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Theorem eluzsub 9870
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 9849 . . . 4 |- ( N e. ( ZZ>= ` ( M + K ) ) -> N e. ZZ )
213ad2ant3 1047 . . 3 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> N e. ZZ )
3 simp2 1025 . . 3 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> K e. ZZ )
42, 3zsubcld 9691 . 2 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> ( N - K ) e. ZZ )
5 simp3 1026 . . . . 5 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> N e. ( ZZ>= ` ( M + K ) ) )
6 simp1 1024 . . . . . . 7 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> M e. ZZ )
76, 3zaddcld 9690 . . . . . 6 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> ( M + K ) e. ZZ )
8 eluz1 9843 . . . . . 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 9686 . . . 4 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> M e. RR )
133zred 9686 . . . 4 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> K e. RR )
142zred 9686 . . . 4 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> N e. RR )
15 leaddsub 8700 . . . 4 |- ( ( M e. RR /\ K e. RR /\ N e. RR ) -> ( ( M + K ) <_ N <-> M <_ ( N - K ) ) )
1612, 13, 14, 15syl3anc 1274 . . 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 9843 . . 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 953 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 1005   e. wcel 2203   class class class wbr 4102   ` cfv 5343  (class class class)co 6041   RRcr 8114   + caddc 8118   <_ cle 8297   - cmin 8432   ZZcz 9563   ZZ>=cuz 9839
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4221  ax-pow 4279  ax-pr 4314  ax-un 4545  ax-setind 4650  ax-cnex 8206  ax-resscn 8207  ax-1cn 8208  ax-1re 8209  ax-icn 8210  ax-addcl 8211  ax-addrcl 8212  ax-mulcl 8213  ax-addcom 8215  ax-addass 8217  ax-distr 8219  ax-i2m1 8220  ax-0lt1 8221  ax-0id 8223  ax-rnegex 8224  ax-cnre 8226  ax-pre-ltirr 8227  ax-pre-ltwlin 8228  ax-pre-lttrn 8229  ax-pre-ltadd 8231
This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3667  df-sn 3688  df-pr 3689  df-op 3691  df-uni 3908  df-int 3943  df-br 4103  df-opab 4165  df-mpt 4166  df-id 4405  df-xp 4746  df-rel 4747  df-cnv 4748  df-co 4749  df-dm 4750  df-rn 4751  df-res 4752  df-ima 4753  df-iota 5303  df-fun 5345  df-fn 5346  df-f 5347  df-fv 5351  df-riota 5994  df-ov 6044  df-oprab 6045  df-mpo 6046  df-pnf 8298  df-mnf 8299  df-xr 8300  df-ltxr 8301  df-le 8302  df-sub 8434  df-neg 8435  df-inn 9226  df-n0 9485  df-z 9564  df-uz 9840
This theorem is referenced by:  fzoss2  10494  shftuz  11480  climshftlemg  11965  isumshft  12154
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