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Theorem elfzolborelfzop1 40685
Description: An element of a half-open integer interval is either equal to the left bound of the interval or an element of a half-open integer interval with a left bound increased by 1. (Contributed by AV, 2-Jun-2020.)
Assertion
Ref Expression
elfzolborelfzop1  |-  ( K  e.  ( M..^ N
)  ->  ( K  =  M  \/  K  e.  ( ( M  + 
1 )..^ N ) ) )

Proof of Theorem elfzolborelfzop1
StepHypRef Expression
1 elfzo2 11960 . 2  |-  ( K  e.  ( M..^ N
)  <->  ( K  e.  ( ZZ>= `  M )  /\  N  e.  ZZ  /\  K  <  N ) )
2 eluz2 11199 . . . 4  |-  ( K  e.  ( ZZ>= `  M
)  <->  ( M  e.  ZZ  /\  K  e.  ZZ  /\  M  <_  K ) )
3 zre 10975 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  RR )
4 zre 10975 . . . . . . 7  |-  ( K  e.  ZZ  ->  K  e.  RR )
5 leloe 9751 . . . . . . 7  |-  ( ( M  e.  RR  /\  K  e.  RR )  ->  ( M  <_  K  <->  ( M  <  K  \/  M  =  K )
) )
63, 4, 5syl2an 484 . . . . . 6  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  <_  K  <->  ( M  <  K  \/  M  =  K )
) )
7 peano2z 11012 . . . . . . . . . . . . . . . 16  |-  ( M  e.  ZZ  ->  ( M  +  1 )  e.  ZZ )
87adantr 471 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  +  1 )  e.  ZZ )
98ad2antrl 739 . . . . . . . . . . . . . 14  |-  ( ( M  <  K  /\  ( ( M  e.  ZZ  /\  K  e.  ZZ )  /\  N  e.  ZZ ) )  -> 
( M  +  1 )  e.  ZZ )
10 simprlr 778 . . . . . . . . . . . . . 14  |-  ( ( M  <  K  /\  ( ( M  e.  ZZ  /\  K  e.  ZZ )  /\  N  e.  ZZ ) )  ->  K  e.  ZZ )
11 simpl 463 . . . . . . . . . . . . . . 15  |-  ( ( M  <  K  /\  ( ( M  e.  ZZ  /\  K  e.  ZZ )  /\  N  e.  ZZ ) )  ->  M  <  K )
12 zltp1le 11020 . . . . . . . . . . . . . . . 16  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  <  K  <->  ( M  +  1 )  <_  K ) )
1312ad2antrl 739 . . . . . . . . . . . . . . 15  |-  ( ( M  <  K  /\  ( ( M  e.  ZZ  /\  K  e.  ZZ )  /\  N  e.  ZZ ) )  -> 
( M  <  K  <->  ( M  +  1 )  <_  K ) )
1411, 13mpbid 215 . . . . . . . . . . . . . 14  |-  ( ( M  <  K  /\  ( ( M  e.  ZZ  /\  K  e.  ZZ )  /\  N  e.  ZZ ) )  -> 
( M  +  1 )  <_  K )
159, 10, 143jca 1194 . . . . . . . . . . . . 13  |-  ( ( M  <  K  /\  ( ( M  e.  ZZ  /\  K  e.  ZZ )  /\  N  e.  ZZ ) )  -> 
( ( M  + 
1 )  e.  ZZ  /\  K  e.  ZZ  /\  ( M  +  1
)  <_  K )
)
1615adantr 471 . . . . . . . . . . . 12  |-  ( ( ( M  <  K  /\  ( ( M  e.  ZZ  /\  K  e.  ZZ )  /\  N  e.  ZZ ) )  /\  K  <  N )  -> 
( ( M  + 
1 )  e.  ZZ  /\  K  e.  ZZ  /\  ( M  +  1
)  <_  K )
)
17 simplrr 776 . . . . . . . . . . . 12  |-  ( ( ( M  <  K  /\  ( ( M  e.  ZZ  /\  K  e.  ZZ )  /\  N  e.  ZZ ) )  /\  K  <  N )  ->  N  e.  ZZ )
18 simpr 467 . . . . . . . . . . . 12  |-  ( ( ( M  <  K  /\  ( ( M  e.  ZZ  /\  K  e.  ZZ )  /\  N  e.  ZZ ) )  /\  K  <  N )  ->  K  <  N )
19 elfzo2 11960 . . . . . . . . . . . . 13  |-  ( K  e.  ( ( M  +  1 )..^ N
)  <->  ( K  e.  ( ZZ>= `  ( M  +  1 ) )  /\  N  e.  ZZ  /\  K  <  N ) )
20 eluz2 11199 . . . . . . . . . . . . . 14  |-  ( K  e.  ( ZZ>= `  ( M  +  1 ) )  <->  ( ( M  +  1 )  e.  ZZ  /\  K  e.  ZZ  /\  ( M  +  1 )  <_  K ) )
21203anbi1i 1205 . . . . . . . . . . . . 13  |-  ( ( K  e.  ( ZZ>= `  ( M  +  1
) )  /\  N  e.  ZZ  /\  K  < 
N )  <->  ( (
( M  +  1 )  e.  ZZ  /\  K  e.  ZZ  /\  ( M  +  1 )  <_  K )  /\  N  e.  ZZ  /\  K  <  N ) )
2219, 21bitri 257 . . . . . . . . . . . 12  |-  ( K  e.  ( ( M  +  1 )..^ N
)  <->  ( ( ( M  +  1 )  e.  ZZ  /\  K  e.  ZZ  /\  ( M  +  1 )  <_  K )  /\  N  e.  ZZ  /\  K  < 
N ) )
2316, 17, 18, 22syl3anbrc 1198 . . . . . . . . . . 11  |-  ( ( ( M  <  K  /\  ( ( M  e.  ZZ  /\  K  e.  ZZ )  /\  N  e.  ZZ ) )  /\  K  <  N )  ->  K  e.  ( ( M  +  1 )..^ N ) )
2423olcd 399 . . . . . . . . . 10  |-  ( ( ( M  <  K  /\  ( ( M  e.  ZZ  /\  K  e.  ZZ )  /\  N  e.  ZZ ) )  /\  K  <  N )  -> 
( K  =  M  \/  K  e.  ( ( M  +  1 )..^ N ) ) )
2524exp31 613 . . . . . . . . 9  |-  ( M  <  K  ->  (
( ( M  e.  ZZ  /\  K  e.  ZZ )  /\  N  e.  ZZ )  ->  ( K  <  N  ->  ( K  =  M  \/  K  e.  ( ( M  +  1 )..^ N ) ) ) ) )
26 orc 391 . . . . . . . . . . 11  |-  ( K  =  M  ->  ( K  =  M  \/  K  e.  ( ( M  +  1 )..^ N ) ) )
2726eqcoms 2470 . . . . . . . . . 10  |-  ( M  =  K  ->  ( K  =  M  \/  K  e.  ( ( M  +  1 )..^ N ) ) )
28272a1d 27 . . . . . . . . 9  |-  ( M  =  K  ->  (
( ( M  e.  ZZ  /\  K  e.  ZZ )  /\  N  e.  ZZ )  ->  ( K  <  N  ->  ( K  =  M  \/  K  e.  ( ( M  +  1 )..^ N ) ) ) ) )
2925, 28jaoi 385 . . . . . . . 8  |-  ( ( M  <  K  \/  M  =  K )  ->  ( ( ( M  e.  ZZ  /\  K  e.  ZZ )  /\  N  e.  ZZ )  ->  ( K  <  N  ->  ( K  =  M  \/  K  e.  ( ( M  +  1 )..^ N ) ) ) ) )
3029expd 442 . . . . . . 7  |-  ( ( M  <  K  \/  M  =  K )  ->  ( ( M  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  e.  ZZ  ->  ( K  <  N  -> 
( K  =  M  \/  K  e.  ( ( M  +  1 )..^ N ) ) ) ) ) )
3130com12 32 . . . . . 6  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ )  ->  ( ( M  < 
K  \/  M  =  K )  ->  ( N  e.  ZZ  ->  ( K  <  N  -> 
( K  =  M  \/  K  e.  ( ( M  +  1 )..^ N ) ) ) ) ) )
326, 31sylbid 223 . . . . 5  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  <_  K  ->  ( N  e.  ZZ  ->  ( K  <  N  ->  ( K  =  M  \/  K  e.  ( ( M  +  1 )..^ N ) ) ) ) ) )
33323impia 1212 . . . 4  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ  /\  M  <_  K )  ->  ( N  e.  ZZ  ->  ( K  <  N  -> 
( K  =  M  \/  K  e.  ( ( M  +  1 )..^ N ) ) ) ) )
342, 33sylbi 200 . . 3  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( N  e.  ZZ  ->  ( K  <  N  ->  ( K  =  M  \/  K  e.  ( ( M  + 
1 )..^ N ) ) ) ) )
35343imp 1208 . 2  |-  ( ( K  e.  ( ZZ>= `  M )  /\  N  e.  ZZ  /\  K  < 
N )  ->  ( K  =  M  \/  K  e.  ( ( M  +  1 )..^ N ) ) )
361, 35sylbi 200 1  |-  ( K  e.  ( M..^ N
)  ->  ( K  =  M  \/  K  e.  ( ( M  + 
1 )..^ N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 374    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898   class class class wbr 4418   ` cfv 5605  (class class class)co 6320   RRcr 9569   1c1 9571    + caddc 9573    < clt 9706    <_ cle 9707   ZZcz 10971   ZZ>=cuz 11193  ..^cfzo 11952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-1st 6825  df-2nd 6826  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-er 7394  df-en 7601  df-dom 7602  df-sdom 7603  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-nn 10643  df-n0 10904  df-z 10972  df-uz 11194  df-fz 11820  df-fzo 11953
This theorem is referenced by:  nnpw2blenfzo2  40762
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