. S /\ x < ( abs ` D ) ) <-> E! x ( x e. ZZ /\ E. q e. ZZ ( 0 <_ x /\ x < ( abs ` D ) /\ N = ( ( q x. D ) + x ) ) ) )
33 df-reu 2919 . . . 4 |- ( E! x e. S x < ( abs ` D ) <-> E! x ( x e. S /\ x < ( abs ` D ) ) )
34 df-reu 2919 . . . 4 |- ( E! x e. ZZ E. q e. ZZ ( 0 <_ x /\ x < ( abs ` D ) /\ N = ( ( q x. D ) + x ) ) <-> E! x ( x e. ZZ /\ E. q e. ZZ ( 0 <_ x /\ x < ( abs ` D ) /\ N = ( ( q x. D ) + x ) ) ) )
3532, 33, 343bitr4i 292 . . 3 |- ( E! x e. S x < ( abs ` D ) <-> E! x e. ZZ E. q e. ZZ ( 0 <_ x /\ x < ( abs ` D ) /\ N = ( ( q x. D ) + x ) ) )
366, 35mpbi 220 . 2 |- E! x e. ZZ E. q e. ZZ ( 0 <_ x /\ x < ( abs ` D ) /\ N = ( ( q x. D ) + x ) )
37 breq2 4657 . . . . 5 |- ( x = r -> ( 0 <_ x <-> 0 <_ r ) )
38 breq1 4656 . . . . 5 |- ( x = r -> ( x < ( abs ` D ) <-> r < ( abs ` D ) ) )
39 oveq2 6658 . . . . . 6 |- ( x = r -> ( ( q x. D ) + x ) = ( ( q x. D ) + r ) )
4039eqeq2d 2632 . . . . 5 |- ( x = r -> ( N = ( ( q x. D ) + x ) <-> N = ( ( q x. D ) + r ) ) )
4137, 38, 403anbi123d 1399 . . . 4 |- ( x = r -> ( ( 0 <_ x /\ x < ( abs ` D ) /\ N = ( ( q x. D ) + x ) ) <-> ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) ) )
4241rexbidv 3052 . . 3 |- ( x = r -> ( E. q e. ZZ ( 0 <_ x /\ x < ( abs ` D ) /\ N = ( ( q x. D ) + x ) ) <-> E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) ) )
4342cbvreuv 3173 . 2 |- ( E! x e. ZZ E. q e. ZZ ( 0 <_ x /\ x < ( abs ` D ) /\ N = ( ( q x. D ) + x ) ) <-> E! r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) )
4436, 43mpbi 220 1 |- E! r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) )
Colors of variables: wff setvar class
Syntax hints:   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E!weu 2470   =/= wne 2794   E.wrex 2913   E!wreu 2914   {crab 2916   class class class wbr 4653   ` cfv 5888  (class class class)co 6650  infcinf 8347   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   NN0cn0 11292   ZZcz 11377   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-1st 7168  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-fz 12327  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:  divalg  15126
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