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Theorem modabsdifz 37553
Description: Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Assertion
Ref Expression
modabsdifz |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> ( ( N - ( N mod ( abs ` M ) ) ) / M ) e. ZZ )
Proof of Theorem modabsdifz
StepHypRef Expression
1 simp1 1061 . . 3 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> N e. RR )
2 simp2 1062 . . . . 5 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> M e. RR )
32recnd 10068 . . . 4 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> M e. CC )
4 simp3 1063 . . . 4 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> M =/= 0 )
53, 4absrpcld 14187 . . 3 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> ( abs ` M ) e. RR+ )
6 moddifz 12682 . . 3 |- ( ( N e. RR /\ ( abs ` M ) e. RR+ ) -> ( ( N - ( N mod ( abs ` M ) ) ) / ( abs ` M ) ) e. ZZ )
71, 5, 6syl2anc 693 . 2 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> ( ( N - ( N mod ( abs ` M ) ) ) / ( abs ` M ) ) e. ZZ )
8 absidm 14063 . . . . . . 7 |- ( M e. CC -> ( abs ` ( abs ` M ) ) = ( abs ` M ) )
93, 8syl 17 . . . . . 6 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> ( abs ` ( abs ` M ) ) = ( abs ` M ) )
109oveq2d 6666 . . . . 5 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> ( ( abs ` ( N - ( N mod ( abs ` M ) ) ) ) / ( abs ` ( abs ` M ) ) ) = ( ( abs ` ( N - ( N mod ( abs ` M ) ) ) ) / ( abs ` M ) ) )
111, 5modcld 12674 . . . . . . . 8 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> ( N mod ( abs ` M ) ) e. RR )
121, 11resubcld 10458 . . . . . . 7 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> ( N - ( N mod ( abs ` M ) ) ) e. RR )
1312recnd 10068 . . . . . 6 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> ( N - ( N mod ( abs ` M ) ) ) e. CC )
143abscld 14175 . . . . . . 7 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> ( abs ` M ) e. RR )
1514recnd 10068 . . . . . 6 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> ( abs ` M ) e. CC )
165rpne0d 11877 . . . . . 6 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> ( abs ` M ) =/= 0 )
1713, 15, 16absdivd 14194 . . . . 5 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> ( abs ` ( ( N - ( N mod ( abs ` M ) ) ) / ( abs ` M ) ) ) = ( ( abs ` ( N - ( N mod ( abs ` M ) ) ) ) / ( abs ` ( abs ` M ) ) ) )
1813, 3, 4absdivd 14194 . . . . 5 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> ( abs ` ( ( N - ( N mod ( abs ` M ) ) ) / M ) ) = ( ( abs ` ( N - ( N mod ( abs ` M ) ) ) ) / ( abs ` M ) ) )
1910, 17, 183eqtr4d 2666 . . . 4 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> ( abs ` ( ( N - ( N mod ( abs ` M ) ) ) / ( abs ` M ) ) ) = ( abs ` ( ( N - ( N mod ( abs ` M ) ) ) / M ) ) )
2019eleq1d 2686 . . 3 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> ( ( abs ` ( ( N - ( N mod ( abs ` M ) ) ) / ( abs ` M ) ) ) e. ZZ <-> ( abs ` ( ( N - ( N mod ( abs ` M ) ) ) / M ) ) e. ZZ ) )
2112, 14, 16redivcld 10853 . . . 4 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> ( ( N - ( N mod ( abs ` M ) ) ) / ( abs ` M ) ) e. RR )
22 absz 14051 . . . 4 |- ( ( ( N - ( N mod ( abs ` M ) ) ) / ( abs ` M ) ) e. RR -> ( ( ( N - ( N mod ( abs ` M ) ) ) / ( abs ` M ) ) e. ZZ <-> ( abs ` ( ( N - ( N mod ( abs ` M ) ) ) / ( abs ` M ) ) ) e. ZZ ) )
2321, 22syl 17 . . 3 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> ( ( ( N - ( N mod ( abs ` M ) ) ) / ( abs ` M ) ) e. ZZ <-> ( abs ` ( ( N - ( N mod ( abs ` M ) ) ) / ( abs ` M ) ) ) e. ZZ ) )
2412, 2, 4redivcld 10853 . . . 4 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> ( ( N - ( N mod ( abs ` M ) ) ) / M ) e. RR )
25 absz 14051 . . . 4 |- ( ( ( N - ( N mod ( abs ` M ) ) ) / M ) e. RR -> ( ( ( N - ( N mod ( abs ` M ) ) ) / M ) e. ZZ <-> ( abs ` ( ( N - ( N mod ( abs ` M ) ) ) / M ) ) e. ZZ ) )
2624, 25syl 17 . . 3 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> ( ( ( N - ( N mod ( abs ` M ) ) ) / M ) e. ZZ <-> ( abs ` ( ( N - ( N mod ( abs ` M ) ) ) / M ) ) e. ZZ ) )
2720, 23, 263bitr4d 300 . 2 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> ( ( ( N - ( N mod ( abs ` M ) ) ) / ( abs ` M ) ) e. ZZ <-> ( ( N - ( N mod ( abs ` M ) ) ) / M ) e. ZZ ) )
287, 27mpbid 222 1 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> ( ( N - ( N mod ( abs ` M ) ) ) / M ) e. ZZ )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   - cmin 10266   / cdiv 10684   ZZcz 11377   RR+crp 11832   mod cmo 12668   abscabs 13974
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-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976
This theorem is referenced by:  jm2.19  37560
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