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Theorem fsum0diaglem 11461
Description: Lemma for fisum0diag 11462. (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 10040 . . . . . . 7 |- ( j e. ( 0 ... N ) -> 0 <_ j )
21adantr 276 . . . . . 6 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> 0 <_ j )
3 elfz3nn0 10128 . . . . . . . . . 10 |- ( j e. ( 0 ... N ) -> N e. NN0 )
43adantr 276 . . . . . . . . 9 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> N e. NN0 )
54nn0zd 9386 . . . . . . . 8 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> N e. ZZ )
65zred 9388 . . . . . . 7 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> N e. RR )
7 elfzelz 10038 . . . . . . . . 9 |- ( j e. ( 0 ... N ) -> j e. ZZ )
87adantr 276 . . . . . . . 8 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> j e. ZZ )
98zred 9388 . . . . . . 7 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> j e. RR )
106, 9subge02d 8507 . . . . . 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 9393 . . . . . 6 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> ( N - j ) e. ZZ )
13 eluz 9554 . . . . . 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 10077 . . . 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 3168 . 2 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> k e. ( 0 ... N ) )
20 elfzelz 10038 . . . . . 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 9388 . . . 4 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> k e. RR )
23 elfzle2 10041 . . . . 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 8519 . . 3 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> j <_ ( N - k ) )
26 elfzuz 10034 . . . . 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 9393 . . . 4 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> ( N - k ) e. ZZ )
29 elfz5 10030 . . . 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 2158   C_ wss 3141   class class class wbr 4015   ` cfv 5228  (class class class)co 5888   0cc0 7824   <_ cle 8006   - cmin 8141   NN0cn0 9189   ZZcz 9266   ZZ>=cuz 9541   ...cfz 10021
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-ltadd 7940
This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-inn 8933  df-n0 9190  df-z 9267  df-uz 9542  df-fz 10022
This theorem is referenced by:  fisum0diag  11462  fprod0diagfz  11649
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