MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nn0disj Structured version   Unicode version

Theorem nn0disj 11795
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 3673 . . . . . . 7  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  <->  ( k  e.  ( 0 ... N
)  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) ) )
21simprbi 462 . . . . . 6  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  ( ZZ>= `  ( N  +  1 ) ) )
3 eluzle 11094 . . . . . 6  |-  ( k  e.  ( ZZ>= `  ( N  +  1 ) )  ->  ( N  +  1 )  <_ 
k )
42, 3syl 16 . . . . 5  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( N  +  1 )  <_ 
k )
5 eluzel2 11087 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  ( N  +  1 ) )  ->  ( N  +  1 )  e.  ZZ )
62, 5syl 16 . . . . . 6  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( N  +  1 )  e.  ZZ )
7 eluzelz 11091 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  ( N  +  1 ) )  ->  k  e.  ZZ )
82, 7syl 16 . . . . . 6  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  ZZ )
9 zlem1lt 10911 . . . . . 6  |-  ( ( ( N  +  1 )  e.  ZZ  /\  k  e.  ZZ )  ->  ( ( N  + 
1 )  <_  k  <->  ( ( N  +  1 )  -  1 )  <  k ) )
106, 8, 9syl2anc 659 . . . . 5  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( ( N  +  1 )  <_  k  <->  ( ( N  +  1 )  -  1 )  < 
k ) )
114, 10mpbid 210 . . . 4  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( ( N  +  1 )  -  1 )  < 
k )
121simplbi 458 . . . . . 6  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  ( 0 ... N
) )
13 elfzle2 11693 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  <_  N )
1412, 13syl 16 . . . . 5  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  <_  N )
158zred 10965 . . . . . . 7  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  RR )
16 elfzel2 11689 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... N )  ->  N  e.  ZZ )
1716adantr 463 . . . . . . . . 9  |-  ( ( k  e.  ( 0 ... N )  /\  k  e.  ( ZZ>= `  ( N  +  1
) ) )  ->  N  e.  ZZ )
181, 17sylbi 195 . . . . . . . 8  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  N  e.  ZZ )
1918zred 10965 . . . . . . 7  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  N  e.  RR )
2015, 19lenltd 9720 . . . . . 6  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( k  <_  N  <->  -.  N  <  k ) )
2118zcnd 10966 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  N  e.  CC )
22 pncan1 9979 . . . . . . . . . 10  |-  ( N  e.  CC  ->  (
( N  +  1 )  -  1 )  =  N )
2321, 22syl 16 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( ( N  +  1 )  -  1 )  =  N )
2423eqcomd 2462 . . . . . . . 8  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  N  =  ( ( N  + 
1 )  -  1 ) )
2524breq1d 4449 . . . . . . 7  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( N  <  k  <->  ( ( N  +  1 )  - 
1 )  <  k
) )
2625notbid 292 . . . . . 6  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( -.  N  <  k  <->  -.  (
( N  +  1 )  -  1 )  <  k ) )
2720, 26bitrd 253 . . . . 5  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( k  <_  N  <->  -.  ( ( N  +  1 )  -  1 )  < 
k ) )
2814, 27mpbid 210 . . . 4  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  -.  (
( N  +  1 )  -  1 )  <  k )
2911, 28pm2.21dd 174 . . 3  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  (/) )
3029ssriv 3493 . 2  |-  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  C_  (/)
31 ss0 3815 . 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 184    /\ wa 367    = wceq 1398    e. wcel 1823    i^i cin 3460    C_ wss 3461   (/)c0 3783   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    < clt 9617    <_ cle 9618    - cmin 9796   ZZcz 10860   ZZ>=cuz 11082   ...cfz 11675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676
This theorem is referenced by:  chfacfscmulgsum  19528  chfacfpmmulgsum  19532
  Copyright terms: Public domain W3C validator