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Theorem fzsuc2 9493
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 9052 . 2  |-  ( N  e.  ( ZZ>= `  ( M  -  1 ) )  ->  ( N  =  ( M  - 
1 )  \/  N  e.  ( ZZ>= `  ( ( M  -  1 )  +  1 ) ) ) )
2 zcn 8755 . . . . . . . 8  |-  ( M  e.  ZZ  ->  M  e.  CC )
3 ax-1cn 7438 . . . . . . . 8  |-  1  e.  CC
4 npcan 7691 . . . . . . . 8  |-  ( ( M  e.  CC  /\  1  e.  CC )  ->  ( ( M  - 
1 )  +  1 )  =  M )
52, 3, 4sylancl 404 . . . . . . 7  |-  ( M  e.  ZZ  ->  (
( M  -  1 )  +  1 )  =  M )
65oveq2d 5668 . . . . . 6  |-  ( M  e.  ZZ  ->  ( M ... ( ( M  -  1 )  +  1 ) )  =  ( M ... M
) )
7 uncom 3144 . . . . . . . 8  |-  ( (/)  u. 
{ M } )  =  ( { M }  u.  (/) )
8 un0 3316 . . . . . . . 8  |-  ( { M }  u.  (/) )  =  { M }
97, 8eqtri 2108 . . . . . . 7  |-  ( (/)  u. 
{ M } )  =  { M }
10 zre 8754 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  RR )
1110ltm1d 8393 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  ( M  -  1 )  <  M )
12 peano2zm 8788 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  ( M  -  1 )  e.  ZZ )
13 fzn 9456 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  ( M  -  1
)  e.  ZZ )  ->  ( ( M  -  1 )  < 
M  <->  ( M ... ( M  -  1
) )  =  (/) ) )
1412, 13mpdan 412 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  (
( M  -  1 )  <  M  <->  ( M ... ( M  -  1 ) )  =  (/) ) )
1511, 14mpbid 145 . . . . . . . 8  |-  ( M  e.  ZZ  ->  ( M ... ( M  - 
1 ) )  =  (/) )
165sneqd 3459 . . . . . . . 8  |-  ( M  e.  ZZ  ->  { ( ( M  -  1 )  +  1 ) }  =  { M } )
1715, 16uneq12d 3155 . . . . . . 7  |-  ( M  e.  ZZ  ->  (
( M ... ( M  -  1 ) )  u.  { ( ( M  -  1 )  +  1 ) } )  =  (
(/)  u.  { M } ) )
18 fzsn 9480 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( M ... M )  =  { M } )
199, 17, 183eqtr4a 2146 . . . . . 6  |-  ( M  e.  ZZ  ->  (
( M ... ( M  -  1 ) )  u.  { ( ( M  -  1 )  +  1 ) } )  =  ( M ... M ) )
206, 19eqtr4d 2123 . . . . 5  |-  ( M  e.  ZZ  ->  ( M ... ( ( M  -  1 )  +  1 ) )  =  ( ( M ... ( M  -  1
) )  u.  {
( ( M  - 
1 )  +  1 ) } ) )
21 oveq1 5659 . . . . . . 7  |-  ( N  =  ( M  - 
1 )  ->  ( N  +  1 )  =  ( ( M  -  1 )  +  1 ) )
2221oveq2d 5668 . . . . . 6  |-  ( N  =  ( M  - 
1 )  ->  ( M ... ( N  + 
1 ) )  =  ( M ... (
( M  -  1 )  +  1 ) ) )
23 oveq2 5660 . . . . . . 7  |-  ( N  =  ( M  - 
1 )  ->  ( M ... N )  =  ( M ... ( M  -  1 ) ) )
2421sneqd 3459 . . . . . . 7  |-  ( N  =  ( M  - 
1 )  ->  { ( N  +  1 ) }  =  { ( ( M  -  1 )  +  1 ) } )
2523, 24uneq12d 3155 . . . . . 6  |-  ( N  =  ( M  - 
1 )  ->  (
( M ... N
)  u.  { ( N  +  1 ) } )  =  ( ( M ... ( M  -  1 ) )  u.  { ( ( M  -  1 )  +  1 ) } ) )
2622, 25eqeq12d 2102 . . . . 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 155 . . . 4  |-  ( M  e.  ZZ  ->  ( N  =  ( M  -  1 )  -> 
( M ... ( N  +  1 ) )  =  ( ( M ... N )  u.  { ( N  +  1 ) } ) ) )
2827imp 122 . . 3  |-  ( ( M  e.  ZZ  /\  N  =  ( M  -  1 ) )  ->  ( M ... ( N  +  1
) )  =  ( ( M ... N
)  u.  { ( N  +  1 ) } ) )
295fveq2d 5309 . . . . . 6  |-  ( M  e.  ZZ  ->  ( ZZ>=
`  ( ( M  -  1 )  +  1 ) )  =  ( ZZ>= `  M )
)
3029eleq2d 2157 . . . . 5  |-  ( M  e.  ZZ  ->  ( N  e.  ( ZZ>= `  ( ( M  - 
1 )  +  1 ) )  <->  N  e.  ( ZZ>= `  M )
) )
3130biimpa 290 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>= `  ( ( M  - 
1 )  +  1 ) ) )  ->  N  e.  ( ZZ>= `  M ) )
32 fzsuc 9483 . . . 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 746 . 2  |-  ( ( M  e.  ZZ  /\  ( N  =  ( M  -  1 )  \/  N  e.  (
ZZ>= `  ( ( M  -  1 )  +  1 ) ) ) )  ->  ( M ... ( N  +  1 ) )  =  ( ( M ... N
)  u.  { ( N  +  1 ) } ) )
351, 34sylan2 280 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 102    <-> wb 103    \/ wo 664    = wceq 1289    e. wcel 1438    u. cun 2997   (/)c0 3286   {csn 3446   class class class wbr 3845   ` cfv 5015  (class class class)co 5652   CCcc 7348   1c1 7351    + caddc 7353    < clt 7522    - cmin 7653   ZZcz 8750   ZZ>=cuz 9019   ...cfz 9424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-addass 7447  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-0id 7453  ax-rnegex 7454  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-inn 8423  df-n0 8674  df-z 8751  df-uz 9020  df-fz 9425
This theorem is referenced by:  fseq1p1m1  9508  frecfzennn  9833  zfz1isolemsplit  10243  fsumm1  10810
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