Theorem subfzo0 10012

Description: The difference between two elements in a half-open range of nonnegative integers is greater than the negation of the upper bound and less than the upper bound of the range. (Contributed by AV, 20-Mar-2021.)

Assertion

Ref	Expression
subfzo0	|- ( ( I e. ( 0..^ N ) /\ J e. ( 0..^ N ) ) -> ( -u N < ( I - J ) /\ ( I - J ) < N ) )

Proof of Theorem subfzo0

Step	Hyp	Ref	Expression
1		elfzo0 9952	. . 3 |- ( I e. ( 0..^ N ) <-> ( I e. NN0 /\ N e. NN /\ I < N ) )
2		elfzo0 9952	. . . . 5 |- ( J e. ( 0..^ N ) <-> ( J e. NN0 /\ N e. NN /\ J < N ) )
3		nn0re 8979	. . . . . . . . . . . 12 |- ( I e. NN0 -> I e. RR )
4	3	adantr 274	. . . . . . . . . . 11 |- ( ( I e. NN0 /\ I < N ) -> I e. RR )
5		nnre 8720	. . . . . . . . . . . . . 14 |- ( N e. NN -> N e. RR )
6		nn0re 8979	. . . . . . . . . . . . . 14 |- ( J e. NN0 -> J e. RR )
7		resubcl 8019	. . . . . . . . . . . . . 14 |- ( ( N e. RR /\ J e. RR ) -> ( N - J ) e. RR )
8	5, 6, 7	syl2an 287	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ J e. NN0 ) -> ( N - J ) e. RR )
9	8	ancoms 266	. . . . . . . . . . . 12 |- ( ( J e. NN0 /\ N e. NN ) -> ( N - J ) e. RR )
10	9	3adant3 1001	. . . . . . . . . . 11 |- ( ( J e. NN0 /\ N e. NN /\ J < N ) -> ( N - J ) e. RR )
11	4, 10	anim12i 336	. . . . . . . . . 10 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( I e. RR /\ ( N - J ) e. RR ) )
12		nn0ge0 8995	. . . . . . . . . . . 12 |- ( I e. NN0 -> 0 <_ I )
13	12	adantr 274	. . . . . . . . . . 11 |- ( ( I e. NN0 /\ I < N ) -> 0 <_ I )
14		posdif 8210	. . . . . . . . . . . . 13 |- ( ( J e. RR /\ N e. RR ) -> ( J < N <-> 0 < ( N - J ) ) )
15	6, 5, 14	syl2an 287	. . . . . . . . . . . 12 |- ( ( J e. NN0 /\ N e. NN ) -> ( J < N <-> 0 < ( N - J ) ) )
16	15	biimp3a 1323	. . . . . . . . . . 11 |- ( ( J e. NN0 /\ N e. NN /\ J < N ) -> 0 < ( N - J ) )
17	13, 16	anim12i 336	. . . . . . . . . 10 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( 0 <_ I /\ 0 < ( N - J ) ) )
18		addgegt0 8204	. . . . . . . . . 10 |- ( ( ( I e. RR /\ ( N - J ) e. RR ) /\ ( 0 <_ I /\ 0 < ( N - J ) ) ) -> 0 < ( I + ( N - J ) ) )
19	11, 17, 18	syl2anc 408	. . . . . . . . 9 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> 0 < ( I + ( N - J ) ) )
20		nn0cn 8980	. . . . . . . . . . . 12 |- ( I e. NN0 -> I e. CC )
21	20	adantr 274	. . . . . . . . . . 11 |- ( ( I e. NN0 /\ I < N ) -> I e. CC )
22	21	adantr 274	. . . . . . . . . 10 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> I e. CC )
23		nn0cn 8980	. . . . . . . . . . . 12 |- ( J e. NN0 -> J e. CC )
24	23	3ad2ant1 1002	. . . . . . . . . . 11 |- ( ( J e. NN0 /\ N e. NN /\ J < N ) -> J e. CC )
25	24	adantl 275	. . . . . . . . . 10 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> J e. CC )
26		nncn 8721	. . . . . . . . . . . 12 |- ( N e. NN -> N e. CC )
27	26	3ad2ant2 1003	. . . . . . . . . . 11 |- ( ( J e. NN0 /\ N e. NN /\ J < N ) -> N e. CC )
28	27	adantl 275	. . . . . . . . . 10 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> N e. CC )
29	22, 25, 28	subadd23d 8088	. . . . . . . . 9 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( ( I - J ) + N ) = ( I + ( N - J ) ) )
30	19, 29	breqtrrd 3951	. . . . . . . 8 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> 0 < ( ( I - J ) + N ) )
31	6	3ad2ant1 1002	. . . . . . . . . 10 |- ( ( J e. NN0 /\ N e. NN /\ J < N ) -> J e. RR )
32		resubcl 8019	. . . . . . . . . 10 |- ( ( I e. RR /\ J e. RR ) -> ( I - J ) e. RR )
33	4, 31, 32	syl2an 287	. . . . . . . . 9 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( I - J ) e. RR )
34	5	3ad2ant2 1003	. . . . . . . . . 10 |- ( ( J e. NN0 /\ N e. NN /\ J < N ) -> N e. RR )
35	34	adantl 275	. . . . . . . . 9 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> N e. RR )
36	33, 35	possumd 8324	. . . . . . . 8 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( 0 < ( ( I - J ) + N ) <-> -u N < ( I - J ) ) )
37	30, 36	mpbid 146	. . . . . . 7 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> -u N < ( I - J ) )
38	3	adantr 274	. . . . . . . . . . . 12 |- ( ( I e. NN0 /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> I e. RR )
39	34	adantl 275	. . . . . . . . . . . 12 |- ( ( I e. NN0 /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> N e. RR )
40		readdcl 7739	. . . . . . . . . . . . . . 15 |- ( ( J e. RR /\ N e. RR ) -> ( J + N ) e. RR )
41	6, 5, 40	syl2an 287	. . . . . . . . . . . . . 14 |- ( ( J e. NN0 /\ N e. NN ) -> ( J + N ) e. RR )
42	41	3adant3 1001	. . . . . . . . . . . . 13 |- ( ( J e. NN0 /\ N e. NN /\ J < N ) -> ( J + N ) e. RR )
43	42	adantl 275	. . . . . . . . . . . 12 |- ( ( I e. NN0 /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( J + N ) e. RR )
44	38, 39, 43	3jca 1161	. . . . . . . . . . 11 |- ( ( I e. NN0 /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( I e. RR /\ N e. RR /\ ( J + N ) e. RR ) )
45		nn0ge0 8995	. . . . . . . . . . . . . 14 |- ( J e. NN0 -> 0 <_ J )
46	45	3ad2ant1 1002	. . . . . . . . . . . . 13 |- ( ( J e. NN0 /\ N e. NN /\ J < N ) -> 0 <_ J )
47	46	adantl 275	. . . . . . . . . . . 12 |- ( ( I e. NN0 /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> 0 <_ J )
48	5, 6	anim12i 336	. . . . . . . . . . . . . . . 16 |- ( ( N e. NN /\ J e. NN0 ) -> ( N e. RR /\ J e. RR ) )
49	48	ancoms 266	. . . . . . . . . . . . . . 15 |- ( ( J e. NN0 /\ N e. NN ) -> ( N e. RR /\ J e. RR ) )
50	49	3adant3 1001	. . . . . . . . . . . . . 14 |- ( ( J e. NN0 /\ N e. NN /\ J < N ) -> ( N e. RR /\ J e. RR ) )
51	50	adantl 275	. . . . . . . . . . . . 13 |- ( ( I e. NN0 /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( N e. RR /\ J e. RR ) )
52		addge02 8228	. . . . . . . . . . . . 13 |- ( ( N e. RR /\ J e. RR ) -> ( 0 <_ J <-> N <_ ( J + N ) ) )
53	51, 52	syl 14	. . . . . . . . . . . 12 |- ( ( I e. NN0 /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( 0 <_ J <-> N <_ ( J + N ) ) )
54	47, 53	mpbid 146	. . . . . . . . . . 11 |- ( ( I e. NN0 /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> N <_ ( J + N ) )
55	44, 54	lelttrdi 8181	. . . . . . . . . 10 |- ( ( I e. NN0 /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( I < N -> I < ( J + N ) ) )
56	55	impancom 258	. . . . . . . . 9 |- ( ( I e. NN0 /\ I < N ) -> ( ( J e. NN0 /\ N e. NN /\ J < N ) -> I < ( J + N ) ) )
57	56	imp 123	. . . . . . . 8 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> I < ( J + N ) )
58	4	adantr 274	. . . . . . . . 9 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> I e. RR )
59	31	adantl 275	. . . . . . . . 9 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> J e. RR )
60	58, 59, 35	ltsubadd2d 8298	. . . . . . . 8 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( ( I - J ) < N <-> I < ( J + N ) ) )
61	57, 60	mpbird 166	. . . . . . 7 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( I - J ) < N )
62	37, 61	jca 304	. . . . . 6 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( -u N < ( I - J ) /\ ( I - J ) < N ) )
63	62	ex 114	. . . . 5 |- ( ( I e. NN0 /\ I < N ) -> ( ( J e. NN0 /\ N e. NN /\ J < N ) -> ( -u N < ( I - J ) /\ ( I - J ) < N ) ) )
64	2, 63	syl5bi 151	. . . 4 |- ( ( I e. NN0 /\ I < N ) -> ( J e. ( 0..^ N ) -> ( -u N < ( I - J ) /\ ( I - J ) < N ) ) )
65	64	3adant2 1000	. . 3 |- ( ( I e. NN0 /\ N e. NN /\ I < N ) -> ( J e. ( 0..^ N ) -> ( -u N < ( I - J ) /\ ( I - J ) < N ) ) )
66	1, 65	sylbi 120	. 2 |- ( I e. ( 0..^ N ) -> ( J e. ( 0..^ N ) -> ( -u N < ( I - J ) /\ ( I - J ) < N ) ) )
67	66	imp 123	1 |- ( ( I e. ( 0..^ N ) /\ J e. ( 0..^ N ) ) -> ( -u N < ( I - J ) /\ ( I - J ) < N ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   e. wcel 1480   class class class wbr 3924  (class class class)co 5767   CCcc 7611   RRcr 7612   0cc0 7613   + caddc 7616   < clt 7793   <_ cle 7794   - cmin 7926   -ucneg 7927   NNcn 8713   NN0cn0 8970  ..^cfzo 9912

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-ltadd 7729

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-inn 8714  df-n0 8971  df-z 9048  df-uz 9320  df-fz 9784  df-fzo 9913

This theorem is referenced by:  addmodlteq  10164