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Theorem fzrev3 9860
Description: The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
Assertion
Ref Expression
fzrev3  |-  ( K  e.  ZZ  ->  ( K  e.  ( M ... N )  <->  ( ( M  +  N )  -  K )  e.  ( M ... N ) ) )

Proof of Theorem fzrev3
StepHypRef Expression
1 simpl 108 . . 3  |-  ( ( K  e.  ZZ  /\  K  e.  ( M ... N ) )  ->  K  e.  ZZ )
2 elfzel1 9798 . . . 4  |-  ( K  e.  ( M ... N )  ->  M  e.  ZZ )
32adantl 275 . . 3  |-  ( ( K  e.  ZZ  /\  K  e.  ( M ... N ) )  ->  M  e.  ZZ )
4 elfzel2 9797 . . . 4  |-  ( K  e.  ( M ... N )  ->  N  e.  ZZ )
54adantl 275 . . 3  |-  ( ( K  e.  ZZ  /\  K  e.  ( M ... N ) )  ->  N  e.  ZZ )
61, 3, 53jca 1161 . 2  |-  ( ( K  e.  ZZ  /\  K  e.  ( M ... N ) )  -> 
( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )
)
7 simpl 108 . . 3  |-  ( ( K  e.  ZZ  /\  ( ( M  +  N )  -  K
)  e.  ( M ... N ) )  ->  K  e.  ZZ )
8 elfzel1 9798 . . . 4  |-  ( ( ( M  +  N
)  -  K )  e.  ( M ... N )  ->  M  e.  ZZ )
98adantl 275 . . 3  |-  ( ( K  e.  ZZ  /\  ( ( M  +  N )  -  K
)  e.  ( M ... N ) )  ->  M  e.  ZZ )
10 elfzel2 9797 . . . 4  |-  ( ( ( M  +  N
)  -  K )  e.  ( M ... N )  ->  N  e.  ZZ )
1110adantl 275 . . 3  |-  ( ( K  e.  ZZ  /\  ( ( M  +  N )  -  K
)  e.  ( M ... N ) )  ->  N  e.  ZZ )
127, 9, 113jca 1161 . 2  |-  ( ( K  e.  ZZ  /\  ( ( M  +  N )  -  K
)  e.  ( M ... N ) )  ->  ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ ) )
13 zcn 9052 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  CC )
14 zcn 9052 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
15 pncan 7961 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( ( M  +  N )  -  N
)  =  M )
16 pncan2 7962 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( ( M  +  N )  -  M
)  =  N )
1715, 16oveq12d 5785 . . . . . 6  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( ( ( M  +  N )  -  N ) ... (
( M  +  N
)  -  M ) )  =  ( M ... N ) )
1813, 14, 17syl2an 287 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( M  +  N )  -  N ) ... (
( M  +  N
)  -  M ) )  =  ( M ... N ) )
1918eleq2d 2207 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( ( ( M  +  N )  -  N
) ... ( ( M  +  N )  -  M ) )  <->  K  e.  ( M ... N ) ) )
20193adant1 999 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( (
( M  +  N
)  -  N ) ... ( ( M  +  N )  -  M ) )  <->  K  e.  ( M ... N ) ) )
21 3simpc 980 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
22 zaddcl 9087 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  +  N
)  e.  ZZ )
23223adant1 999 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  +  N )  e.  ZZ )
24 simp1 981 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  ZZ )
25 fzrev 9857 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( M  +  N )  e.  ZZ  /\  K  e.  ZZ ) )  -> 
( K  e.  ( ( ( M  +  N )  -  N
) ... ( ( M  +  N )  -  M ) )  <->  ( ( M  +  N )  -  K )  e.  ( M ... N ) ) )
2621, 23, 24, 25syl12anc 1214 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( (
( M  +  N
)  -  N ) ... ( ( M  +  N )  -  M ) )  <->  ( ( M  +  N )  -  K )  e.  ( M ... N ) ) )
2720, 26bitr3d 189 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N )  <->  ( ( M  +  N )  -  K )  e.  ( M ... N ) ) )
286, 12, 27pm5.21nd 901 1  |-  ( K  e.  ZZ  ->  ( K  e.  ( M ... N )  <->  ( ( M  +  N )  -  K )  e.  ( M ... N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480  (class class class)co 5767   CCcc 7611    + caddc 7616    - cmin 7926   ZZcz 9047   ...cfz 9783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-ltadd 7729
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-inn 8714  df-n0 8971  df-z 9048  df-uz 9320  df-fz 9784
This theorem is referenced by:  fzrev3i  9861
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