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Theorem fsum0diaglem 13557
Description: Lemma for fsum0diag 13558. (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 11690 . . . . . . 7  |-  ( j  e.  ( 0 ... N )  ->  0  <_  j )
21adantr 465 . . . . . 6  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  0  <_  j
)
3 elfz3nn0 11772 . . . . . . . . . 10  |-  ( j  e.  ( 0 ... N )  ->  N  e.  NN0 )
43adantr 465 . . . . . . . . 9  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  N  e.  NN0 )
54nn0zd 10965 . . . . . . . 8  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  N  e.  ZZ )
65zred 10967 . . . . . . 7  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  N  e.  RR )
7 elfzelz 11689 . . . . . . . . 9  |-  ( j  e.  ( 0 ... N )  ->  j  e.  ZZ )
87adantr 465 . . . . . . . 8  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  j  e.  ZZ )
98zred 10967 . . . . . . 7  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  j  e.  RR )
106, 9subge02d 10145 . . . . . 6  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( 0  <_ 
j  <->  ( N  -  j )  <_  N
) )
112, 10mpbid 210 . . . . 5  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( N  -  j )  <_  N
)
125, 8zsubcld 10972 . . . . . 6  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( N  -  j )  e.  ZZ )
13 eluz 11096 . . . . . 6  |-  ( ( ( N  -  j
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  e.  (
ZZ>= `  ( N  -  j ) )  <->  ( N  -  j )  <_  N ) )
1412, 5, 13syl2anc 661 . . . . 5  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( N  e.  ( ZZ>= `  ( N  -  j ) )  <-> 
( N  -  j
)  <_  N )
)
1511, 14mpbird 232 . . . 4  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  N  e.  (
ZZ>= `  ( N  -  j ) ) )
16 fzss2 11724 . . . 4  |-  ( N  e.  ( ZZ>= `  ( N  -  j )
)  ->  ( 0 ... ( N  -  j ) )  C_  ( 0 ... N
) )
1715, 16syl 16 . . 3  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( 0 ... ( N  -  j
) )  C_  (
0 ... N ) )
18 simpr 461 . . 3  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  k  e.  ( 0 ... ( N  -  j ) ) )
1917, 18sseldd 3505 . 2  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  k  e.  ( 0 ... N ) )
20 elfzelz 11689 . . . . . 6  |-  ( k  e.  ( 0 ... ( N  -  j
) )  ->  k  e.  ZZ )
2120adantl 466 . . . . 5  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  k  e.  ZZ )
2221zred 10967 . . . 4  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  k  e.  RR )
23 elfzle2 11691 . . . . 5  |-  ( k  e.  ( 0 ... ( N  -  j
) )  ->  k  <_  ( N  -  j
) )
2423adantl 466 . . . 4  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  k  <_  ( N  -  j )
)
2522, 6, 9, 24lesubd 10157 . . 3  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  j  <_  ( N  -  k )
)
26 elfzuz 11685 . . . . 5  |-  ( j  e.  ( 0 ... N )  ->  j  e.  ( ZZ>= `  0 )
)
2726adantr 465 . . . 4  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  j  e.  (
ZZ>= `  0 ) )
285, 21zsubcld 10972 . . . 4  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( N  -  k )  e.  ZZ )
29 elfz5 11681 . . . 4  |-  ( ( j  e.  ( ZZ>= ` 
0 )  /\  ( N  -  k )  e.  ZZ )  ->  (
j  e.  ( 0 ... ( N  -  k ) )  <->  j  <_  ( N  -  k ) ) )
3027, 28, 29syl2anc 661 . . 3  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( j  e.  ( 0 ... ( N  -  k )
)  <->  j  <_  ( N  -  k )
) )
3125, 30mpbird 232 . 2  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  j  e.  ( 0 ... ( N  -  k ) ) )
3219, 31jca 532 1  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( k  e.  ( 0 ... N
)  /\  j  e.  ( 0 ... ( N  -  k )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1767    C_ wss 3476   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   0cc0 9493    <_ cle 9630    - cmin 9806   NN0cn0 10796   ZZcz 10865   ZZ>=cuz 11083   ...cfz 11673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674
This theorem is referenced by:  fsum0diag  13558  fprod0diag  28969
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