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Theorem uzsplit 11528
Description: Express an upper integer set as the disjoint (see uzdisj 11529) union of the first  N values and the rest. (Contributed by Mario Carneiro, 24-Apr-2014.)
Assertion
Ref Expression
uzsplit  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  M )  =  ( ( M ... ( N  -  1 ) )  u.  ( ZZ>= `  N ) ) )

Proof of Theorem uzsplit
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 eluzelre 10869 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  RR )
2 eluzelre 10869 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  RR )
3 lelttric 9479 . . . . . . . 8  |-  ( ( N  e.  RR  /\  k  e.  RR )  ->  ( N  <_  k  \/  k  <  N ) )
41, 2, 3syl2an 477 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( N  <_  k  \/  k  < 
N ) )
5 eluzelz 10868 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
6 eluzelz 10868 . . . . . . . . 9  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
7 eluz 10872 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  k  e.  ZZ )  ->  ( k  e.  (
ZZ>= `  N )  <->  N  <_  k ) )
85, 6, 7syl2an 477 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( k  e.  ( ZZ>= `  N )  <->  N  <_  k ) )
9 eluzel2 10864 . . . . . . . . . 10  |-  ( k  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
10 elfzm11 11526 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( k  e.  ( M ... ( N  -  1 ) )  <-> 
( k  e.  ZZ  /\  M  <_  k  /\  k  <  N ) ) )
11 df-3an 967 . . . . . . . . . . 11  |-  ( ( k  e.  ZZ  /\  M  <_  k  /\  k  <  N )  <->  ( (
k  e.  ZZ  /\  M  <_  k )  /\  k  <  N ) )
1210, 11syl6bb 261 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( k  e.  ( M ... ( N  -  1 ) )  <-> 
( ( k  e.  ZZ  /\  M  <_ 
k )  /\  k  <  N ) ) )
139, 5, 12syl2anr 478 . . . . . . . . 9  |-  ( ( N  e.  ( ZZ>= `  M )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( k  e.  ( M ... ( N  -  1 ) )  <->  ( ( k  e.  ZZ  /\  M  <_  k )  /\  k  <  N ) ) )
14 eluzle 10871 . . . . . . . . . . . 12  |-  ( k  e.  ( ZZ>= `  M
)  ->  M  <_  k )
156, 14jca 532 . . . . . . . . . . 11  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( k  e.  ZZ  /\  M  <_ 
k ) )
1615adantl 466 . . . . . . . . . 10  |-  ( ( N  e.  ( ZZ>= `  M )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( k  e.  ZZ  /\  M  <_ 
k ) )
1716biantrurd 508 . . . . . . . . 9  |-  ( ( N  e.  ( ZZ>= `  M )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( k  <  N  <->  ( ( k  e.  ZZ  /\  M  <_  k )  /\  k  <  N ) ) )
1813, 17bitr4d 256 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( k  e.  ( M ... ( N  -  1 ) )  <->  k  <  N
) )
198, 18orbi12d 709 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
k  e.  ( ZZ>= `  N )  \/  k  e.  ( M ... ( N  -  1 ) ) )  <->  ( N  <_  k  \/  k  < 
N ) ) )
204, 19mpbird 232 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( k  e.  ( ZZ>= `  N )  \/  k  e.  ( M ... ( N  - 
1 ) ) ) )
2120orcomd 388 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( k  e.  ( M ... ( N  -  1 ) )  \/  k  e.  ( ZZ>= `  N )
) )
2221ex 434 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( k  e.  ( ZZ>= `  M )  ->  ( k  e.  ( M ... ( N  -  1 ) )  \/  k  e.  (
ZZ>= `  N ) ) ) )
23 elfzuz 11447 . . . . . 6  |-  ( k  e.  ( M ... ( N  -  1
) )  ->  k  e.  ( ZZ>= `  M )
)
2423a1i 11 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( k  e.  ( M ... ( N  -  1 ) )  ->  k  e.  ( ZZ>= `  M )
) )
25 uztrn 10875 . . . . . 6  |-  ( ( k  e.  ( ZZ>= `  N )  /\  N  e.  ( ZZ>= `  M )
)  ->  k  e.  ( ZZ>= `  M )
)
2625expcom 435 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( k  e.  ( ZZ>= `  N )  ->  k  e.  ( ZZ>= `  M ) ) )
2724, 26jaod 380 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( (
k  e.  ( M ... ( N  - 
1 ) )  \/  k  e.  ( ZZ>= `  N ) )  -> 
k  e.  ( ZZ>= `  M ) ) )
2822, 27impbid 191 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( k  e.  ( ZZ>= `  M )  <->  ( k  e.  ( M ... ( N  - 
1 ) )  \/  k  e.  ( ZZ>= `  N ) ) ) )
29 elun 3495 . . 3  |-  ( k  e.  ( ( M ... ( N  - 
1 ) )  u.  ( ZZ>= `  N )
)  <->  ( k  e.  ( M ... ( N  -  1 ) )  \/  k  e.  ( ZZ>= `  N )
) )
3028, 29syl6bbr 263 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( k  e.  ( ZZ>= `  M )  <->  k  e.  ( ( M ... ( N  - 
1 ) )  u.  ( ZZ>= `  N )
) ) )
3130eqrdv 2439 1  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  M )  =  ( ( M ... ( N  -  1 ) )  u.  ( ZZ>= `  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    u. cun 3324   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   RRcr 9279   1c1 9281    < clt 9416    <_ cle 9417    - cmin 9593   ZZcz 10644   ZZ>=cuz 10859   ...cfz 11435
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 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436
This theorem is referenced by:  uniioombllem3  21063  uniioombllem4  21064  plyaddlem1  21679  plymullem1  21680
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