ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fsum0diaglem Unicode version
Theorem fsum0diaglem 11583
Description: Lemma for fisum0diag 11584. (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 10093 . . . . . . 7 |- ( j e. ( 0 ... N ) -> 0 <_ j )
21adantr 276 . . . . . 6 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> 0 <_ j )
3 elfz3nn0 10181 . . . . . . . . . 10 |- ( j e. ( 0 ... N ) -> N e. NN0 )
43adantr 276 . . . . . . . . 9 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> N e. NN0 )
54nn0zd 9437 . . . . . . . 8 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> N e. ZZ )
65zred 9439 . . . . . . 7 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> N e. RR )
7 elfzelz 10091 . . . . . . . . 9 |- ( j e. ( 0 ... N ) -> j e. ZZ )
87adantr 276 . . . . . . . 8 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> j e. ZZ )
98zred 9439 . . . . . . 7 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> j e. RR )
106, 9subge02d 8556 . . . . . 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 9444 . . . . . 6 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> ( N - j ) e. ZZ )
13 eluz 9605 . . . . . 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 10130 . . . 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 3180 . 2 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> k e. ( 0 ... N ) )
20 elfzelz 10091 . . . . . 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 9439 . . . 4 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> k e. RR )
23 elfzle2 10094 . . . . 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 8568 . . 3 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> j <_ ( N - k ) )
26 elfzuz 10087 . . . . 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 9444 . . . 4 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> ( N - k ) e. ZZ )
29 elfz5 10083 . . . 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 2164   C_ wss 3153   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   0cc0 7872   <_ cle 8055   - cmin 8190   NN0cn0 9240   ZZcz 9317   ZZ>=cuz 9592   ...cfz 10074
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 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-ltadd 7988
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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-n0 9241  df-z 9318  df-uz 9593  df-fz 10075
This theorem is referenced by:  fisum0diag  11584  fprod0diagfz  11771
  Copyright terms: Public domain W3C validator