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Theorem fzsuc2 11630
Description: Join a successor to the end of a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Mar-2014.)
Assertion
Ref Expression
fzsuc2  |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>= `  ( M  -  1
) ) )  -> 
( M ... ( N  +  1 ) )  =  ( ( M ... N )  u.  { ( N  +  1 ) } ) )

Proof of Theorem fzsuc2
StepHypRef Expression
1 uzp1 11004 . 2  |-  ( N  e.  ( ZZ>= `  ( M  -  1 ) )  ->  ( N  =  ( M  - 
1 )  \/  N  e.  ( ZZ>= `  ( ( M  -  1 )  +  1 ) ) ) )
2 zcn 10761 . . . . . . . 8  |-  ( M  e.  ZZ  ->  M  e.  CC )
3 ax-1cn 9450 . . . . . . . 8  |-  1  e.  CC
4 npcan 9729 . . . . . . . 8  |-  ( ( M  e.  CC  /\  1  e.  CC )  ->  ( ( M  - 
1 )  +  1 )  =  M )
52, 3, 4sylancl 662 . . . . . . 7  |-  ( M  e.  ZZ  ->  (
( M  -  1 )  +  1 )  =  M )
65oveq2d 6215 . . . . . 6  |-  ( M  e.  ZZ  ->  ( M ... ( ( M  -  1 )  +  1 ) )  =  ( M ... M
) )
7 uncom 3607 . . . . . . . 8  |-  ( (/)  u. 
{ M } )  =  ( { M }  u.  (/) )
8 un0 3769 . . . . . . . 8  |-  ( { M }  u.  (/) )  =  { M }
97, 8eqtri 2483 . . . . . . 7  |-  ( (/)  u. 
{ M } )  =  { M }
10 zre 10760 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  RR )
1110ltm1d 10375 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  ( M  -  1 )  <  M )
12 peano2zm 10798 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  ( M  -  1 )  e.  ZZ )
13 fzn 11582 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  ( M  -  1
)  e.  ZZ )  ->  ( ( M  -  1 )  < 
M  <->  ( M ... ( M  -  1
) )  =  (/) ) )
1412, 13mpdan 668 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  (
( M  -  1 )  <  M  <->  ( M ... ( M  -  1 ) )  =  (/) ) )
1511, 14mpbid 210 . . . . . . . 8  |-  ( M  e.  ZZ  ->  ( M ... ( M  - 
1 ) )  =  (/) )
165sneqd 3996 . . . . . . . 8  |-  ( M  e.  ZZ  ->  { ( ( M  -  1 )  +  1 ) }  =  { M } )
1715, 16uneq12d 3618 . . . . . . 7  |-  ( M  e.  ZZ  ->  (
( M ... ( M  -  1 ) )  u.  { ( ( M  -  1 )  +  1 ) } )  =  (
(/)  u.  { M } ) )
18 fzsn 11616 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( M ... M )  =  { M } )
199, 17, 183eqtr4a 2521 . . . . . 6  |-  ( M  e.  ZZ  ->  (
( M ... ( M  -  1 ) )  u.  { ( ( M  -  1 )  +  1 ) } )  =  ( M ... M ) )
206, 19eqtr4d 2498 . . . . 5  |-  ( M  e.  ZZ  ->  ( M ... ( ( M  -  1 )  +  1 ) )  =  ( ( M ... ( M  -  1
) )  u.  {
( ( M  - 
1 )  +  1 ) } ) )
21 oveq1 6206 . . . . . . 7  |-  ( N  =  ( M  - 
1 )  ->  ( N  +  1 )  =  ( ( M  -  1 )  +  1 ) )
2221oveq2d 6215 . . . . . 6  |-  ( N  =  ( M  - 
1 )  ->  ( M ... ( N  + 
1 ) )  =  ( M ... (
( M  -  1 )  +  1 ) ) )
23 oveq2 6207 . . . . . . 7  |-  ( N  =  ( M  - 
1 )  ->  ( M ... N )  =  ( M ... ( M  -  1 ) ) )
2421sneqd 3996 . . . . . . 7  |-  ( N  =  ( M  - 
1 )  ->  { ( N  +  1 ) }  =  { ( ( M  -  1 )  +  1 ) } )
2523, 24uneq12d 3618 . . . . . 6  |-  ( N  =  ( M  - 
1 )  ->  (
( M ... N
)  u.  { ( N  +  1 ) } )  =  ( ( M ... ( M  -  1 ) )  u.  { ( ( M  -  1 )  +  1 ) } ) )
2622, 25eqeq12d 2476 . . . . 5  |-  ( N  =  ( M  - 
1 )  ->  (
( M ... ( N  +  1 ) )  =  ( ( M ... N )  u.  { ( N  +  1 ) } )  <->  ( M ... ( ( M  - 
1 )  +  1 ) )  =  ( ( M ... ( M  -  1 ) )  u.  { ( ( M  -  1 )  +  1 ) } ) ) )
2720, 26syl5ibrcom 222 . . . 4  |-  ( M  e.  ZZ  ->  ( N  =  ( M  -  1 )  -> 
( M ... ( N  +  1 ) )  =  ( ( M ... N )  u.  { ( N  +  1 ) } ) ) )
2827imp 429 . . 3  |-  ( ( M  e.  ZZ  /\  N  =  ( M  -  1 ) )  ->  ( M ... ( N  +  1
) )  =  ( ( M ... N
)  u.  { ( N  +  1 ) } ) )
295fveq2d 5802 . . . . . 6  |-  ( M  e.  ZZ  ->  ( ZZ>=
`  ( ( M  -  1 )  +  1 ) )  =  ( ZZ>= `  M )
)
3029eleq2d 2524 . . . . 5  |-  ( M  e.  ZZ  ->  ( N  e.  ( ZZ>= `  ( ( M  - 
1 )  +  1 ) )  <->  N  e.  ( ZZ>= `  M )
) )
3130biimpa 484 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>= `  ( ( M  - 
1 )  +  1 ) ) )  ->  N  e.  ( ZZ>= `  M ) )
32 fzsuc 11618 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M ... ( N  +  1 ) )  =  ( ( M ... N
)  u.  { ( N  +  1 ) } ) )
3331, 32syl 16 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>= `  ( ( M  - 
1 )  +  1 ) ) )  -> 
( M ... ( N  +  1 ) )  =  ( ( M ... N )  u.  { ( N  +  1 ) } ) )
3428, 33jaodan 783 . 2  |-  ( ( M  e.  ZZ  /\  ( N  =  ( M  -  1 )  \/  N  e.  (
ZZ>= `  ( ( M  -  1 )  +  1 ) ) ) )  ->  ( M ... ( N  +  1 ) )  =  ( ( M ... N
)  u.  { ( N  +  1 ) } ) )
351, 34sylan2 474 1  |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>= `  ( M  -  1
) ) )  -> 
( M ... ( N  +  1 ) )  =  ( ( M ... N )  u.  { ( N  +  1 ) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    u. cun 3433   (/)c0 3744   {csn 3984   class class class wbr 4399   ` cfv 5525  (class class class)co 6199   CCcc 9390   1c1 9393    + caddc 9395    < clt 9528    - cmin 9705   ZZcz 10756   ZZ>=cuz 10971   ...cfz 11553
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 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-n0 10690  df-z 10757  df-uz 10972  df-fz 11554
This theorem is referenced by:  fseq1p1m1  11650  fzennn  11906  fsumm1  13337  prmreclem4  14097  ppiprm  22621  ppinprm  22622  chtprm  22623  chtnprm  22624  eupap1  23748  eupath2lem3  23751  fprodm1  27620  mapfzcons  29199
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