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Theorem fzsplit3 26216
Description: Split a finite interval of integers into two parts. (Contributed by Thierry Arnoux, 2-May-2017.)
Assertion
Ref Expression
fzsplit3  |-  ( K  e.  ( M ... N )  ->  ( M ... N )  =  ( ( M ... ( K  -  1
) )  u.  ( K ... N ) ) )

Proof of Theorem fzsplit3
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elfzelz 11563 . . . . . . 7  |-  ( x  e.  ( M ... N )  ->  x  e.  ZZ )
21zred 10851 . . . . . 6  |-  ( x  e.  ( M ... N )  ->  x  e.  RR )
3 elfzelz 11563 . . . . . . . 8  |-  ( K  e.  ( M ... N )  ->  K  e.  ZZ )
43zred 10851 . . . . . . 7  |-  ( K  e.  ( M ... N )  ->  K  e.  RR )
5 1re 9489 . . . . . . . 8  |-  1  e.  RR
65a1i 11 . . . . . . 7  |-  ( K  e.  ( M ... N )  ->  1  e.  RR )
74, 6resubcld 9880 . . . . . 6  |-  ( K  e.  ( M ... N )  ->  ( K  -  1 )  e.  RR )
8 lelttric 9585 . . . . . 6  |-  ( ( x  e.  RR  /\  ( K  -  1
)  e.  RR )  ->  ( x  <_ 
( K  -  1 )  \/  ( K  -  1 )  < 
x ) )
92, 7, 8syl2anr 478 . . . . 5  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... N ) )  -> 
( x  <_  ( K  -  1 )  \/  ( K  - 
1 )  <  x
) )
10 elfzuz 11559 . . . . . . 7  |-  ( x  e.  ( M ... N )  ->  x  e.  ( ZZ>= `  M )
)
11 1z 10780 . . . . . . . . 9  |-  1  e.  ZZ
1211a1i 11 . . . . . . . 8  |-  ( K  e.  ( M ... N )  ->  1  e.  ZZ )
133, 12zsubcld 10856 . . . . . . 7  |-  ( K  e.  ( M ... N )  ->  ( K  -  1 )  e.  ZZ )
14 elfz5 11555 . . . . . . 7  |-  ( ( x  e.  ( ZZ>= `  M )  /\  ( K  -  1 )  e.  ZZ )  -> 
( x  e.  ( M ... ( K  -  1 ) )  <-> 
x  <_  ( K  -  1 ) ) )
1510, 13, 14syl2anr 478 . . . . . 6  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... N ) )  -> 
( x  e.  ( M ... ( K  -  1 ) )  <-> 
x  <_  ( K  -  1 ) ) )
16 elfzuz3 11560 . . . . . . . . 9  |-  ( x  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  x )
)
1716adantl 466 . . . . . . . 8  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... N ) )  ->  N  e.  ( ZZ>= `  x ) )
18 elfzuzb 11557 . . . . . . . . 9  |-  ( x  e.  ( K ... N )  <->  ( x  e.  ( ZZ>= `  K )  /\  N  e.  ( ZZ>=
`  x ) ) )
1918rbaib 898 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  x
)  ->  ( x  e.  ( K ... N
)  <->  x  e.  ( ZZ>=
`  K ) ) )
2017, 19syl 16 . . . . . . 7  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... N ) )  -> 
( x  e.  ( K ... N )  <-> 
x  e.  ( ZZ>= `  K ) ) )
21 eluz 10978 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  x  e.  ZZ )  ->  ( x  e.  (
ZZ>= `  K )  <->  K  <_  x ) )
223, 1, 21syl2an 477 . . . . . . 7  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... N ) )  -> 
( x  e.  (
ZZ>= `  K )  <->  K  <_  x ) )
23 zlem1lt 10800 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  x  e.  ZZ )  ->  ( K  <_  x  <->  ( K  -  1 )  <  x ) )
243, 1, 23syl2an 477 . . . . . . 7  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... N ) )  -> 
( K  <_  x  <->  ( K  -  1 )  <  x ) )
2520, 22, 243bitrd 279 . . . . . 6  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... N ) )  -> 
( x  e.  ( K ... N )  <-> 
( K  -  1 )  <  x ) )
2615, 25orbi12d 709 . . . . 5  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... N ) )  -> 
( ( x  e.  ( M ... ( K  -  1 ) )  \/  x  e.  ( K ... N
) )  <->  ( x  <_  ( K  -  1 )  \/  ( K  -  1 )  < 
x ) ) )
279, 26mpbird 232 . . . 4  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... N ) )  -> 
( x  e.  ( M ... ( K  -  1 ) )  \/  x  e.  ( K ... N ) ) )
28 elfzuz 11559 . . . . . . 7  |-  ( x  e.  ( M ... ( K  -  1
) )  ->  x  e.  ( ZZ>= `  M )
)
2928adantl 466 . . . . . 6  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... ( K  -  1 ) ) )  ->  x  e.  ( ZZ>= `  M ) )
30 elfzuz3 11560 . . . . . . . 8  |-  ( K  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  K )
)
3130adantr 465 . . . . . . 7  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... ( K  -  1 ) ) )  ->  N  e.  ( ZZ>= `  K ) )
32 elfzuz3 11560 . . . . . . . . . 10  |-  ( x  e.  ( M ... ( K  -  1
) )  ->  ( K  -  1 )  e.  ( ZZ>= `  x
) )
3332adantl 466 . . . . . . . . 9  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... ( K  -  1 ) ) )  -> 
( K  -  1 )  e.  ( ZZ>= `  x ) )
34 peano2uz 11012 . . . . . . . . 9  |-  ( ( K  -  1 )  e.  ( ZZ>= `  x
)  ->  ( ( K  -  1 )  +  1 )  e.  ( ZZ>= `  x )
)
3533, 34syl 16 . . . . . . . 8  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... ( K  -  1 ) ) )  -> 
( ( K  - 
1 )  +  1 )  e.  ( ZZ>= `  x ) )
364recnd 9516 . . . . . . . . . . 11  |-  ( K  e.  ( M ... N )  ->  K  e.  CC )
376recnd 9516 . . . . . . . . . . 11  |-  ( K  e.  ( M ... N )  ->  1  e.  CC )
3836, 37npcand 9827 . . . . . . . . . 10  |-  ( K  e.  ( M ... N )  ->  (
( K  -  1 )  +  1 )  =  K )
3938eleq1d 2520 . . . . . . . . 9  |-  ( K  e.  ( M ... N )  ->  (
( ( K  - 
1 )  +  1 )  e.  ( ZZ>= `  x )  <->  K  e.  ( ZZ>= `  x )
) )
4039adantr 465 . . . . . . . 8  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... ( K  -  1 ) ) )  -> 
( ( ( K  -  1 )  +  1 )  e.  (
ZZ>= `  x )  <->  K  e.  ( ZZ>= `  x )
) )
4135, 40mpbid 210 . . . . . . 7  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... ( K  -  1 ) ) )  ->  K  e.  ( ZZ>= `  x ) )
42 uztrn 10981 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  x )
)  ->  N  e.  ( ZZ>= `  x )
)
4331, 41, 42syl2anc 661 . . . . . 6  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... ( K  -  1 ) ) )  ->  N  e.  ( ZZ>= `  x ) )
44 elfzuzb 11557 . . . . . 6  |-  ( x  e.  ( M ... N )  <->  ( x  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>=
`  x ) ) )
4529, 43, 44sylanbrc 664 . . . . 5  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... ( K  -  1 ) ) )  ->  x  e.  ( M ... N ) )
46 elfzuz 11559 . . . . . . 7  |-  ( x  e.  ( K ... N )  ->  x  e.  ( ZZ>= `  K )
)
47 elfzuz 11559 . . . . . . 7  |-  ( K  e.  ( M ... N )  ->  K  e.  ( ZZ>= `  M )
)
48 uztrn 10981 . . . . . . 7  |-  ( ( x  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  x  e.  ( ZZ>= `  M )
)
4946, 47, 48syl2anr 478 . . . . . 6  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( K ... N ) )  ->  x  e.  ( ZZ>= `  M ) )
50 elfzuz3 11560 . . . . . . 7  |-  ( x  e.  ( K ... N )  ->  N  e.  ( ZZ>= `  x )
)
5150adantl 466 . . . . . 6  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( K ... N ) )  ->  N  e.  ( ZZ>= `  x ) )
5249, 51, 44sylanbrc 664 . . . . 5  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( K ... N ) )  ->  x  e.  ( M ... N ) )
5345, 52jaodan 783 . . . 4  |-  ( ( K  e.  ( M ... N )  /\  ( x  e.  ( M ... ( K  - 
1 ) )  \/  x  e.  ( K ... N ) ) )  ->  x  e.  ( M ... N ) )
5427, 53impbida 828 . . 3  |-  ( K  e.  ( M ... N )  ->  (
x  e.  ( M ... N )  <->  ( x  e.  ( M ... ( K  -  1 ) )  \/  x  e.  ( K ... N
) ) ) )
55 elun 3598 . . 3  |-  ( x  e.  ( ( M ... ( K  - 
1 ) )  u.  ( K ... N
) )  <->  ( x  e.  ( M ... ( K  -  1 ) )  \/  x  e.  ( K ... N
) ) )
5654, 55syl6bbr 263 . 2  |-  ( K  e.  ( M ... N )  ->  (
x  e.  ( M ... N )  <->  x  e.  ( ( M ... ( K  -  1
) )  u.  ( K ... N ) ) ) )
5756eqrdv 2448 1  |-  ( K  e.  ( M ... N )  ->  ( M ... N )  =  ( ( M ... ( K  -  1
) )  u.  ( K ... N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    u. cun 3427   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   RRcr 9385   1c1 9387    + caddc 9389    < clt 9522    <_ cle 9523    - cmin 9699   ZZcz 10750   ZZ>=cuz 10965   ...cfz 11547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548
This theorem is referenced by:  ballotlemfrceq  27048
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