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Theorem fzrev3 11749
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 455 . . 3  |-  ( ( K  e.  ZZ  /\  K  e.  ( M ... N ) )  ->  K  e.  ZZ )
2 elfzel1 11690 . . . 4  |-  ( K  e.  ( M ... N )  ->  M  e.  ZZ )
32adantl 464 . . 3  |-  ( ( K  e.  ZZ  /\  K  e.  ( M ... N ) )  ->  M  e.  ZZ )
4 elfzel2 11689 . . . 4  |-  ( K  e.  ( M ... N )  ->  N  e.  ZZ )
54adantl 464 . . 3  |-  ( ( K  e.  ZZ  /\  K  e.  ( M ... N ) )  ->  N  e.  ZZ )
61, 3, 53jca 1174 . 2  |-  ( ( K  e.  ZZ  /\  K  e.  ( M ... N ) )  -> 
( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )
)
7 simpl 455 . . 3  |-  ( ( K  e.  ZZ  /\  ( ( M  +  N )  -  K
)  e.  ( M ... N ) )  ->  K  e.  ZZ )
8 elfzel1 11690 . . . 4  |-  ( ( ( M  +  N
)  -  K )  e.  ( M ... N )  ->  M  e.  ZZ )
98adantl 464 . . 3  |-  ( ( K  e.  ZZ  /\  ( ( M  +  N )  -  K
)  e.  ( M ... N ) )  ->  M  e.  ZZ )
10 elfzel2 11689 . . . 4  |-  ( ( ( M  +  N
)  -  K )  e.  ( M ... N )  ->  N  e.  ZZ )
1110adantl 464 . . 3  |-  ( ( K  e.  ZZ  /\  ( ( M  +  N )  -  K
)  e.  ( M ... N ) )  ->  N  e.  ZZ )
127, 9, 113jca 1174 . 2  |-  ( ( K  e.  ZZ  /\  ( ( M  +  N )  -  K
)  e.  ( M ... N ) )  ->  ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ ) )
13 zcn 10865 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  CC )
14 zcn 10865 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
15 pncan 9817 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( ( M  +  N )  -  N
)  =  M )
16 pncan2 9818 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( ( M  +  N )  -  M
)  =  N )
1715, 16oveq12d 6288 . . . . . 6  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( ( ( M  +  N )  -  N ) ... (
( M  +  N
)  -  M ) )  =  ( M ... N ) )
1813, 14, 17syl2an 475 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( M  +  N )  -  N ) ... (
( M  +  N
)  -  M ) )  =  ( M ... N ) )
1918eleq2d 2524 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( ( ( M  +  N )  -  N
) ... ( ( M  +  N )  -  M ) )  <->  K  e.  ( M ... N ) ) )
20193adant1 1012 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( (
( M  +  N
)  -  N ) ... ( ( M  +  N )  -  M ) )  <->  K  e.  ( M ... N ) ) )
21 3simpc 993 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
22 zaddcl 10900 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  +  N
)  e.  ZZ )
23223adant1 1012 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  +  N )  e.  ZZ )
24 simp1 994 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  ZZ )
25 fzrev 11746 . . . 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 1224 . . 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 255 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N )  <->  ( ( M  +  N )  -  K )  e.  ( M ... N ) ) )
286, 12, 27pm5.21nd 898 1  |-  ( K  e.  ZZ  ->  ( K  e.  ( M ... N )  <->  ( ( M  +  N )  -  K )  e.  ( M ... N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823  (class class class)co 6270   CCcc 9479    + caddc 9484    - cmin 9796   ZZcz 10860   ...cfz 11675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676
This theorem is referenced by:  fzrev3i  11750
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