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Theorem fsum0diaglem 10895
Description: Lemma for fisum0diag 10896. (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 9502 . . . . . . 7  |-  ( j  e.  ( 0 ... N )  ->  0  <_  j )
21adantr 271 . . . . . 6  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  0  <_  j
)
3 elfz3nn0 9590 . . . . . . . . . 10  |-  ( j  e.  ( 0 ... N )  ->  N  e.  NN0 )
43adantr 271 . . . . . . . . 9  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  N  e.  NN0 )
54nn0zd 8927 . . . . . . . 8  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  N  e.  ZZ )
65zred 8929 . . . . . . 7  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  N  e.  RR )
7 elfzelz 9501 . . . . . . . . 9  |-  ( j  e.  ( 0 ... N )  ->  j  e.  ZZ )
87adantr 271 . . . . . . . 8  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  j  e.  ZZ )
98zred 8929 . . . . . . 7  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  j  e.  RR )
106, 9subge02d 8075 . . . . . 6  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( 0  <_ 
j  <->  ( N  -  j )  <_  N
) )
112, 10mpbid 146 . . . . 5  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( N  -  j )  <_  N
)
125, 8zsubcld 8934 . . . . . 6  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( N  -  j )  e.  ZZ )
13 eluz 9093 . . . . . 6  |-  ( ( ( N  -  j
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  e.  (
ZZ>= `  ( N  -  j ) )  <->  ( N  -  j )  <_  N ) )
1412, 5, 13syl2anc 404 . . . . 5  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( N  e.  ( ZZ>= `  ( N  -  j ) )  <-> 
( N  -  j
)  <_  N )
)
1511, 14mpbird 166 . . . 4  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  N  e.  (
ZZ>= `  ( N  -  j ) ) )
16 fzss2 9539 . . . 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 109 . . 3  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  k  e.  ( 0 ... ( N  -  j ) ) )
1917, 18sseldd 3027 . 2  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  k  e.  ( 0 ... N ) )
20 elfzelz 9501 . . . . . 6  |-  ( k  e.  ( 0 ... ( N  -  j
) )  ->  k  e.  ZZ )
2120adantl 272 . . . . 5  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  k  e.  ZZ )
2221zred 8929 . . . 4  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  k  e.  RR )
23 elfzle2 9503 . . . . 5  |-  ( k  e.  ( 0 ... ( N  -  j
) )  ->  k  <_  ( N  -  j
) )
2423adantl 272 . . . 4  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  k  <_  ( N  -  j )
)
2522, 6, 9, 24lesubd 8087 . . 3  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  j  <_  ( N  -  k )
)
26 elfzuz 9497 . . . . 5  |-  ( j  e.  ( 0 ... N )  ->  j  e.  ( ZZ>= `  0 )
)
2726adantr 271 . . . 4  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  j  e.  (
ZZ>= `  0 ) )
285, 21zsubcld 8934 . . . 4  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( N  -  k )  e.  ZZ )
29 elfz5 9493 . . . 4  |-  ( ( j  e.  ( ZZ>= ` 
0 )  /\  ( N  -  k )  e.  ZZ )  ->  (
j  e.  ( 0 ... ( N  -  k ) )  <->  j  <_  ( N  -  k ) ) )
3027, 28, 29syl2anc 404 . . 3  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( j  e.  ( 0 ... ( N  -  k )
)  <->  j  <_  ( N  -  k )
) )
3125, 30mpbird 166 . 2  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  j  e.  ( 0 ... ( N  -  k ) ) )
3219, 31jca 301 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 103    <-> wb 104    e. wcel 1439    C_ wss 3000   class class class wbr 3851   ` cfv 5028  (class class class)co 5666   0cc0 7411    <_ cle 7584    - cmin 7714   NN0cn0 8734   ZZcz 8811   ZZ>=cuz 9080   ...cfz 9485
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-addcom 7506  ax-addass 7508  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-0id 7514  ax-rnegex 7515  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-ltadd 7522
This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-inn 8484  df-n0 8735  df-z 8812  df-uz 9081  df-fz 9486
This theorem is referenced by:  fisum0diag  10896
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