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Theorem fzrev3 11543
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 11473 . . . 4  |-  ( K  e.  ( M ... N )  ->  M  e.  ZZ )
32adantl 466 . . 3  |-  ( ( K  e.  ZZ  /\  K  e.  ( M ... N ) )  ->  M  e.  ZZ )
4 elfzel2 11472 . . . 4  |-  ( K  e.  ( M ... N )  ->  N  e.  ZZ )
54adantl 466 . . 3  |-  ( ( K  e.  ZZ  /\  K  e.  ( M ... N ) )  ->  N  e.  ZZ )
61, 3, 53jca 1168 . 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 11473 . . . 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 11472 . . . 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 1168 . 2  |-  ( ( K  e.  ZZ  /\  ( ( M  +  N )  -  K
)  e.  ( M ... N ) )  ->  ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ ) )
13 zcn 10672 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  CC )
14 zcn 10672 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
15 pncan 9637 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( ( M  +  N )  -  N
)  =  M )
16 pncan2 9638 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( ( M  +  N )  -  M
)  =  N )
1715, 16oveq12d 6130 . . . . . 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 2510 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( ( ( M  +  N )  -  N
) ... ( ( M  +  N )  -  M ) )  <->  K  e.  ( M ... N ) ) )
20193adant1 1006 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( (
( M  +  N
)  -  N ) ... ( ( M  +  N )  -  M ) )  <->  K  e.  ( M ... N ) ) )
21 3simpc 987 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
22 zaddcl 10706 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  +  N
)  e.  ZZ )
23223adant1 1006 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  +  N )  e.  ZZ )
24 simp1 988 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  ZZ )
25 fzrev 11540 . . . 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 1216 . . 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 893 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 965    = wceq 1369    e. wcel 1756  (class class class)co 6112   CCcc 9301    + caddc 9306    - cmin 9616   ZZcz 10667   ...cfz 11458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-n0 10601  df-z 10668  df-uz 10883  df-fz 11459
This theorem is referenced by:  fzrev3i  11544
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