MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fzsuc2 Structured version   Unicode version

Theorem fzsuc2 11737
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 11115 . 2  |-  ( N  e.  ( ZZ>= `  ( M  -  1 ) )  ->  ( N  =  ( M  - 
1 )  \/  N  e.  ( ZZ>= `  ( ( M  -  1 )  +  1 ) ) ) )
2 zcn 10869 . . . . . . . 8  |-  ( M  e.  ZZ  ->  M  e.  CC )
3 ax-1cn 9550 . . . . . . . 8  |-  1  e.  CC
4 npcan 9829 . . . . . . . 8  |-  ( ( M  e.  CC  /\  1  e.  CC )  ->  ( ( M  - 
1 )  +  1 )  =  M )
52, 3, 4sylancl 662 . . . . . . 7  |-  ( M  e.  ZZ  ->  (
( M  -  1 )  +  1 )  =  M )
65oveq2d 6300 . . . . . 6  |-  ( M  e.  ZZ  ->  ( M ... ( ( M  -  1 )  +  1 ) )  =  ( M ... M
) )
7 uncom 3648 . . . . . . . 8  |-  ( (/)  u. 
{ M } )  =  ( { M }  u.  (/) )
8 un0 3810 . . . . . . . 8  |-  ( { M }  u.  (/) )  =  { M }
97, 8eqtri 2496 . . . . . . 7  |-  ( (/)  u. 
{ M } )  =  { M }
10 zre 10868 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  RR )
1110ltm1d 10478 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  ( M  -  1 )  <  M )
12 peano2zm 10906 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  ( M  -  1 )  e.  ZZ )
13 fzn 11702 . . . . . . . . . 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 4039 . . . . . . . 8  |-  ( M  e.  ZZ  ->  { ( ( M  -  1 )  +  1 ) }  =  { M } )
1715, 16uneq12d 3659 . . . . . . 7  |-  ( M  e.  ZZ  ->  (
( M ... ( M  -  1 ) )  u.  { ( ( M  -  1 )  +  1 ) } )  =  (
(/)  u.  { M } ) )
18 fzsn 11725 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( M ... M )  =  { M } )
199, 17, 183eqtr4a 2534 . . . . . 6  |-  ( M  e.  ZZ  ->  (
( M ... ( M  -  1 ) )  u.  { ( ( M  -  1 )  +  1 ) } )  =  ( M ... M ) )
206, 19eqtr4d 2511 . . . . 5  |-  ( M  e.  ZZ  ->  ( M ... ( ( M  -  1 )  +  1 ) )  =  ( ( M ... ( M  -  1
) )  u.  {
( ( M  - 
1 )  +  1 ) } ) )
21 oveq1 6291 . . . . . . 7  |-  ( N  =  ( M  - 
1 )  ->  ( N  +  1 )  =  ( ( M  -  1 )  +  1 ) )
2221oveq2d 6300 . . . . . 6  |-  ( N  =  ( M  - 
1 )  ->  ( M ... ( N  + 
1 ) )  =  ( M ... (
( M  -  1 )  +  1 ) ) )
23 oveq2 6292 . . . . . . 7  |-  ( N  =  ( M  - 
1 )  ->  ( M ... N )  =  ( M ... ( M  -  1 ) ) )
2421sneqd 4039 . . . . . . 7  |-  ( N  =  ( M  - 
1 )  ->  { ( N  +  1 ) }  =  { ( ( M  -  1 )  +  1 ) } )
2523, 24uneq12d 3659 . . . . . 6  |-  ( N  =  ( M  - 
1 )  ->  (
( M ... N
)  u.  { ( N  +  1 ) } )  =  ( ( M ... ( M  -  1 ) )  u.  { ( ( M  -  1 )  +  1 ) } ) )
2622, 25eqeq12d 2489 . . . . 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 5870 . . . . . 6  |-  ( M  e.  ZZ  ->  ( ZZ>=
`  ( ( M  -  1 )  +  1 ) )  =  ( ZZ>= `  M )
)
3029eleq2d 2537 . . . . 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 11727 . . . 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 1379    e. wcel 1767    u. cun 3474   (/)c0 3785   {csn 4027   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   CCcc 9490   1c1 9493    + caddc 9495    < clt 9628    - cmin 9805   ZZcz 10864   ZZ>=cuz 11082   ...cfz 11672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673
This theorem is referenced by:  fseq1p1m1  11752  fzennn  12046  fsumm1  13529  prmreclem4  14296  ppiprm  23181  ppinprm  23182  chtprm  23183  chtnprm  23184  eupap1  24680  eupath2lem3  24683  fprodm1  28701  mapfzcons  30280
  Copyright terms: Public domain W3C validator