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Theorem fzdisj 11852
Description: Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.)
Assertion
Ref Expression
fzdisj  |-  ( K  <  M  ->  (
( J ... K
)  i^i  ( M ... N ) )  =  (/) )

Proof of Theorem fzdisj
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elin 3608 . . . 4  |-  ( x  e.  ( ( J ... K )  i^i  ( M ... N
) )  <->  ( x  e.  ( J ... K
)  /\  x  e.  ( M ... N ) ) )
2 elfzel1 11825 . . . . . . . 8  |-  ( x  e.  ( M ... N )  ->  M  e.  ZZ )
32adantl 473 . . . . . . 7  |-  ( ( x  e.  ( J ... K )  /\  x  e.  ( M ... N ) )  ->  M  e.  ZZ )
43zred 11063 . . . . . 6  |-  ( ( x  e.  ( J ... K )  /\  x  e.  ( M ... N ) )  ->  M  e.  RR )
5 elfzelz 11826 . . . . . . . 8  |-  ( x  e.  ( M ... N )  ->  x  e.  ZZ )
65zred 11063 . . . . . . 7  |-  ( x  e.  ( M ... N )  ->  x  e.  RR )
76adantl 473 . . . . . 6  |-  ( ( x  e.  ( J ... K )  /\  x  e.  ( M ... N ) )  ->  x  e.  RR )
8 elfzel2 11824 . . . . . . . 8  |-  ( x  e.  ( J ... K )  ->  K  e.  ZZ )
98adantr 472 . . . . . . 7  |-  ( ( x  e.  ( J ... K )  /\  x  e.  ( M ... N ) )  ->  K  e.  ZZ )
109zred 11063 . . . . . 6  |-  ( ( x  e.  ( J ... K )  /\  x  e.  ( M ... N ) )  ->  K  e.  RR )
11 elfzle1 11828 . . . . . . 7  |-  ( x  e.  ( M ... N )  ->  M  <_  x )
1211adantl 473 . . . . . 6  |-  ( ( x  e.  ( J ... K )  /\  x  e.  ( M ... N ) )  ->  M  <_  x )
13 elfzle2 11829 . . . . . . 7  |-  ( x  e.  ( J ... K )  ->  x  <_  K )
1413adantr 472 . . . . . 6  |-  ( ( x  e.  ( J ... K )  /\  x  e.  ( M ... N ) )  ->  x  <_  K )
154, 7, 10, 12, 14letrd 9809 . . . . 5  |-  ( ( x  e.  ( J ... K )  /\  x  e.  ( M ... N ) )  ->  M  <_  K )
164, 10lenltd 9798 . . . . 5  |-  ( ( x  e.  ( J ... K )  /\  x  e.  ( M ... N ) )  -> 
( M  <_  K  <->  -.  K  <  M ) )
1715, 16mpbid 215 . . . 4  |-  ( ( x  e.  ( J ... K )  /\  x  e.  ( M ... N ) )  ->  -.  K  <  M )
181, 17sylbi 200 . . 3  |-  ( x  e.  ( ( J ... K )  i^i  ( M ... N
) )  ->  -.  K  <  M )
1918con2i 124 . 2  |-  ( K  <  M  ->  -.  x  e.  ( ( J ... K )  i^i  ( M ... N
) ) )
2019eq0rdv 3773 1  |-  ( K  <  M  ->  (
( J ... K
)  i^i  ( M ... N ) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    i^i cin 3389   (/)c0 3722   class class class wbr 4395  (class class class)co 6308   RRcr 9556    < clt 9693    <_ cle 9694   ZZcz 10961   ...cfz 11810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-pre-lttri 9631  ax-pre-lttrn 9632
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-neg 9883  df-z 10962  df-uz 11183  df-fz 11811
This theorem is referenced by:  fsumm1  13889  fsum1p  13891  o1fsum  13950  climcndslem1  13984  climcndslem2  13985  mertenslem1  14017  fprod1p  14099  fprodeq0  14106  fallfacval4  14173  prmreclem5  14943  strleun  15298  uniioombllem3  22622  mtest  23438  birthdaylem2  23957  fsumharmonic  24016  ftalem5  24080  ftalem5OLD  24082  chtdif  24164  ppidif  24169  lgsquadlem2  24362  dchrisum0lem1b  24432  dchrisum0lem3  24436  pntrsumbnd2  24484  pntrlog2bndlem6  24500  pntpbnd2  24504  pntlemf  24522  axlowdimlem2  25052  axlowdimlem16  25066  constr3trllem3  25459  esumpmono  28974  ballotlemfrceq  29434  ballotlemfrceqOLD  29472  poimirlem1  32005  poimirlem2  32006  poimirlem3  32007  poimirlem4  32008  poimirlem6  32010  poimirlem7  32011  poimirlem11  32015  poimirlem12  32016  poimirlem16  32020  poimirlem17  32021  poimirlem19  32023  poimirlem20  32024  poimirlem23  32027  poimirlem24  32028  poimirlem25  32029  poimirlem28  32032  poimirlem29  32033  poimirlem31  32035  eldioph2lem1  35673  stoweidlem11  37983
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