Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fzdifsuc2 Structured version   Visualization version   Unicode version

Theorem fzdifsuc2 37530
Description: Remove a successor from the end of a finite set of sequential integers. Similar to fzdifsuc 11855, but with a weaker condition. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Assertion
Ref Expression
fzdifsuc2  |-  ( N  e.  ( ZZ>= `  ( M  -  1 ) )  ->  ( M ... N )  =  ( ( M ... ( N  +  1 ) )  \  { ( N  +  1 ) } ) )

Proof of Theorem fzdifsuc2
StepHypRef Expression
1 simpr 463 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  N  =  ( M  -  1
) )
2 zre 10941 . . . . . . . 8  |-  ( M  e.  ZZ  ->  M  e.  RR )
32ad2antlr 733 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  M  e.  RR )
43ltm1d 10539 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( M  -  1 )  < 
M )
51, 4eqbrtrd 4423 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  N  <  M )
6 simplr 762 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  M  e.  ZZ )
7 eluzelz 11168 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  ( M  -  1 ) )  ->  N  e.  ZZ )
87ad2antrr 732 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  N  e.  ZZ )
9 fzn 11815 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <  M  <->  ( M ... N )  =  (/) ) )
106, 8, 9syl2anc 667 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( N  <  M  <->  ( M ... N )  =  (/) ) )
115, 10mpbid 214 . . . 4  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( M ... N )  =  (/) )
12 difid 3835 . . . . . 6  |-  ( { M }  \  { M } )  =  (/)
1312a1i 11 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( { M }  \  { M } )  =  (/) )
1413eqcomd 2457 . . . 4  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  (/)  =  ( { M }  \  { M } ) )
15 oveq1 6297 . . . . . . . . 9  |-  ( N  =  ( M  - 
1 )  ->  ( N  +  1 )  =  ( ( M  -  1 )  +  1 ) )
1615adantl 468 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( N  +  1 )  =  ( ( M  - 
1 )  +  1 ) )
172recnd 9669 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
1817ad2antlr 733 . . . . . . . . 9  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  M  e.  CC )
19 1cnd 9659 . . . . . . . . 9  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  1  e.  CC )
2018, 19npcand 9990 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( ( M  -  1 )  +  1 )  =  M )
2116, 20eqtrd 2485 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( N  +  1 )  =  M )
2221oveq2d 6306 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( M ... ( N  +  1 ) )  =  ( M ... M ) )
23 fzsn 11840 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( M ... M )  =  { M } )
2423ad2antlr 733 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( M ... M )  =  { M } )
2522, 24eqtr2d 2486 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  { M }  =  ( M ... ( N  +  1 ) ) )
2621eqcomd 2457 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  M  =  ( N  +  1
) )
2726sneqd 3980 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  { M }  =  { ( N  +  1 ) } )
2825, 27difeq12d 3552 . . . 4  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( { M }  \  { M } )  =  ( ( M ... ( N  +  1 ) )  \  { ( N  +  1 ) } ) )
2911, 14, 283eqtrd 2489 . . 3  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( M ... N )  =  ( ( M ... ( N  +  1 ) )  \  { ( N  +  1 ) } ) )
30 simplr 762 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  M  e.  ZZ )
317ad2antrr 732 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  N  e.  ZZ )
322ad2antlr 733 . . . . . . . . 9  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  M  e.  RR )
33 1red 9658 . . . . . . . . 9  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  1  e.  RR )
3432, 33resubcld 10047 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  ( M  -  1 )  e.  RR )
3531zred 11040 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  N  e.  RR )
36 eluzle 11171 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  ( M  -  1 ) )  ->  ( M  -  1 )  <_  N )
3736ad2antrr 732 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  ( M  -  1 )  <_  N )
38 neqne 37374 . . . . . . . . 9  |-  ( -.  N  =  ( M  -  1 )  ->  N  =/=  ( M  - 
1 ) )
3938adantl 468 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  N  =/=  ( M  -  1
) )
4034, 35, 37, 39leneltd 9789 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  ( M  -  1 )  < 
N )
41 zlem1lt 10988 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  <->  ( M  -  1 )  <  N ) )
4230, 31, 41syl2anc 667 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  ( M  <_  N  <->  ( M  - 
1 )  <  N
) )
4340, 42mpbird 236 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  M  <_  N )
4430, 31, 433jca 1188 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <_  N ) )
45 eluz2 11165 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  <->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <_  N ) )
4644, 45sylibr 216 . . . 4  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  N  e.  ( ZZ>= `  M )
)
47 fzdifsuc 11855 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M ... N )  =  ( ( M ... ( N  +  1 ) )  \  { ( N  +  1 ) } ) )
4846, 47syl 17 . . 3  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  ( M ... N )  =  ( ( M ... ( N  +  1 ) )  \  { ( N  +  1 ) } ) )
4929, 48pm2.61dan 800 . 2  |-  ( ( N  e.  ( ZZ>= `  ( M  -  1
) )  /\  M  e.  ZZ )  ->  ( M ... N )  =  ( ( M ... ( N  +  1
) )  \  {
( N  +  1 ) } ) )
50 eluzel2 11164 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
5150con3i 141 . . . . . 6  |-  ( -.  M  e.  ZZ  ->  -.  N  e.  ( ZZ>= `  M ) )
52 fzn0 11813 . . . . . 6  |-  ( ( M ... N )  =/=  (/)  <->  N  e.  ( ZZ>=
`  M ) )
5351, 52sylnibr 307 . . . . 5  |-  ( -.  M  e.  ZZ  ->  -.  ( M ... N
)  =/=  (/) )
54 nne 2628 . . . . 5  |-  ( -.  ( M ... N
)  =/=  (/)  <->  ( M ... N )  =  (/) )
5553, 54sylib 200 . . . 4  |-  ( -.  M  e.  ZZ  ->  ( M ... N )  =  (/) )
56 eluzel2 11164 . . . . . . . . 9  |-  ( ( N  +  1 )  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
5756con3i 141 . . . . . . . 8  |-  ( -.  M  e.  ZZ  ->  -.  ( N  +  1 )  e.  ( ZZ>= `  M ) )
58 fzn0 11813 . . . . . . . 8  |-  ( ( M ... ( N  +  1 ) )  =/=  (/)  <->  ( N  + 
1 )  e.  (
ZZ>= `  M ) )
5957, 58sylnibr 307 . . . . . . 7  |-  ( -.  M  e.  ZZ  ->  -.  ( M ... ( N  +  1 ) )  =/=  (/) )
60 nne 2628 . . . . . . 7  |-  ( -.  ( M ... ( N  +  1 ) )  =/=  (/)  <->  ( M ... ( N  +  1 ) )  =  (/) )
6159, 60sylib 200 . . . . . 6  |-  ( -.  M  e.  ZZ  ->  ( M ... ( N  +  1 ) )  =  (/) )
6261difeq1d 3550 . . . . 5  |-  ( -.  M  e.  ZZ  ->  ( ( M ... ( N  +  1 ) )  \  { ( N  +  1 ) } )  =  (
(/)  \  { ( N  +  1 ) } ) )
63 0dif 3838 . . . . . 6  |-  ( (/)  \  { ( N  + 
1 ) } )  =  (/)
6463a1i 11 . . . . 5  |-  ( -.  M  e.  ZZ  ->  (
(/)  \  { ( N  +  1 ) } )  =  (/) )
6562, 64eqtr2d 2486 . . . 4  |-  ( -.  M  e.  ZZ  ->  (/)  =  ( ( M ... ( N  + 
1 ) )  \  { ( N  + 
1 ) } ) )
6655, 65eqtrd 2485 . . 3  |-  ( -.  M  e.  ZZ  ->  ( M ... N )  =  ( ( M ... ( N  + 
1 ) )  \  { ( N  + 
1 ) } ) )
6766adantl 468 . 2  |-  ( ( N  e.  ( ZZ>= `  ( M  -  1
) )  /\  -.  M  e.  ZZ )  ->  ( M ... N
)  =  ( ( M ... ( N  +  1 ) ) 
\  { ( N  +  1 ) } ) )
6849, 67pm2.61dan 800 1  |-  ( N  e.  ( ZZ>= `  ( M  -  1 ) )  ->  ( M ... N )  =  ( ( M ... ( N  +  1 ) )  \  { ( N  +  1 ) } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622    \ cdif 3401   (/)c0 3731   {csn 3968   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   1c1 9540    + caddc 9542    < clt 9675    <_ cle 9676    - cmin 9860   ZZcz 10937   ZZ>=cuz 11159   ...cfz 11784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785
This theorem is referenced by:  dvnmul  37818
  Copyright terms: Public domain W3C validator