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Theorem fzdifsuc2 37618
Description: Remove a successor from the end of a finite set of sequential integers. Similar to fzdifsuc 11881, 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 468 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  N  =  ( M  -  1
) )
2 zre 10965 . . . . . . . 8  |-  ( M  e.  ZZ  ->  M  e.  RR )
32ad2antlr 741 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  M  e.  RR )
43ltm1d 10561 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( M  -  1 )  < 
M )
51, 4eqbrtrd 4416 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  N  <  M )
6 simplr 770 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  M  e.  ZZ )
7 eluzelz 11192 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  ( M  -  1 ) )  ->  N  e.  ZZ )
87ad2antrr 740 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  N  e.  ZZ )
9 fzn 11841 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <  M  <->  ( M ... N )  =  (/) ) )
106, 8, 9syl2anc 673 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( N  <  M  <->  ( M ... N )  =  (/) ) )
115, 10mpbid 215 . . . 4  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( M ... N )  =  (/) )
12 difid 3747 . . . . . 6  |-  ( { M }  \  { M } )  =  (/)
1312a1i 11 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( { M }  \  { M } )  =  (/) )
1413eqcomd 2477 . . . 4  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  (/)  =  ( { M }  \  { M } ) )
15 oveq1 6315 . . . . . . . . 9  |-  ( N  =  ( M  - 
1 )  ->  ( N  +  1 )  =  ( ( M  -  1 )  +  1 ) )
1615adantl 473 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( N  +  1 )  =  ( ( M  - 
1 )  +  1 ) )
172recnd 9687 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
1817ad2antlr 741 . . . . . . . . 9  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  M  e.  CC )
19 1cnd 9677 . . . . . . . . 9  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  1  e.  CC )
2018, 19npcand 10009 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( ( M  -  1 )  +  1 )  =  M )
2116, 20eqtrd 2505 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( N  +  1 )  =  M )
2221oveq2d 6324 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( M ... ( N  +  1 ) )  =  ( M ... M ) )
23 fzsn 11866 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( M ... M )  =  { M } )
2423ad2antlr 741 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( M ... M )  =  { M } )
2522, 24eqtr2d 2506 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  { M }  =  ( M ... ( N  +  1 ) ) )
2621eqcomd 2477 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  M  =  ( N  +  1
) )
2726sneqd 3971 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  { M }  =  { ( N  +  1 ) } )
2825, 27difeq12d 3541 . . . 4  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( { M }  \  { M } )  =  ( ( M ... ( N  +  1 ) )  \  { ( N  +  1 ) } ) )
2911, 14, 283eqtrd 2509 . . 3  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( M ... N )  =  ( ( M ... ( N  +  1 ) )  \  { ( N  +  1 ) } ) )
30 simplr 770 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  M  e.  ZZ )
317ad2antrr 740 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  N  e.  ZZ )
322ad2antlr 741 . . . . . . . . 9  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  M  e.  RR )
33 1red 9676 . . . . . . . . 9  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  1  e.  RR )
3432, 33resubcld 10068 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  ( M  -  1 )  e.  RR )
3531zred 11063 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  N  e.  RR )
36 eluzle 11195 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  ( M  -  1 ) )  ->  ( M  -  1 )  <_  N )
3736ad2antrr 740 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  ( M  -  1 )  <_  N )
38 neqne 2651 . . . . . . . . 9  |-  ( -.  N  =  ( M  -  1 )  ->  N  =/=  ( M  - 
1 ) )
3938adantl 473 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  N  =/=  ( M  -  1
) )
4034, 35, 37, 39leneltd 9806 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  ( M  -  1 )  < 
N )
41 zlem1lt 11012 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  <->  ( M  -  1 )  <  N ) )
4230, 31, 41syl2anc 673 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  ( M  <_  N  <->  ( M  - 
1 )  <  N
) )
4340, 42mpbird 240 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  M  <_  N )
4430, 31, 433jca 1210 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <_  N ) )
45 eluz2 11188 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  <->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <_  N ) )
4644, 45sylibr 217 . . . 4  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  N  e.  ( ZZ>= `  M )
)
47 fzdifsuc 11881 . . . 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 808 . 2  |-  ( ( N  e.  ( ZZ>= `  ( M  -  1
) )  /\  M  e.  ZZ )  ->  ( M ... N )  =  ( ( M ... ( N  +  1
) )  \  {
( N  +  1 ) } ) )
50 eluzel2 11187 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
5150con3i 142 . . . . . 6  |-  ( -.  M  e.  ZZ  ->  -.  N  e.  ( ZZ>= `  M ) )
52 fzn0 11839 . . . . . 6  |-  ( ( M ... N )  =/=  (/)  <->  N  e.  ( ZZ>=
`  M ) )
5351, 52sylnibr 312 . . . . 5  |-  ( -.  M  e.  ZZ  ->  -.  ( M ... N
)  =/=  (/) )
54 nne 2647 . . . . 5  |-  ( -.  ( M ... N
)  =/=  (/)  <->  ( M ... N )  =  (/) )
5553, 54sylib 201 . . . 4  |-  ( -.  M  e.  ZZ  ->  ( M ... N )  =  (/) )
56 eluzel2 11187 . . . . . . . . 9  |-  ( ( N  +  1 )  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
5756con3i 142 . . . . . . . 8  |-  ( -.  M  e.  ZZ  ->  -.  ( N  +  1 )  e.  ( ZZ>= `  M ) )
58 fzn0 11839 . . . . . . . 8  |-  ( ( M ... ( N  +  1 ) )  =/=  (/)  <->  ( N  + 
1 )  e.  (
ZZ>= `  M ) )
5957, 58sylnibr 312 . . . . . . 7  |-  ( -.  M  e.  ZZ  ->  -.  ( M ... ( N  +  1 ) )  =/=  (/) )
60 nne 2647 . . . . . . 7  |-  ( -.  ( M ... ( N  +  1 ) )  =/=  (/)  <->  ( M ... ( N  +  1 ) )  =  (/) )
6159, 60sylib 201 . . . . . 6  |-  ( -.  M  e.  ZZ  ->  ( M ... ( N  +  1 ) )  =  (/) )
6261difeq1d 3539 . . . . 5  |-  ( -.  M  e.  ZZ  ->  ( ( M ... ( N  +  1 ) )  \  { ( N  +  1 ) } )  =  (
(/)  \  { ( N  +  1 ) } ) )
63 0dif 3771 . . . . . 6  |-  ( (/)  \  { ( N  + 
1 ) } )  =  (/)
6463a1i 11 . . . . 5  |-  ( -.  M  e.  ZZ  ->  (
(/)  \  { ( N  +  1 ) } )  =  (/) )
6562, 64eqtr2d 2506 . . . 4  |-  ( -.  M  e.  ZZ  ->  (/)  =  ( ( M ... ( N  + 
1 ) )  \  { ( N  + 
1 ) } ) )
6655, 65eqtrd 2505 . . 3  |-  ( -.  M  e.  ZZ  ->  ( M ... N )  =  ( ( M ... ( N  + 
1 ) )  \  { ( N  + 
1 ) } ) )
6766adantl 473 . 2  |-  ( ( N  e.  ( ZZ>= `  ( M  -  1
) )  /\  -.  M  e.  ZZ )  ->  ( M ... N
)  =  ( ( M ... ( N  +  1 ) ) 
\  { ( N  +  1 ) } ) )
6849, 67pm2.61dan 808 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641    \ cdif 3387   (/)c0 3722   {csn 3959   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   1c1 9558    + caddc 9560    < clt 9693    <_ cle 9694    - cmin 9880   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811
This theorem is referenced by:  dvnmul  37915
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