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Theorem fzsuc2 10842
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 10261 . 2  |-  ( N  e.  ( ZZ>= `  ( M  -  1 ) )  ->  ( N  =  ( M  - 
1 )  \/  N  e.  ( ZZ>= `  ( ( M  -  1 )  +  1 ) ) ) )
2 zcn 10029 . . . . . . . 8  |-  ( M  e.  ZZ  ->  M  e.  CC )
3 ax-1cn 8795 . . . . . . . 8  |-  1  e.  CC
4 npcan 9060 . . . . . . . 8  |-  ( ( M  e.  CC  /\  1  e.  CC )  ->  ( ( M  - 
1 )  +  1 )  =  M )
52, 3, 4sylancl 643 . . . . . . 7  |-  ( M  e.  ZZ  ->  (
( M  -  1 )  +  1 )  =  M )
65oveq2d 5874 . . . . . 6  |-  ( M  e.  ZZ  ->  ( M ... ( ( M  -  1 )  +  1 ) )  =  ( M ... M
) )
7 uncom 3319 . . . . . . . 8  |-  ( (/)  u. 
{ M } )  =  ( { M }  u.  (/) )
8 un0 3479 . . . . . . . 8  |-  ( { M }  u.  (/) )  =  { M }
97, 8eqtri 2303 . . . . . . 7  |-  ( (/)  u. 
{ M } )  =  { M }
10 zre 10028 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  RR )
1110ltm1d 9689 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  ( M  -  1 )  <  M )
12 peano2zm 10062 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  ( M  -  1 )  e.  ZZ )
13 fzn 10810 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  ( M  -  1
)  e.  ZZ )  ->  ( ( M  -  1 )  < 
M  <->  ( M ... ( M  -  1
) )  =  (/) ) )
1412, 13mpdan 649 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  (
( M  -  1 )  <  M  <->  ( M ... ( M  -  1 ) )  =  (/) ) )
1511, 14mpbid 201 . . . . . . . 8  |-  ( M  e.  ZZ  ->  ( M ... ( M  - 
1 ) )  =  (/) )
165sneqd 3653 . . . . . . . 8  |-  ( M  e.  ZZ  ->  { ( ( M  -  1 )  +  1 ) }  =  { M } )
1715, 16uneq12d 3330 . . . . . . 7  |-  ( M  e.  ZZ  ->  (
( M ... ( M  -  1 ) )  u.  { ( ( M  -  1 )  +  1 ) } )  =  (
(/)  u.  { M } ) )
18 fzsn 10833 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( M ... M )  =  { M } )
199, 17, 183eqtr4a 2341 . . . . . 6  |-  ( M  e.  ZZ  ->  (
( M ... ( M  -  1 ) )  u.  { ( ( M  -  1 )  +  1 ) } )  =  ( M ... M ) )
206, 19eqtr4d 2318 . . . . 5  |-  ( M  e.  ZZ  ->  ( M ... ( ( M  -  1 )  +  1 ) )  =  ( ( M ... ( M  -  1
) )  u.  {
( ( M  - 
1 )  +  1 ) } ) )
21 oveq1 5865 . . . . . . 7  |-  ( N  =  ( M  - 
1 )  ->  ( N  +  1 )  =  ( ( M  -  1 )  +  1 ) )
2221oveq2d 5874 . . . . . 6  |-  ( N  =  ( M  - 
1 )  ->  ( M ... ( N  + 
1 ) )  =  ( M ... (
( M  -  1 )  +  1 ) ) )
23 oveq2 5866 . . . . . . 7  |-  ( N  =  ( M  - 
1 )  ->  ( M ... N )  =  ( M ... ( M  -  1 ) ) )
2421sneqd 3653 . . . . . . 7  |-  ( N  =  ( M  - 
1 )  ->  { ( N  +  1 ) }  =  { ( ( M  -  1 )  +  1 ) } )
2523, 24uneq12d 3330 . . . . . 6  |-  ( N  =  ( M  - 
1 )  ->  (
( M ... N
)  u.  { ( N  +  1 ) } )  =  ( ( M ... ( M  -  1 ) )  u.  { ( ( M  -  1 )  +  1 ) } ) )
2622, 25eqeq12d 2297 . . . . 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 213 . . . 4  |-  ( M  e.  ZZ  ->  ( N  =  ( M  -  1 )  -> 
( M ... ( N  +  1 ) )  =  ( ( M ... N )  u.  { ( N  +  1 ) } ) ) )
2827imp 418 . . 3  |-  ( ( M  e.  ZZ  /\  N  =  ( M  -  1 ) )  ->  ( M ... ( N  +  1
) )  =  ( ( M ... N
)  u.  { ( N  +  1 ) } ) )
295fveq2d 5529 . . . . . 6  |-  ( M  e.  ZZ  ->  ( ZZ>=
`  ( ( M  -  1 )  +  1 ) )  =  ( ZZ>= `  M )
)
3029eleq2d 2350 . . . . 5  |-  ( M  e.  ZZ  ->  ( N  e.  ( ZZ>= `  ( ( M  - 
1 )  +  1 ) )  <->  N  e.  ( ZZ>= `  M )
) )
3130biimpa 470 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>= `  ( ( M  - 
1 )  +  1 ) ) )  ->  N  e.  ( ZZ>= `  M ) )
32 fzsuc 10835 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M ... ( N  +  1 ) )  =  ( ( M ... N
)  u.  { ( N  +  1 ) } ) )
3331, 32syl 15 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>= `  ( ( M  - 
1 )  +  1 ) ) )  -> 
( M ... ( N  +  1 ) )  =  ( ( M ... N )  u.  { ( N  +  1 ) } ) )
3428, 33jaodan 760 . 2  |-  ( ( M  e.  ZZ  /\  ( N  =  ( M  -  1 )  \/  N  e.  (
ZZ>= `  ( ( M  -  1 )  +  1 ) ) ) )  ->  ( M ... ( N  +  1 ) )  =  ( ( M ... N
)  u.  { ( N  +  1 ) } ) )
351, 34sylan2 460 1  |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>= `  ( M  -  1
) ) )  -> 
( M ... ( N  +  1 ) )  =  ( ( M ... N )  u.  { ( N  +  1 ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    u. cun 3150   (/)c0 3455   {csn 3640   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738    + caddc 8740    < clt 8867    - cmin 9037   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782
This theorem is referenced by:  fseq1p1m1  10857  fzennn  11030  fsumm1  12216  prmreclem4  12966  ppiprm  20389  ppinprm  20390  chtprm  20391  chtnprm  20392  eupap1  23900  eupath2lem3  23903  mapfzcons  26793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783
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