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Theorem fzsuc2 9871
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 9371 . 2  |-  ( N  e.  ( ZZ>= `  ( M  -  1 ) )  ->  ( N  =  ( M  - 
1 )  \/  N  e.  ( ZZ>= `  ( ( M  -  1 )  +  1 ) ) ) )
2 zcn 9071 . . . . . . . 8  |-  ( M  e.  ZZ  ->  M  e.  CC )
3 ax-1cn 7725 . . . . . . . 8  |-  1  e.  CC
4 npcan 7983 . . . . . . . 8  |-  ( ( M  e.  CC  /\  1  e.  CC )  ->  ( ( M  - 
1 )  +  1 )  =  M )
52, 3, 4sylancl 409 . . . . . . 7  |-  ( M  e.  ZZ  ->  (
( M  -  1 )  +  1 )  =  M )
65oveq2d 5790 . . . . . 6  |-  ( M  e.  ZZ  ->  ( M ... ( ( M  -  1 )  +  1 ) )  =  ( M ... M
) )
7 uncom 3220 . . . . . . . 8  |-  ( (/)  u. 
{ M } )  =  ( { M }  u.  (/) )
8 un0 3396 . . . . . . . 8  |-  ( { M }  u.  (/) )  =  { M }
97, 8eqtri 2160 . . . . . . 7  |-  ( (/)  u. 
{ M } )  =  { M }
10 zre 9070 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  RR )
1110ltm1d 8702 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  ( M  -  1 )  <  M )
12 peano2zm 9104 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  ( M  -  1 )  e.  ZZ )
13 fzn 9834 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  ( M  -  1
)  e.  ZZ )  ->  ( ( M  -  1 )  < 
M  <->  ( M ... ( M  -  1
) )  =  (/) ) )
1412, 13mpdan 417 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  (
( M  -  1 )  <  M  <->  ( M ... ( M  -  1 ) )  =  (/) ) )
1511, 14mpbid 146 . . . . . . . 8  |-  ( M  e.  ZZ  ->  ( M ... ( M  - 
1 ) )  =  (/) )
165sneqd 3540 . . . . . . . 8  |-  ( M  e.  ZZ  ->  { ( ( M  -  1 )  +  1 ) }  =  { M } )
1715, 16uneq12d 3231 . . . . . . 7  |-  ( M  e.  ZZ  ->  (
( M ... ( M  -  1 ) )  u.  { ( ( M  -  1 )  +  1 ) } )  =  (
(/)  u.  { M } ) )
18 fzsn 9858 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( M ... M )  =  { M } )
199, 17, 183eqtr4a 2198 . . . . . 6  |-  ( M  e.  ZZ  ->  (
( M ... ( M  -  1 ) )  u.  { ( ( M  -  1 )  +  1 ) } )  =  ( M ... M ) )
206, 19eqtr4d 2175 . . . . 5  |-  ( M  e.  ZZ  ->  ( M ... ( ( M  -  1 )  +  1 ) )  =  ( ( M ... ( M  -  1
) )  u.  {
( ( M  - 
1 )  +  1 ) } ) )
21 oveq1 5781 . . . . . . 7  |-  ( N  =  ( M  - 
1 )  ->  ( N  +  1 )  =  ( ( M  -  1 )  +  1 ) )
2221oveq2d 5790 . . . . . 6  |-  ( N  =  ( M  - 
1 )  ->  ( M ... ( N  + 
1 ) )  =  ( M ... (
( M  -  1 )  +  1 ) ) )
23 oveq2 5782 . . . . . . 7  |-  ( N  =  ( M  - 
1 )  ->  ( M ... N )  =  ( M ... ( M  -  1 ) ) )
2421sneqd 3540 . . . . . . 7  |-  ( N  =  ( M  - 
1 )  ->  { ( N  +  1 ) }  =  { ( ( M  -  1 )  +  1 ) } )
2523, 24uneq12d 3231 . . . . . 6  |-  ( N  =  ( M  - 
1 )  ->  (
( M ... N
)  u.  { ( N  +  1 ) } )  =  ( ( M ... ( M  -  1 ) )  u.  { ( ( M  -  1 )  +  1 ) } ) )
2622, 25eqeq12d 2154 . . . . 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 156 . . . 4  |-  ( M  e.  ZZ  ->  ( N  =  ( M  -  1 )  -> 
( M ... ( N  +  1 ) )  =  ( ( M ... N )  u.  { ( N  +  1 ) } ) ) )
2827imp 123 . . 3  |-  ( ( M  e.  ZZ  /\  N  =  ( M  -  1 ) )  ->  ( M ... ( N  +  1
) )  =  ( ( M ... N
)  u.  { ( N  +  1 ) } ) )
295fveq2d 5425 . . . . . 6  |-  ( M  e.  ZZ  ->  ( ZZ>=
`  ( ( M  -  1 )  +  1 ) )  =  ( ZZ>= `  M )
)
3029eleq2d 2209 . . . . 5  |-  ( M  e.  ZZ  ->  ( N  e.  ( ZZ>= `  ( ( M  - 
1 )  +  1 ) )  <->  N  e.  ( ZZ>= `  M )
) )
3130biimpa 294 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>= `  ( ( M  - 
1 )  +  1 ) ) )  ->  N  e.  ( ZZ>= `  M ) )
32 fzsuc 9861 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M ... ( N  +  1 ) )  =  ( ( M ... N
)  u.  { ( N  +  1 ) } ) )
3331, 32syl 14 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>= `  ( ( M  - 
1 )  +  1 ) ) )  -> 
( M ... ( N  +  1 ) )  =  ( ( M ... N )  u.  { ( N  +  1 ) } ) )
3428, 33jaodan 786 . 2  |-  ( ( M  e.  ZZ  /\  ( N  =  ( M  -  1 )  \/  N  e.  (
ZZ>= `  ( ( M  -  1 )  +  1 ) ) ) )  ->  ( M ... ( N  +  1 ) )  =  ( ( M ... N
)  u.  { ( N  +  1 ) } ) )
351, 34sylan2 284 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    /\ wa 103    <-> wb 104    \/ wo 697    = wceq 1331    e. wcel 1480    u. cun 3069   (/)c0 3363   {csn 3527   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7630   1c1 7633    + caddc 7635    < clt 7812    - cmin 7945   ZZcz 9066   ZZ>=cuz 9338   ...cfz 9802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-addass 7734  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-inn 8733  df-n0 8990  df-z 9067  df-uz 9339  df-fz 9803
This theorem is referenced by:  fseq1p1m1  9886  frecfzennn  10211  zfz1isolemsplit  10593  fsumm1  11197
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