Theorem fsum0diaglem 11443

Description: Lemma for fisum0diag 11444. (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 10024	. . . . . . 7 |- ( j e. ( 0 ... N ) -> 0 <_ j )
2	1	adantr 276	. . . . . 6 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> 0 <_ j )
3		elfz3nn0 10112	. . . . . . . . . 10 |- ( j e. ( 0 ... N ) -> N e. NN0 )
4	3	adantr 276	. . . . . . . . 9 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> N e. NN0 )
5	4	nn0zd 9371	. . . . . . . 8 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> N e. ZZ )
6	5	zred 9373	. . . . . . 7 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> N e. RR )
7		elfzelz 10022	. . . . . . . . 9 |- ( j e. ( 0 ... N ) -> j e. ZZ )
8	7	adantr 276	. . . . . . . 8 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> j e. ZZ )
9	8	zred 9373	. . . . . . 7 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> j e. RR )
10	6, 9	subge02d 8492	. . . . . 6 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> ( 0 <_ j <-> ( N - j ) <_ N ) )
11	2, 10	mpbid 147	. . . . 5 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> ( N - j ) <_ N )
12	5, 8	zsubcld 9378	. . . . . 6 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> ( N - j ) e. ZZ )
13		eluz 9539	. . . . . 6 |- ( ( ( N - j ) e. ZZ /\ N e. ZZ ) -> ( N e. ( ZZ>= ` ( N - j ) ) <-> ( N - j ) <_ N ) )
14	12, 5, 13	syl2anc 411	. . . . 5 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> ( N e. ( ZZ>= ` ( N - j ) ) <-> ( N - j ) <_ N ) )
15	11, 14	mpbird 167	. . . 4 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> N e. ( ZZ>= ` ( N - j ) ) )
16		fzss2 10061	. . . 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 110	. . 3 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> k e. ( 0 ... ( N - j ) ) )
19	17, 18	sseldd 3156	. 2 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> k e. ( 0 ... N ) )
20		elfzelz 10022	. . . . . 6 |- ( k e. ( 0 ... ( N - j ) ) -> k e. ZZ )
21	20	adantl 277	. . . . 5 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> k e. ZZ )
22	21	zred 9373	. . . 4 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> k e. RR )
23		elfzle2 10025	. . . . 5 |- ( k e. ( 0 ... ( N - j ) ) -> k <_ ( N - j ) )
24	23	adantl 277	. . . 4 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> k <_ ( N - j ) )
25	22, 6, 9, 24	lesubd 8504	. . 3 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> j <_ ( N - k ) )
26		elfzuz 10018	. . . . 5 |- ( j e. ( 0 ... N ) -> j e. ( ZZ>= ` 0 ) )
27	26	adantr 276	. . . 4 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> j e. ( ZZ>= ` 0 ) )
28	5, 21	zsubcld 9378	. . . 4 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> ( N - k ) e. ZZ )
29		elfz5 10014	. . . 4 |- ( ( j e. ( ZZ>= ` 0 ) /\ ( N - k ) e. ZZ ) -> ( j e. ( 0 ... ( N - k ) ) <-> j <_ ( N - k ) ) )
30	27, 28, 29	syl2anc 411	. . 3 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> ( j e. ( 0 ... ( N - k ) ) <-> j <_ ( N - k ) ) )
31	25, 30	mpbird 167	. 2 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> j e. ( 0 ... ( N - k ) ) )
32	19, 31	jca 306	1 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> ( k e. ( 0 ... N ) /\ j e. ( 0 ... ( N - k ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2148   C_ wss 3129   class class class wbr 4003   ` cfv 5216  (class class class)co 5874   0cc0 7810   <_ cle 7991   - cmin 8126   NN0cn0 9174   ZZcz 9251   ZZ>=cuz 9526   ...cfz 10006

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-ltadd 7926

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-inn 8918  df-n0 9175  df-z 9252  df-uz 9527  df-fz 10007

This theorem is referenced by:  fisum0diag  11444  fprod0diagfz  11631