Theorem fsum0diaglem 11403

Description: Lemma for fisum0diag 11404. (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 9983	. . . . . . 7 |- ( j e. ( 0 ... N ) -> 0 <_ j )
2	1	adantr 274	. . . . . 6 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> 0 <_ j )
3		elfz3nn0 10071	. . . . . . . . . 10 |- ( j e. ( 0 ... N ) -> N e. NN0 )
4	3	adantr 274	. . . . . . . . 9 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> N e. NN0 )
5	4	nn0zd 9332	. . . . . . . 8 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> N e. ZZ )
6	5	zred 9334	. . . . . . 7 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> N e. RR )
7		elfzelz 9981	. . . . . . . . 9 |- ( j e. ( 0 ... N ) -> j e. ZZ )
8	7	adantr 274	. . . . . . . 8 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> j e. ZZ )
9	8	zred 9334	. . . . . . 7 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> j e. RR )
10	6, 9	subge02d 8456	. . . . . 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 9339	. . . . . 6 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> ( N - j ) e. ZZ )
13		eluz 9500	. . . . . 6 |- ( ( ( N - j ) e. ZZ /\ N e. ZZ ) -> ( N e. ( ZZ>= ` ( N - j ) ) <-> ( N - j ) <_ N ) )
14	12, 5, 13	syl2anc 409	. . . . 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 10020	. . . 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 3148	. 2 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> k e. ( 0 ... N ) )
20		elfzelz 9981	. . . . . 6 |- ( k e. ( 0 ... ( N - j ) ) -> k e. ZZ )
21	20	adantl 275	. . . . 5 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> k e. ZZ )
22	21	zred 9334	. . . 4 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> k e. RR )
23		elfzle2 9984	. . . . 5 |- ( k e. ( 0 ... ( N - j ) ) -> k <_ ( N - j ) )
24	23	adantl 275	. . . 4 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> k <_ ( N - j ) )
25	22, 6, 9, 24	lesubd 8468	. . 3 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> j <_ ( N - k ) )
26		elfzuz 9977	. . . . 5 |- ( j e. ( 0 ... N ) -> j e. ( ZZ>= ` 0 ) )
27	26	adantr 274	. . . 4 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> j e. ( ZZ>= ` 0 ) )
28	5, 21	zsubcld 9339	. . . 4 |- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> ( N - k ) e. ZZ )
29		elfz5 9973	. . . 4 |- ( ( j e. ( ZZ>= ` 0 ) /\ ( N - k ) e. ZZ ) -> ( j e. ( 0 ... ( N - k ) ) <-> j <_ ( N - k ) ) )
30	27, 28, 29	syl2anc 409	. . 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 304	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 2141   C_ wss 3121   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   0cc0 7774   <_ cle 7955   - cmin 8090   NN0cn0 9135   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966

This theorem is referenced by:  fisum0diag  11404  fprod0diagfz  11591