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Theorem divalg 15126
Description: The division algorithm (theorem). Dividing an integer N by a nonzero integer D produces a (unique) quotient q and a unique remainder 0 <_ r < ( abs ` D ). Theorem 1.14 in [ApostolNT] p. 19. The proof does not use /, |_ or mod. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
divalg |- ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) -> 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
Proof of Theorem divalg
StepHypRef Expression
1 eqeq1 2626 . . . . . 6 |- ( N = if ( N e. ZZ , N , 1 ) -> ( N = ( ( q x. D ) + r ) <-> if ( N e. ZZ , N , 1 ) = ( ( q x. D ) + r ) ) )
213anbi3d 1405 . . . . 5 |- ( N = if ( N e. ZZ , N , 1 ) -> ( ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) <-> ( 0 <_ r /\ r < ( abs ` D ) /\ if ( N e. ZZ , N , 1 ) = ( ( q x. D ) + r ) ) ) )
32rexbidv 3052 . . . 4 |- ( N = if ( N e. ZZ , N , 1 ) -> ( E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) <-> E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ if ( N e. ZZ , N , 1 ) = ( ( q x. D ) + r ) ) ) )
43reubidv 3126 . . 3 |- ( N = if ( N e. ZZ , N , 1 ) -> ( E! r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) <-> E! r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ if ( N e. ZZ , N , 1 ) = ( ( q x. D ) + r ) ) ) )
5 fveq2 6191 . . . . . . 7 |- ( D = if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) -> ( abs ` D ) = ( abs ` if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) ) )
65breq2d 4665 . . . . . 6 |- ( D = if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) -> ( r < ( abs ` D ) <-> r < ( abs ` if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) ) ) )
7 oveq2 6658 . . . . . . . 8 |- ( D = if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) -> ( q x. D ) = ( q x. if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) ) )
87oveq1d 6665 . . . . . . 7 |- ( D = if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) -> ( ( q x. D ) + r ) = ( ( q x. if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) ) + r ) )
98eqeq2d 2632 . . . . . 6 |- ( D = if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) -> ( if ( N e. ZZ , N , 1 ) = ( ( q x. D ) + r ) <-> if ( N e. ZZ , N , 1 ) = ( ( q x. if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) ) + r ) ) )
106, 93anbi23d 1402 . . . . 5 |- ( D = if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) -> ( ( 0 <_ r /\ r < ( abs ` D ) /\ if ( N e. ZZ , N , 1 ) = ( ( q x. D ) + r ) ) <-> ( 0 <_ r /\ r < ( abs ` if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) ) /\ if ( N e. ZZ , N , 1 ) = ( ( q x. if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) ) + r ) ) ) )
1110rexbidv 3052 . . . 4 |- ( D = if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) -> ( E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ if ( N e. ZZ , N , 1 ) = ( ( q x. D ) + r ) ) <-> E. q e. ZZ ( 0 <_ r /\ r < ( abs ` if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) ) /\ if ( N e. ZZ , N , 1 ) = ( ( q x. if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) ) + r ) ) ) )
1211reubidv 3126 . . 3 |- ( D = if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) -> ( E! r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ if ( N e. ZZ , N , 1 ) = ( ( q x. D ) + r ) ) <-> E! r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) ) /\ if ( N e. ZZ , N , 1 ) = ( ( q x. if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) ) + r ) ) ) )
13 1z 11407 . . . . 5 |- 1 e. ZZ
1413elimel 4150 . . . 4 |- if ( N e. ZZ , N , 1 ) e. ZZ
15 simpl 473 . . . . 5 |- ( ( D e. ZZ /\ D =/= 0 ) -> D e. ZZ )
16 eleq1 2689 . . . . 5 |- ( D = if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) -> ( D e. ZZ <-> if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) e. ZZ ) )
17 eleq1 2689 . . . . 5 |- ( 1 = if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) -> ( 1 e. ZZ <-> if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) e. ZZ ) )
1815, 16, 17, 13elimdhyp 4151 . . . 4 |- if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) e. ZZ
19 simpr 477 . . . . 5 |- ( ( D e. ZZ /\ D =/= 0 ) -> D =/= 0 )
20 neeq1 2856 . . . . 5 |- ( D = if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) -> ( D =/= 0 <-> if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) =/= 0 ) )
21 neeq1 2856 . . . . 5 |- ( 1 = if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) -> ( 1 =/= 0 <-> if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) =/= 0 ) )
22 ax-1ne0 10005 . . . . 5 |- 1 =/= 0
2319, 20, 21, 22elimdhyp 4151 . . . 4 |- if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) =/= 0
24 eqid 2622 . . . 4 |- { r e. NN0 | if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) || ( if ( N e. ZZ , N , 1 ) - r ) } = { r e. NN0 | if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) || ( if ( N e. ZZ , N , 1 ) - r ) }
2514, 18, 23, 24divalglem10 15125 . . 3 |- E! r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) ) /\ if ( N e. ZZ , N , 1 ) = ( ( q x. if ( ( D e. ZZ /\ D =/= 0 ) , D , 1 ) ) + r ) )
264, 12, 25dedth2h 4140 . 2 |- ( ( N e. ZZ /\ ( D e. ZZ /\ D =/= 0 ) ) -> E! r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) )
27263impb 1260 1 |- ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) -> 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:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   E!wreu 2914   {crab 2916   ifcif 4086   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + 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:  divalg2  15128
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