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Theorem fzdifsuc2 31717
Description: Remove a successor from the end of a finite set of sequential integers. Similar to fzdifsuc 11683, 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 459 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  N  =  ( M  -  1
) )
2 zre 10807 . . . . . . . 8  |-  ( M  e.  ZZ  ->  M  e.  RR )
32ad2antlr 724 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  M  e.  RR )
43ltm1d 10416 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( M  -  1 )  < 
M )
51, 4eqbrtrd 4404 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  N  <  M )
6 simplr 753 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  M  e.  ZZ )
7 eluzelz 11032 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  ( M  -  1 ) )  ->  N  e.  ZZ )
87ad2antrr 723 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  N  e.  ZZ )
9 fzn 11645 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <  M  <->  ( M ... N )  =  (/) ) )
106, 8, 9syl2anc 659 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( N  <  M  <->  ( M ... N )  =  (/) ) )
115, 10mpbid 210 . . . 4  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( M ... N )  =  (/) )
12 difid 3829 . . . . . 6  |-  ( { M }  \  { M } )  =  (/)
1312a1i 11 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( { M }  \  { M } )  =  (/) )
1413eqcomd 2404 . . . 4  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  (/)  =  ( { M }  \  { M } ) )
15 oveq1 6225 . . . . . . . . 9  |-  ( N  =  ( M  - 
1 )  ->  ( N  +  1 )  =  ( ( M  -  1 )  +  1 ) )
1615adantl 464 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( N  +  1 )  =  ( ( M  - 
1 )  +  1 ) )
172recnd 9555 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
1817ad2antlr 724 . . . . . . . . 9  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  M  e.  CC )
19 1cnd 9545 . . . . . . . . 9  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  1  e.  CC )
2018, 19npcand 9870 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( ( M  -  1 )  +  1 )  =  M )
2116, 20eqtrd 2437 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( N  +  1 )  =  M )
2221oveq2d 6234 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( M ... ( N  +  1 ) )  =  ( M ... M ) )
23 fzsn 11669 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( M ... M )  =  { M } )
2423ad2antlr 724 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( M ... M )  =  { M } )
2522, 24eqtr2d 2438 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  { M }  =  ( M ... ( N  +  1 ) ) )
2621eqcomd 2404 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  M  =  ( N  +  1
) )
2726sneqd 3973 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  { M }  =  { ( N  +  1 ) } )
2825, 27difeq12d 3554 . . . 4  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( { M }  \  { M } )  =  ( ( M ... ( N  +  1 ) )  \  { ( N  +  1 ) } ) )
2911, 14, 283eqtrd 2441 . . 3  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  N  =  ( M  -  1 ) )  ->  ( M ... N )  =  ( ( M ... ( N  +  1 ) )  \  { ( N  +  1 ) } ) )
30 simplr 753 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  M  e.  ZZ )
317ad2antrr 723 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  N  e.  ZZ )
322ad2antlr 724 . . . . . . . . 9  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  M  e.  RR )
33 1red 9544 . . . . . . . . 9  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  1  e.  RR )
3432, 33resubcld 9927 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  ( M  -  1 )  e.  RR )
3531zred 10906 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  N  e.  RR )
36 eluzle 11035 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  ( M  -  1 ) )  ->  ( M  -  1 )  <_  N )
3736ad2antrr 723 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  ( M  -  1 )  <_  N )
38 neqne 31641 . . . . . . . . 9  |-  ( -.  N  =  ( M  -  1 )  ->  N  =/=  ( M  - 
1 ) )
3938adantl 464 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  N  =/=  ( M  -  1
) )
4034, 35, 37, 39leneltd 31699 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  ( M  -  1 )  < 
N )
41 zlem1lt 10854 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  <->  ( M  -  1 )  <  N ) )
4230, 31, 41syl2anc 659 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  ( M  <_  N  <->  ( M  - 
1 )  <  N
) )
4340, 42mpbird 232 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  M  <_  N )
4430, 31, 433jca 1174 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <_  N ) )
45 eluz2 11029 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  <->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <_  N ) )
4644, 45sylibr 212 . . . 4  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  N  e.  ( ZZ>= `  M )
)
47 fzdifsuc 11683 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M ... N )  =  ( ( M ... ( N  +  1 ) )  \  { ( N  +  1 ) } ) )
4846, 47syl 16 . . 3  |-  ( ( ( N  e.  (
ZZ>= `  ( M  - 
1 ) )  /\  M  e.  ZZ )  /\  -.  N  =  ( M  -  1 ) )  ->  ( M ... N )  =  ( ( M ... ( N  +  1 ) )  \  { ( N  +  1 ) } ) )
4929, 48pm2.61dan 789 . 2  |-  ( ( N  e.  ( ZZ>= `  ( M  -  1
) )  /\  M  e.  ZZ )  ->  ( M ... N )  =  ( ( M ... ( N  +  1
) )  \  {
( N  +  1 ) } ) )
50 eluzel2 11028 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
5150con3i 135 . . . . . 6  |-  ( -.  M  e.  ZZ  ->  -.  N  e.  ( ZZ>= `  M ) )
52 fzn0 11643 . . . . . 6  |-  ( ( M ... N )  =/=  (/)  <->  N  e.  ( ZZ>=
`  M ) )
5351, 52sylnibr 303 . . . . 5  |-  ( -.  M  e.  ZZ  ->  -.  ( M ... N
)  =/=  (/) )
54 nne 2597 . . . . 5  |-  ( -.  ( M ... N
)  =/=  (/)  <->  ( M ... N )  =  (/) )
5553, 54sylib 196 . . . 4  |-  ( -.  M  e.  ZZ  ->  ( M ... N )  =  (/) )
56 eluzel2 11028 . . . . . . . . 9  |-  ( ( N  +  1 )  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
5756con3i 135 . . . . . . . 8  |-  ( -.  M  e.  ZZ  ->  -.  ( N  +  1 )  e.  ( ZZ>= `  M ) )
58 fzn0 11643 . . . . . . . 8  |-  ( ( M ... ( N  +  1 ) )  =/=  (/)  <->  ( N  + 
1 )  e.  (
ZZ>= `  M ) )
5957, 58sylnibr 303 . . . . . . 7  |-  ( -.  M  e.  ZZ  ->  -.  ( M ... ( N  +  1 ) )  =/=  (/) )
60 nne 2597 . . . . . . 7  |-  ( -.  ( M ... ( N  +  1 ) )  =/=  (/)  <->  ( M ... ( N  +  1 ) )  =  (/) )
6159, 60sylib 196 . . . . . 6  |-  ( -.  M  e.  ZZ  ->  ( M ... ( N  +  1 ) )  =  (/) )
6261difeq1d 3552 . . . . 5  |-  ( -.  M  e.  ZZ  ->  ( ( M ... ( N  +  1 ) )  \  { ( N  +  1 ) } )  =  (
(/)  \  { ( N  +  1 ) } ) )
63 0dif 3832 . . . . . 6  |-  ( (/)  \  { ( N  + 
1 ) } )  =  (/)
6463a1i 11 . . . . 5  |-  ( -.  M  e.  ZZ  ->  (
(/)  \  { ( N  +  1 ) } )  =  (/) )
6562, 64eqtr2d 2438 . . . 4  |-  ( -.  M  e.  ZZ  ->  (/)  =  ( ( M ... ( N  + 
1 ) )  \  { ( N  + 
1 ) } ) )
6655, 65eqtrd 2437 . . 3  |-  ( -.  M  e.  ZZ  ->  ( M ... N )  =  ( ( M ... ( N  + 
1 ) )  \  { ( N  + 
1 ) } ) )
6766adantl 464 . 2  |-  ( ( N  e.  ( ZZ>= `  ( M  -  1
) )  /\  -.  M  e.  ZZ )  ->  ( M ... N
)  =  ( ( M ... ( N  +  1 ) ) 
\  { ( N  +  1 ) } ) )
6849, 67pm2.61dan 789 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 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1836    =/= wne 2591    \ cdif 3403   (/)c0 3728   {csn 3961   class class class wbr 4384   ` cfv 5513  (class class class)co 6218   CCcc 9423   RRcr 9424   1c1 9426    + caddc 9428    < clt 9561    <_ cle 9562    - cmin 9740   ZZcz 10803   ZZ>=cuz 11023   ...cfz 11615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-cnex 9481  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-addrcl 9486  ax-mulcl 9487  ax-mulrcl 9488  ax-mulcom 9489  ax-addass 9490  ax-mulass 9491  ax-distr 9492  ax-i2m1 9493  ax-1ne0 9494  ax-1rid 9495  ax-rnegex 9496  ax-rrecex 9497  ax-cnre 9498  ax-pre-lttri 9499  ax-pre-lttrn 9500  ax-pre-ltadd 9501  ax-pre-mulgt0 9502
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-tr 4478  df-eprel 4722  df-id 4726  df-po 4731  df-so 4732  df-fr 4769  df-we 4771  df-ord 4812  df-on 4813  df-lim 4814  df-suc 4815  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-om 6622  df-1st 6721  df-2nd 6722  df-recs 6982  df-rdg 7016  df-er 7251  df-en 7458  df-dom 7459  df-sdom 7460  df-pnf 9563  df-mnf 9564  df-xr 9565  df-ltxr 9566  df-le 9567  df-sub 9742  df-neg 9743  df-nn 10475  df-n0 10735  df-z 10804  df-uz 11024  df-fz 11616
This theorem is referenced by:  dvnmul  31945
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