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Theorem fzdisj 11823
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 3616 . . . 4  |-  ( x  e.  ( ( J ... K )  i^i  ( M ... N
) )  <->  ( x  e.  ( J ... K
)  /\  x  e.  ( M ... N ) ) )
2 elfzel1 11796 . . . . . . . 8  |-  ( x  e.  ( M ... N )  ->  M  e.  ZZ )
32adantl 468 . . . . . . 7  |-  ( ( x  e.  ( J ... K )  /\  x  e.  ( M ... N ) )  ->  M  e.  ZZ )
43zred 11037 . . . . . 6  |-  ( ( x  e.  ( J ... K )  /\  x  e.  ( M ... N ) )  ->  M  e.  RR )
5 elfzelz 11797 . . . . . . . 8  |-  ( x  e.  ( M ... N )  ->  x  e.  ZZ )
65zred 11037 . . . . . . 7  |-  ( x  e.  ( M ... N )  ->  x  e.  RR )
76adantl 468 . . . . . 6  |-  ( ( x  e.  ( J ... K )  /\  x  e.  ( M ... N ) )  ->  x  e.  RR )
8 elfzel2 11795 . . . . . . . 8  |-  ( x  e.  ( J ... K )  ->  K  e.  ZZ )
98adantr 467 . . . . . . 7  |-  ( ( x  e.  ( J ... K )  /\  x  e.  ( M ... N ) )  ->  K  e.  ZZ )
109zred 11037 . . . . . 6  |-  ( ( x  e.  ( J ... K )  /\  x  e.  ( M ... N ) )  ->  K  e.  RR )
11 elfzle1 11799 . . . . . . 7  |-  ( x  e.  ( M ... N )  ->  M  <_  x )
1211adantl 468 . . . . . 6  |-  ( ( x  e.  ( J ... K )  /\  x  e.  ( M ... N ) )  ->  M  <_  x )
13 elfzle2 11800 . . . . . . 7  |-  ( x  e.  ( J ... K )  ->  x  <_  K )
1413adantr 467 . . . . . 6  |-  ( ( x  e.  ( J ... K )  /\  x  e.  ( M ... N ) )  ->  x  <_  K )
154, 7, 10, 12, 14letrd 9789 . . . . 5  |-  ( ( x  e.  ( J ... K )  /\  x  e.  ( M ... N ) )  ->  M  <_  K )
164, 10lenltd 9778 . . . . 5  |-  ( ( x  e.  ( J ... K )  /\  x  e.  ( M ... N ) )  -> 
( M  <_  K  <->  -.  K  <  M ) )
1715, 16mpbid 214 . . . 4  |-  ( ( x  e.  ( J ... K )  /\  x  e.  ( M ... N ) )  ->  -.  K  <  M )
181, 17sylbi 199 . . 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 3768 1  |-  ( K  <  M  ->  (
( J ... K
)  i^i  ( M ... N ) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    = wceq 1443    e. wcel 1886    i^i cin 3402   (/)c0 3730   class class class wbr 4401  (class class class)co 6288   RRcr 9535    < clt 9672    <_ cle 9673   ZZcz 10934   ...cfz 11781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-pre-lttri 9610  ax-pre-lttrn 9611
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-1st 6790  df-2nd 6791  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-neg 9860  df-z 10935  df-uz 11157  df-fz 11782
This theorem is referenced by:  fsumm1  13805  fsum1p  13807  o1fsum  13866  climcndslem1  13900  climcndslem2  13901  mertenslem1  13933  fprod1p  14015  fprodeq0  14022  fallfacval4  14089  prmreclem5  14857  strleun  15213  uniioombllem3  22536  mtest  23352  birthdaylem2  23871  fsumharmonic  23930  ftalem5  23994  ftalem5OLD  23996  chtdif  24078  ppidif  24083  lgsquadlem2  24276  dchrisum0lem1b  24346  dchrisum0lem3  24350  pntrsumbnd2  24398  pntrlog2bndlem6  24414  pntpbnd2  24418  pntlemf  24436  axlowdimlem2  24966  axlowdimlem16  24980  constr3trllem3  25373  esumpmono  28893  ballotlemfrceq  29354  ballotlemfrceqOLD  29392  poimirlem1  31934  poimirlem2  31935  poimirlem3  31936  poimirlem4  31937  poimirlem6  31939  poimirlem7  31940  poimirlem11  31944  poimirlem12  31945  poimirlem16  31949  poimirlem17  31950  poimirlem19  31952  poimirlem20  31953  poimirlem23  31956  poimirlem24  31957  poimirlem25  31958  poimirlem28  31961  poimirlem29  31962  poimirlem31  31964  eldioph2lem1  35596  stoweidlem11  37865
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