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Theorem nn0disj 11932
Description: The first  N  + 
1 elements of the set of nonnegative integers are distinct from any later members. (Contributed by AV, 8-Nov-2019.)
Assertion
Ref Expression
nn0disj  |-  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  =  (/)

Proof of Theorem nn0disj
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elin 3608 . . . . . . 7  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  <->  ( k  e.  ( 0 ... N
)  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) ) )
21simprbi 471 . . . . . 6  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  ( ZZ>= `  ( N  +  1 ) ) )
3 eluzle 11195 . . . . . 6  |-  ( k  e.  ( ZZ>= `  ( N  +  1 ) )  ->  ( N  +  1 )  <_ 
k )
42, 3syl 17 . . . . 5  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( N  +  1 )  <_ 
k )
5 eluzel2 11187 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  ( N  +  1 ) )  ->  ( N  +  1 )  e.  ZZ )
62, 5syl 17 . . . . . 6  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( N  +  1 )  e.  ZZ )
7 eluzelz 11192 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  ( N  +  1 ) )  ->  k  e.  ZZ )
82, 7syl 17 . . . . . 6  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  ZZ )
9 zlem1lt 11012 . . . . . 6  |-  ( ( ( N  +  1 )  e.  ZZ  /\  k  e.  ZZ )  ->  ( ( N  + 
1 )  <_  k  <->  ( ( N  +  1 )  -  1 )  <  k ) )
106, 8, 9syl2anc 673 . . . . 5  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( ( N  +  1 )  <_  k  <->  ( ( N  +  1 )  -  1 )  < 
k ) )
114, 10mpbid 215 . . . 4  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( ( N  +  1 )  -  1 )  < 
k )
121simplbi 467 . . . . . 6  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  ( 0 ... N
) )
13 elfzle2 11829 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  <_  N )
1412, 13syl 17 . . . . 5  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  <_  N )
158zred 11063 . . . . . . 7  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  RR )
16 elfzel2 11824 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... N )  ->  N  e.  ZZ )
1716adantr 472 . . . . . . . . 9  |-  ( ( k  e.  ( 0 ... N )  /\  k  e.  ( ZZ>= `  ( N  +  1
) ) )  ->  N  e.  ZZ )
181, 17sylbi 200 . . . . . . . 8  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  N  e.  ZZ )
1918zred 11063 . . . . . . 7  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  N  e.  RR )
2015, 19lenltd 9798 . . . . . 6  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( k  <_  N  <->  -.  N  <  k ) )
2118zcnd 11064 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  N  e.  CC )
22 pncan1 10064 . . . . . . . . . 10  |-  ( N  e.  CC  ->  (
( N  +  1 )  -  1 )  =  N )
2321, 22syl 17 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( ( N  +  1 )  -  1 )  =  N )
2423eqcomd 2477 . . . . . . . 8  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  N  =  ( ( N  + 
1 )  -  1 ) )
2524breq1d 4405 . . . . . . 7  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( N  <  k  <->  ( ( N  +  1 )  - 
1 )  <  k
) )
2625notbid 301 . . . . . 6  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( -.  N  <  k  <->  -.  (
( N  +  1 )  -  1 )  <  k ) )
2720, 26bitrd 261 . . . . 5  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( k  <_  N  <->  -.  ( ( N  +  1 )  -  1 )  < 
k ) )
2814, 27mpbid 215 . . . 4  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  -.  (
( N  +  1 )  -  1 )  <  k )
2911, 28pm2.21dd 179 . . 3  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  (/) )
3029ssriv 3422 . 2  |-  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  C_  (/)
31 ss0 3768 . 2  |-  ( ( ( 0 ... N
)  i^i  ( ZZ>= `  ( N  +  1
) ) )  C_  (/) 
->  ( ( 0 ... N )  i^i  ( ZZ>=
`  ( N  + 
1 ) ) )  =  (/) )
3230, 31ax-mp 5 1  |-  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    i^i cin 3389    C_ wss 3390   (/)c0 3722   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   CCcc 9555   0cc0 9557   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:  chfacfscmulgsum  19961  chfacfpmmulgsum  19965  nnuzdisj  37665
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