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

Step	Hyp	Ref	Expression
1		elfzle1 9502	. . . . . . 7 |- ( j e. ( 0 ... N ) -> 0 <_ j )
2	1	adantr 271	. . . . . 6 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> 0 <_ j )
3		elfz3nn0 9590	. . . . . . . . . 10 |- ( j e. ( 0 ... N ) -> N e. NN0 )
4	3	adantr 271	. . . . . . . . 9 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> N e. NN0 )
5	4	nn0zd 8927	. . . . . . . 8 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> N e. ZZ )
6	5	zred 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 )
8	7	adantr 271	. . . . . . . 8 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> j e. ZZ )
9	8	zred 8929	. . . . . . 7 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> j e. RR )
10	6, 9	subge02d 8075	. . . . . 6 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> ( 0 <_ j <-> ( N - j ) <_ N ) )
11	2, 10	mpbid 146	. . . . 5 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> ( N - j ) <_ N )
12	5, 8	zsubcld 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 ) )
14	12, 5, 13	syl2anc 404	. . . . 5 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> ( N e. ( ZZ>= ` ( N - j ) ) <-> ( N - j ) <_ N ) )
15	11, 14	mpbird 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 ) )
17	15, 16	syl 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 ) ) )
19	17, 18	sseldd 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 )
21	20	adantl 272	. . . . 5 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> k e. ZZ )
22	21	zred 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 ) )
24	23	adantl 272	. . . 4 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> k <_ ( N - j ) )
25	22, 6, 9, 24	lesubd 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 ) )
27	26	adantr 271	. . . 4 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> j e. ( ZZ>= ` 0 ) )
28	5, 21	zsubcld 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 ) ) )
30	27, 28, 29	syl2anc 404	. . 3 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> ( j e. ( 0 ... ( N - k ) ) <-> j <_ ( N - k ) ) )
31	25, 30	mpbird 166	. 2 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> j e. ( 0 ... ( N - k ) ) )
32	19, 31	jca 301	1 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> ( k e. ( 0 ... N ) /\ j e. ( 0 ... ( N - k ) ) ) )

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