Theorem subfzo0 10198

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 10138	. . 3 |- ( I e. ( 0..^ N ) <-> ( I e. NN0 /\ N e. NN /\ I < N ) )
2		elfzo0 10138	. . . . 5 |- ( J e. ( 0..^ N ) <-> ( J e. NN0 /\ N e. NN /\ J < N ) )
3		nn0re 9144	. . . . . . . . . . . 12 |- ( I e. NN0 -> I e. RR )
4	3	adantr 274	. . . . . . . . . . 11 |- ( ( I e. NN0 /\ I < N ) -> I e. RR )
5		nnre 8885	. . . . . . . . . . . . . 14 |- ( N e. NN -> N e. RR )
6		nn0re 9144	. . . . . . . . . . . . . 14 |- ( J e. NN0 -> J e. RR )
7		resubcl 8183	. . . . . . . . . . . . . 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 1012	. . . . . . . . . . 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 9160	. . . . . . . . . . . 12 |- ( I e. NN0 -> 0 <_ I )
13	12	adantr 274	. . . . . . . . . . 11 |- ( ( I e. NN0 /\ I < N ) -> 0 <_ I )
14		posdif 8374	. . . . . . . . . . . . 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 1340	. . . . . . . . . . 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 8368	. . . . . . . . . 10 |- ( ( ( I e. RR /\ ( N - J ) e. RR ) /\ ( 0 <_ I /\ 0 < ( N - J ) ) ) -> 0 < ( I + ( N - J ) ) )
19	11, 17, 18	syl2anc 409	. . . . . . . . 9 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> 0 < ( I + ( N - J ) ) )
20		nn0cn 9145	. . . . . . . . . . . 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 9145	. . . . . . . . . . . 12 |- ( J e. NN0 -> J e. CC )
24	23	3ad2ant1 1013	. . . . . . . . . . 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 8886	. . . . . . . . . . . 12 |- ( N e. NN -> N e. CC )
27	26	3ad2ant2 1014	. . . . . . . . . . 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 8252	. . . . . . . . 9 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( ( I - J ) + N ) = ( I + ( N - J ) ) )
30	19, 29	breqtrrd 4017	. . . . . . . 8 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> 0 < ( ( I - J ) + N ) )
31	6	3ad2ant1 1013	. . . . . . . . . 10 |- ( ( J e. NN0 /\ N e. NN /\ J < N ) -> J e. RR )
32		resubcl 8183	. . . . . . . . . 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 1014	. . . . . . . . . 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 8488	. . . . . . . 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 7900	. . . . . . . . . . . . . . 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 1012	. . . . . . . . . . . . 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 1172	. . . . . . . . . . 11 |- ( ( I e. NN0 /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( I e. RR /\ N e. RR /\ ( J + N ) e. RR ) )
45		nn0ge0 9160	. . . . . . . . . . . . . 14 |- ( J e. NN0 -> 0 <_ J )
46	45	3ad2ant1 1013	. . . . . . . . . . . . 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 1012	. . . . . . . . . . . . . 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 8392	. . . . . . . . . . . . 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 8345	. . . . . . . . . 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 8462	. . . . . . . 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 1011	. . 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 973   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   + caddc 7777   < clt 7954   <_ cle 7955   - cmin 8090   -ucneg 8091   NNcn 8878   NN0cn0 9135  ..^cfzo 10098

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-csb 3050  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-iun 3875  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-1st 6119  df-2nd 6120  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  df-fzo 10099

This theorem is referenced by:  addmodlteq  10354