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Theorem fzrev3 11757
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 457 . . 3  |-  ( ( K  e.  ZZ  /\  K  e.  ( M ... N ) )  ->  K  e.  ZZ )
2 elfzel1 11699 . . . 4  |-  ( K  e.  ( M ... N )  ->  M  e.  ZZ )
32adantl 466 . . 3  |-  ( ( K  e.  ZZ  /\  K  e.  ( M ... N ) )  ->  M  e.  ZZ )
4 elfzel2 11698 . . . 4  |-  ( K  e.  ( M ... N )  ->  N  e.  ZZ )
54adantl 466 . . 3  |-  ( ( K  e.  ZZ  /\  K  e.  ( M ... N ) )  ->  N  e.  ZZ )
61, 3, 53jca 1176 . 2  |-  ( ( K  e.  ZZ  /\  K  e.  ( M ... N ) )  -> 
( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )
)
7 simpl 457 . . 3  |-  ( ( K  e.  ZZ  /\  ( ( M  +  N )  -  K
)  e.  ( M ... N ) )  ->  K  e.  ZZ )
8 elfzel1 11699 . . . 4  |-  ( ( ( M  +  N
)  -  K )  e.  ( M ... N )  ->  M  e.  ZZ )
98adantl 466 . . 3  |-  ( ( K  e.  ZZ  /\  ( ( M  +  N )  -  K
)  e.  ( M ... N ) )  ->  M  e.  ZZ )
10 elfzel2 11698 . . . 4  |-  ( ( ( M  +  N
)  -  K )  e.  ( M ... N )  ->  N  e.  ZZ )
1110adantl 466 . . 3  |-  ( ( K  e.  ZZ  /\  ( ( M  +  N )  -  K
)  e.  ( M ... N ) )  ->  N  e.  ZZ )
127, 9, 113jca 1176 . 2  |-  ( ( K  e.  ZZ  /\  ( ( M  +  N )  -  K
)  e.  ( M ... N ) )  ->  ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ ) )
13 zcn 10881 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  CC )
14 zcn 10881 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
15 pncan 9838 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( ( M  +  N )  -  N
)  =  M )
16 pncan2 9839 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( ( M  +  N )  -  M
)  =  N )
1715, 16oveq12d 6313 . . . . . 6  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( ( ( M  +  N )  -  N ) ... (
( M  +  N
)  -  M ) )  =  ( M ... N ) )
1813, 14, 17syl2an 477 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( M  +  N )  -  N ) ... (
( M  +  N
)  -  M ) )  =  ( M ... N ) )
1918eleq2d 2537 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( ( ( M  +  N )  -  N
) ... ( ( M  +  N )  -  M ) )  <->  K  e.  ( M ... N ) ) )
20193adant1 1014 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( (
( M  +  N
)  -  N ) ... ( ( M  +  N )  -  M ) )  <->  K  e.  ( M ... N ) ) )
21 3simpc 995 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
22 zaddcl 10915 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  +  N
)  e.  ZZ )
23223adant1 1014 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  +  N )  e.  ZZ )
24 simp1 996 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  ZZ )
25 fzrev 11754 . . . 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 1226 . . 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767  (class class class)co 6295   CCcc 9502    + caddc 9507    - cmin 9817   ZZcz 10876   ...cfz 11684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685
This theorem is referenced by:  fzrev3i  11758
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