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Theorem fzrev3 9835
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 9773 . . . 4  |-  ( K  e.  ( M ... N )  ->  M  e.  ZZ )
32adantl 275 . . 3  |-  ( ( K  e.  ZZ  /\  K  e.  ( M ... N ) )  ->  M  e.  ZZ )
4 elfzel2 9772 . . . 4  |-  ( K  e.  ( M ... N )  ->  N  e.  ZZ )
54adantl 275 . . 3  |-  ( ( K  e.  ZZ  /\  K  e.  ( M ... N ) )  ->  N  e.  ZZ )
61, 3, 53jca 1146 . 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 9773 . . . 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 9772 . . . 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 1146 . 2  |-  ( ( K  e.  ZZ  /\  ( ( M  +  N )  -  K
)  e.  ( M ... N ) )  ->  ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ ) )
13 zcn 9027 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  CC )
14 zcn 9027 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
15 pncan 7936 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( ( M  +  N )  -  N
)  =  M )
16 pncan2 7937 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( ( M  +  N )  -  M
)  =  N )
1715, 16oveq12d 5760 . . . . . 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 2187 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( ( ( M  +  N )  -  N
) ... ( ( M  +  N )  -  M ) )  <->  K  e.  ( M ... N ) ) )
20193adant1 984 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( (
( M  +  N
)  -  N ) ... ( ( M  +  N )  -  M ) )  <->  K  e.  ( M ... N ) ) )
21 3simpc 965 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
22 zaddcl 9062 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  +  N
)  e.  ZZ )
23223adant1 984 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  +  N )  e.  ZZ )
24 simp1 966 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  ZZ )
25 fzrev 9832 . . . 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 1199 . . 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 886 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 947    = wceq 1316    e. wcel 1465  (class class class)co 5742   CCcc 7586    + caddc 7591    - cmin 7901   ZZcz 9022   ...cfz 9758
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8689  df-n0 8946  df-z 9023  df-uz 9295  df-fz 9759
This theorem is referenced by:  fzrev3i  9836
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