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Theorem eluzsub 9752
Description: Membership in an earlier upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
eluzsub |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> ( N - K ) e. ( ZZ>= ` M ) )
Proof of Theorem eluzsub
StepHypRef Expression
1 eluzelz 9731 . . . 4 |- ( N e. ( ZZ>= ` ( M + K ) ) -> N e. ZZ )
213ad2ant3 1044 . . 3 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> N e. ZZ )
3 simp2 1022 . . 3 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> K e. ZZ )
42, 3zsubcld 9574 . 2 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> ( N - K ) e. ZZ )
5 simp3 1023 . . . . 5 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> N e. ( ZZ>= ` ( M + K ) ) )
6 simp1 1021 . . . . . . 7 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> M e. ZZ )
76, 3zaddcld 9573 . . . . . 6 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> ( M + K ) e. ZZ )
8 eluz1 9726 . . . . . 6 |- ( ( M + K ) e. ZZ -> ( N e. ( ZZ>= ` ( M + K ) ) <-> ( N e. ZZ /\ ( M + K ) <_ N ) ) )
97, 8syl 14 . . . . 5 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> ( N e. ( ZZ>= ` ( M + K ) ) <-> ( N e. ZZ /\ ( M + K ) <_ N ) ) )
105, 9mpbid 147 . . . 4 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> ( N e. ZZ /\ ( M + K ) <_ N ) )
1110simprd 114 . . 3 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> ( M + K ) <_ N )
126zred 9569 . . . 4 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> M e. RR )
133zred 9569 . . . 4 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> K e. RR )
142zred 9569 . . . 4 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> N e. RR )
15 leaddsub 8585 . . . 4 |- ( ( M e. RR /\ K e. RR /\ N e. RR ) -> ( ( M + K ) <_ N <-> M <_ ( N - K ) ) )
1612, 13, 14, 15syl3anc 1271 . . 3 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> ( ( M + K ) <_ N <-> M <_ ( N - K ) ) )
1711, 16mpbid 147 . 2 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> M <_ ( N - K ) )
18 eluz1 9726 . . 3 |- ( M e. ZZ -> ( ( N - K ) e. ( ZZ>= ` M ) <-> ( ( N - K ) e. ZZ /\ M <_ ( N - K ) ) ) )
196, 18syl 14 . 2 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> ( ( N - K ) e. ( ZZ>= ` M ) <-> ( ( N - K ) e. ZZ /\ M <_ ( N - K ) ) ) )
204, 17, 19mpbir2and 950 1 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> ( N - K ) e. ( ZZ>= ` M ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   e. wcel 2200   class class class wbr 4083   ` cfv 5318  (class class class)co 6001   RRcr 7998   + caddc 8002   <_ cle 8182   - cmin 8317   ZZcz 9446   ZZ>=cuz 9722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-ltadd 8115
This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-n0 9370  df-z 9447  df-uz 9723
This theorem is referenced by:  fzoss2  10370  shftuz  11328  climshftlemg  11813  isumshft  12001
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