ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fsum0diaglem Unicode version

Theorem fsum0diaglem 11468
Description: Lemma for fisum0diag 11469. (Contributed by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.)
Assertion
Ref Expression
fsum0diaglem  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( k  e.  ( 0 ... N
)  /\  j  e.  ( 0 ... ( N  -  k )
) ) )
Distinct variable group:    j, k, N

Proof of Theorem fsum0diaglem
StepHypRef Expression
1 elfzle1 10047 . . . . . . 7  |-  ( j  e.  ( 0 ... N )  ->  0  <_  j )
21adantr 276 . . . . . 6  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  0  <_  j
)
3 elfz3nn0 10135 . . . . . . . . . 10  |-  ( j  e.  ( 0 ... N )  ->  N  e.  NN0 )
43adantr 276 . . . . . . . . 9  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  N  e.  NN0 )
54nn0zd 9393 . . . . . . . 8  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  N  e.  ZZ )
65zred 9395 . . . . . . 7  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  N  e.  RR )
7 elfzelz 10045 . . . . . . . . 9  |-  ( j  e.  ( 0 ... N )  ->  j  e.  ZZ )
87adantr 276 . . . . . . . 8  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  j  e.  ZZ )
98zred 9395 . . . . . . 7  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  j  e.  RR )
106, 9subge02d 8514 . . . . . 6  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( 0  <_ 
j  <->  ( N  -  j )  <_  N
) )
112, 10mpbid 147 . . . . 5  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( N  -  j )  <_  N
)
125, 8zsubcld 9400 . . . . . 6  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( N  -  j )  e.  ZZ )
13 eluz 9561 . . . . . 6  |-  ( ( ( N  -  j
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  e.  (
ZZ>= `  ( N  -  j ) )  <->  ( N  -  j )  <_  N ) )
1412, 5, 13syl2anc 411 . . . . 5  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( N  e.  ( ZZ>= `  ( N  -  j ) )  <-> 
( N  -  j
)  <_  N )
)
1511, 14mpbird 167 . . . 4  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  N  e.  (
ZZ>= `  ( N  -  j ) ) )
16 fzss2 10084 . . . 4  |-  ( N  e.  ( ZZ>= `  ( N  -  j )
)  ->  ( 0 ... ( N  -  j ) )  C_  ( 0 ... N
) )
1715, 16syl 14 . . 3  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( 0 ... ( N  -  j
) )  C_  (
0 ... N ) )
18 simpr 110 . . 3  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  k  e.  ( 0 ... ( N  -  j ) ) )
1917, 18sseldd 3171 . 2  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  k  e.  ( 0 ... N ) )
20 elfzelz 10045 . . . . . 6  |-  ( k  e.  ( 0 ... ( N  -  j
) )  ->  k  e.  ZZ )
2120adantl 277 . . . . 5  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  k  e.  ZZ )
2221zred 9395 . . . 4  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  k  e.  RR )
23 elfzle2 10048 . . . . 5  |-  ( k  e.  ( 0 ... ( N  -  j
) )  ->  k  <_  ( N  -  j
) )
2423adantl 277 . . . 4  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  k  <_  ( N  -  j )
)
2522, 6, 9, 24lesubd 8526 . . 3  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  j  <_  ( N  -  k )
)
26 elfzuz 10041 . . . . 5  |-  ( j  e.  ( 0 ... N )  ->  j  e.  ( ZZ>= `  0 )
)
2726adantr 276 . . . 4  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  j  e.  (
ZZ>= `  0 ) )
285, 21zsubcld 9400 . . . 4  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( N  -  k )  e.  ZZ )
29 elfz5 10037 . . . 4  |-  ( ( j  e.  ( ZZ>= ` 
0 )  /\  ( N  -  k )  e.  ZZ )  ->  (
j  e.  ( 0 ... ( N  -  k ) )  <->  j  <_  ( N  -  k ) ) )
3027, 28, 29syl2anc 411 . . 3  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( j  e.  ( 0 ... ( N  -  k )
)  <->  j  <_  ( N  -  k )
) )
3125, 30mpbird 167 . 2  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  j  e.  ( 0 ... ( N  -  k ) ) )
3219, 31jca 306 1  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( k  e.  ( 0 ... N
)  /\  j  e.  ( 0 ... ( N  -  k )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    e. wcel 2160    C_ wss 3144   class class class wbr 4018   ` cfv 5232  (class class class)co 5892   0cc0 7831    <_ cle 8013    - cmin 8148   NN0cn0 9196   ZZcz 9273   ZZ>=cuz 9548   ...cfz 10028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7922  ax-resscn 7923  ax-1cn 7924  ax-1re 7925  ax-icn 7926  ax-addcl 7927  ax-addrcl 7928  ax-mulcl 7929  ax-addcom 7931  ax-addass 7933  ax-distr 7935  ax-i2m1 7936  ax-0lt1 7937  ax-0id 7939  ax-rnegex 7940  ax-cnre 7942  ax-pre-ltirr 7943  ax-pre-ltwlin 7944  ax-pre-lttrn 7945  ax-pre-ltadd 7947
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-pnf 8014  df-mnf 8015  df-xr 8016  df-ltxr 8017  df-le 8018  df-sub 8150  df-neg 8151  df-inn 8940  df-n0 9197  df-z 9274  df-uz 9549  df-fz 10029
This theorem is referenced by:  fisum0diag  11469  fprod0diagfz  11656
  Copyright terms: Public domain W3C validator