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Theorem uzsplit 11739
Description: Express an upper integer set as the disjoint (see uzdisj 11740) 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 11081 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  RR )
2 eluzelre 11081 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  RR )
3 lelttric 9680 . . . . . . . 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 11080 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
6 eluzelz 11080 . . . . . . . . 9  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
7 eluz 11084 . . . . . . . . 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 11076 . . . . . . . . . 10  |-  ( k  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
10 elfzm11 11738 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( k  e.  ( M ... ( N  -  1 ) )  <-> 
( k  e.  ZZ  /\  M  <_  k  /\  k  <  N ) ) )
11 df-3an 970 . . . . . . . . . . 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 11083 . . . . . . . . . . . 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 11673 . . . . . 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 11087 . . . . . 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 3638 . . 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 2457 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 968    = wceq 1374    e. wcel 1762    u. cun 3467   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   RRcr 9480   1c1 9482    < clt 9617    <_ cle 9618    - cmin 9794   ZZcz 10853   ZZ>=cuz 11071   ...cfz 11661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  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 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662
This theorem is referenced by:  nn0split  11776  uniioombllem3  21722  uniioombllem4  21723  plyaddlem1  22338  plymullem1  22339
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