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Theorem fzdifsuc2 37416
Description: Remove a successor from the end of a finite set of sequential integers. Similar to fzdifsuc 11857, 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 10943 . . . . . . . 8  |-  ( M  e.  ZZ  ->  M  e.  RR )
32ad2antlr 732 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  M  e.  RR )
43ltm1d 10541 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( M  -  1 )  < 
M )
51, 4eqbrtrd 4442 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  N  <  M )
6 simplr 761 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  M  e.  ZZ )
7 eluzelz 11170 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  ( M  -  1 ) )  ->  N  e.  ZZ )
87ad2antrr 731 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  N  e.  ZZ )
9 fzn 11817 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <  M  <->  ( M ... N )  =  (/) ) )
106, 8, 9syl2anc 666 . . . . 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 3864 . . . . . 6  |-  ( { M }  \  { M } )  =  (/)
1312a1i 11 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( { M }  \  { M } )  =  (/) )
1413eqcomd 2431 . . . 4  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  (/)  =  ( { M }  \  { M } ) )
15 oveq1 6310 . . . . . . . . 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 9671 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
1817ad2antlr 732 . . . . . . . . 9  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  M  e.  CC )
19 1cnd 9661 . . . . . . . . 9  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  1  e.  CC )
2018, 19npcand 9992 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( ( M  -  1 )  +  1 )  =  M )
2116, 20eqtrd 2464 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( N  +  1 )  =  M )
2221oveq2d 6319 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( M ... ( N  +  1 ) )  =  ( M ... M ) )
23 fzsn 11842 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( M ... M )  =  { M } )
2423ad2antlr 732 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( M ... M )  =  { M } )
2522, 24eqtr2d 2465 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  { M }  =  ( M ... ( N  +  1 ) ) )
2621eqcomd 2431 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  M  =  ( N  +  1
) )
2726sneqd 4009 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  { M }  =  { ( N  +  1 ) } )
2825, 27difeq12d 3585 . . . 4  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( { M }  \  { M } )  =  ( ( M ... ( N  +  1 ) )  \  { ( N  +  1 ) } ) )
2911, 14, 283eqtrd 2468 . . 3  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( M ... N )  =  ( ( M ... ( N  +  1 ) )  \  { ( N  +  1 ) } ) )
30 simplr 761 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  M  e.  ZZ )
317ad2antrr 731 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  N  e.  ZZ )
322ad2antlr 732 . . . . . . . . 9  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  M  e.  RR )
33 1red 9660 . . . . . . . . 9  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  1  e.  RR )
3432, 33resubcld 10049 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  ( M  -  1 )  e.  RR )
3531zred 11042 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  N  e.  RR )
36 eluzle 11173 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  ( M  -  1 ) )  ->  ( M  -  1 )  <_  N )
3736ad2antrr 731 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  ( M  -  1 )  <_  N )
38 neqne 37279 . . . . . . . . 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 9791 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  ( M  -  1 )  < 
N )
41 zlem1lt 10990 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  <->  ( M  -  1 )  <  N ) )
4230, 31, 41syl2anc 666 . . . . . . 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 1186 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <_  N ) )
45 eluz2 11167 . . . . 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 11857 . . . 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 799 . 2  |-  ( ( N  e.  ( ZZ>= `  ( M  -  1
) )  /\  M  e.  ZZ )  ->  ( M ... N )  =  ( ( M ... ( N  +  1
) )  \  {
( N  +  1 ) } ) )
50 eluzel2 11166 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
5150con3i 141 . . . . . 6  |-  ( -.  M  e.  ZZ  ->  -.  N  e.  ( ZZ>= `  M ) )
52 fzn0 11815 . . . . . 6  |-  ( ( M ... N )  =/=  (/)  <->  N  e.  ( ZZ>=
`  M ) )
5351, 52sylnibr 307 . . . . 5  |-  ( -.  M  e.  ZZ  ->  -.  ( M ... N
)  =/=  (/) )
54 nne 2625 . . . . 5  |-  ( -.  ( M ... N
)  =/=  (/)  <->  ( M ... N )  =  (/) )
5553, 54sylib 200 . . . 4  |-  ( -.  M  e.  ZZ  ->  ( M ... N )  =  (/) )
56 eluzel2 11166 . . . . . . . . 9  |-  ( ( N  +  1 )  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
5756con3i 141 . . . . . . . 8  |-  ( -.  M  e.  ZZ  ->  -.  ( N  +  1 )  e.  ( ZZ>= `  M ) )
58 fzn0 11815 . . . . . . . 8  |-  ( ( M ... ( N  +  1 ) )  =/=  (/)  <->  ( N  + 
1 )  e.  (
ZZ>= `  M ) )
5957, 58sylnibr 307 . . . . . . 7  |-  ( -.  M  e.  ZZ  ->  -.  ( M ... ( N  +  1 ) )  =/=  (/) )
60 nne 2625 . . . . . . 7  |-  ( -.  ( M ... ( N  +  1 ) )  =/=  (/)  <->  ( M ... ( N  +  1 ) )  =  (/) )
6159, 60sylib 200 . . . . . 6  |-  ( -.  M  e.  ZZ  ->  ( M ... ( N  +  1 ) )  =  (/) )
6261difeq1d 3583 . . . . 5  |-  ( -.  M  e.  ZZ  ->  ( ( M ... ( N  +  1 ) )  \  { ( N  +  1 ) } )  =  (
(/)  \  { ( N  +  1 ) } ) )
63 0dif 3867 . . . . . 6  |-  ( (/)  \  { ( N  + 
1 ) } )  =  (/)
6463a1i 11 . . . . 5  |-  ( -.  M  e.  ZZ  ->  (
(/)  \  { ( N  +  1 ) } )  =  (/) )
6562, 64eqtr2d 2465 . . . 4  |-  ( -.  M  e.  ZZ  ->  (/)  =  ( ( M ... ( N  + 
1 ) )  \  { ( N  + 
1 ) } ) )
6655, 65eqtrd 2464 . . 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 799 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 983    = wceq 1438    e. wcel 1869    =/= wne 2619    \ cdif 3434   (/)c0 3762   {csn 3997   class class class wbr 4421   ` cfv 5599  (class class class)co 6303   CCcc 9539   RRcr 9540   1c1 9542    + caddc 9544    < clt 9677    <_ cle 9678    - cmin 9862   ZZcz 10939   ZZ>=cuz 11161   ...cfz 11786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-nn 10612  df-n0 10872  df-z 10940  df-uz 11162  df-fz 11787
This theorem is referenced by:  dvnmul  37682
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