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Theorem eluzsub 9887
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 9866 . . . 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 9708 . 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 9707 . . . . . 6 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> ( M + K ) e. ZZ )
8 eluz1 9860 . . . . . 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 9703 . . . 4 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> M e. RR )
133zred 9703 . . . 4 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> K e. RR )
142zred 9703 . . . 4 |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> N e. RR )
15 leaddsub 8714 . . . 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 9860 . . 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 2205   class class class wbr 4111   ` cfv 5354  (class class class)co 6052   RRcr 8128   + caddc 8132   <_ cle 8311   - cmin 8446   ZZcz 9579   ZZ>=cuz 9856
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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-ltadd 8245
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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-inn 9240  df-n0 9499  df-z 9580  df-uz 9857
This theorem is referenced by:  fzoss2  10512  shftuz  11506  climshftlemg  11991  isumshft  12180
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