Theorem subfzo0 10228

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 10168	. . 3 |- ( I e. ( 0..^ N ) <-> ( I e. NN0 /\ N e. NN /\ I < N ) )
2		elfzo0 10168	. . . . 5 |- ( J e. ( 0..^ N ) <-> ( J e. NN0 /\ N e. NN /\ J < N ) )
3		nn0re 9174	. . . . . . . . . . . 12 |- ( I e. NN0 -> I e. RR )
4	3	adantr 276	. . . . . . . . . . 11 |- ( ( I e. NN0 /\ I < N ) -> I e. RR )
5		nnre 8915	. . . . . . . . . . . . . 14 |- ( N e. NN -> N e. RR )
6		nn0re 9174	. . . . . . . . . . . . . 14 |- ( J e. NN0 -> J e. RR )
7		resubcl 8211	. . . . . . . . . . . . . 14 |- ( ( N e. RR /\ J e. RR ) -> ( N - J ) e. RR )
8	5, 6, 7	syl2an 289	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ J e. NN0 ) -> ( N - J ) e. RR )
9	8	ancoms 268	. . . . . . . . . . . 12 |- ( ( J e. NN0 /\ N e. NN ) -> ( N - J ) e. RR )
10	9	3adant3 1017	. . . . . . . . . . 11 |- ( ( J e. NN0 /\ N e. NN /\ J < N ) -> ( N - J ) e. RR )
11	4, 10	anim12i 338	. . . . . . . . . 10 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( I e. RR /\ ( N - J ) e. RR ) )
12		nn0ge0 9190	. . . . . . . . . . . 12 |- ( I e. NN0 -> 0 <_ I )
13	12	adantr 276	. . . . . . . . . . 11 |- ( ( I e. NN0 /\ I < N ) -> 0 <_ I )
14		posdif 8402	. . . . . . . . . . . . 13 |- ( ( J e. RR /\ N e. RR ) -> ( J < N <-> 0 < ( N - J ) ) )
15	6, 5, 14	syl2an 289	. . . . . . . . . . . 12 |- ( ( J e. NN0 /\ N e. NN ) -> ( J < N <-> 0 < ( N - J ) ) )
16	15	biimp3a 1345	. . . . . . . . . . 11 |- ( ( J e. NN0 /\ N e. NN /\ J < N ) -> 0 < ( N - J ) )
17	13, 16	anim12i 338	. . . . . . . . . 10 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( 0 <_ I /\ 0 < ( N - J ) ) )
18		addgegt0 8396	. . . . . . . . . 10 |- ( ( ( I e. RR /\ ( N - J ) e. RR ) /\ ( 0 <_ I /\ 0 < ( N - J ) ) ) -> 0 < ( I + ( N - J ) ) )
19	11, 17, 18	syl2anc 411	. . . . . . . . 9 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> 0 < ( I + ( N - J ) ) )
20		nn0cn 9175	. . . . . . . . . . . 12 |- ( I e. NN0 -> I e. CC )
21	20	adantr 276	. . . . . . . . . . 11 |- ( ( I e. NN0 /\ I < N ) -> I e. CC )
22	21	adantr 276	. . . . . . . . . 10 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> I e. CC )
23		nn0cn 9175	. . . . . . . . . . . 12 |- ( J e. NN0 -> J e. CC )
24	23	3ad2ant1 1018	. . . . . . . . . . 11 |- ( ( J e. NN0 /\ N e. NN /\ J < N ) -> J e. CC )
25	24	adantl 277	. . . . . . . . . 10 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> J e. CC )
26		nncn 8916	. . . . . . . . . . . 12 |- ( N e. NN -> N e. CC )
27	26	3ad2ant2 1019	. . . . . . . . . . 11 |- ( ( J e. NN0 /\ N e. NN /\ J < N ) -> N e. CC )
28	27	adantl 277	. . . . . . . . . 10 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> N e. CC )
29	22, 25, 28	subadd23d 8280	. . . . . . . . 9 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( ( I - J ) + N ) = ( I + ( N - J ) ) )
30	19, 29	breqtrrd 4028	. . . . . . . 8 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> 0 < ( ( I - J ) + N ) )
31	6	3ad2ant1 1018	. . . . . . . . . 10 |- ( ( J e. NN0 /\ N e. NN /\ J < N ) -> J e. RR )
32		resubcl 8211	. . . . . . . . . 10 |- ( ( I e. RR /\ J e. RR ) -> ( I - J ) e. RR )
33	4, 31, 32	syl2an 289	. . . . . . . . 9 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( I - J ) e. RR )
34	5	3ad2ant2 1019	. . . . . . . . . 10 |- ( ( J e. NN0 /\ N e. NN /\ J < N ) -> N e. RR )
35	34	adantl 277	. . . . . . . . 9 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> N e. RR )
36	33, 35	possumd 8516	. . . . . . . 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 147	. . . . . . 7 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> -u N < ( I - J ) )
38	3	adantr 276	. . . . . . . . . . . 12 |- ( ( I e. NN0 /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> I e. RR )
39	34	adantl 277	. . . . . . . . . . . 12 |- ( ( I e. NN0 /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> N e. RR )
40		readdcl 7928	. . . . . . . . . . . . . . 15 |- ( ( J e. RR /\ N e. RR ) -> ( J + N ) e. RR )
41	6, 5, 40	syl2an 289	. . . . . . . . . . . . . 14 |- ( ( J e. NN0 /\ N e. NN ) -> ( J + N ) e. RR )
42	41	3adant3 1017	. . . . . . . . . . . . 13 |- ( ( J e. NN0 /\ N e. NN /\ J < N ) -> ( J + N ) e. RR )
43	42	adantl 277	. . . . . . . . . . . 12 |- ( ( I e. NN0 /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( J + N ) e. RR )
44	38, 39, 43	3jca 1177	. . . . . . . . . . 11 |- ( ( I e. NN0 /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( I e. RR /\ N e. RR /\ ( J + N ) e. RR ) )
45		nn0ge0 9190	. . . . . . . . . . . . . 14 |- ( J e. NN0 -> 0 <_ J )
46	45	3ad2ant1 1018	. . . . . . . . . . . . 13 |- ( ( J e. NN0 /\ N e. NN /\ J < N ) -> 0 <_ J )
47	46	adantl 277	. . . . . . . . . . . 12 |- ( ( I e. NN0 /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> 0 <_ J )
48	5, 6	anim12i 338	. . . . . . . . . . . . . . . 16 |- ( ( N e. NN /\ J e. NN0 ) -> ( N e. RR /\ J e. RR ) )
49	48	ancoms 268	. . . . . . . . . . . . . . 15 |- ( ( J e. NN0 /\ N e. NN ) -> ( N e. RR /\ J e. RR ) )
50	49	3adant3 1017	. . . . . . . . . . . . . 14 |- ( ( J e. NN0 /\ N e. NN /\ J < N ) -> ( N e. RR /\ J e. RR ) )
51	50	adantl 277	. . . . . . . . . . . . 13 |- ( ( I e. NN0 /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( N e. RR /\ J e. RR ) )
52		addge02 8420	. . . . . . . . . . . . 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 147	. . . . . . . . . . 11 |- ( ( I e. NN0 /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> N <_ ( J + N ) )
55	44, 54	lelttrdi 8373	. . . . . . . . . 10 |- ( ( I e. NN0 /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( I < N -> I < ( J + N ) ) )
56	55	impancom 260	. . . . . . . . 9 |- ( ( I e. NN0 /\ I < N ) -> ( ( J e. NN0 /\ N e. NN /\ J < N ) -> I < ( J + N ) ) )
57	56	imp 124	. . . . . . . 8 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> I < ( J + N ) )
58	4	adantr 276	. . . . . . . . 9 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> I e. RR )
59	31	adantl 277	. . . . . . . . 9 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> J e. RR )
60	58, 59, 35	ltsubadd2d 8490	. . . . . . . 8 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( ( I - J ) < N <-> I < ( J + N ) ) )
61	57, 60	mpbird 167	. . . . . . 7 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( I - J ) < N )
62	37, 61	jca 306	. . . . . 6 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( -u N < ( I - J ) /\ ( I - J ) < N ) )
63	62	ex 115	. . . . 5 |- ( ( I e. NN0 /\ I < N ) -> ( ( J e. NN0 /\ N e. NN /\ J < N ) -> ( -u N < ( I - J ) /\ ( I - J ) < N ) ) )
64	2, 63	biimtrid 152	. . . 4 |- ( ( I e. NN0 /\ I < N ) -> ( J e. ( 0..^ N ) -> ( -u N < ( I - J ) /\ ( I - J ) < N ) ) )
65	64	3adant2 1016	. . 3 |- ( ( I e. NN0 /\ N e. NN /\ I < N ) -> ( J e. ( 0..^ N ) -> ( -u N < ( I - J ) /\ ( I - J ) < N ) ) )
66	1, 65	sylbi 121	. 2 |- ( I e. ( 0..^ N ) -> ( J e. ( 0..^ N ) -> ( -u N < ( I - J ) /\ ( I - J ) < N ) ) )
67	66	imp 124	1 |- ( ( I e. ( 0..^ N ) /\ J e. ( 0..^ N ) ) -> ( -u N < ( I - J ) /\ ( I - J ) < N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   e. wcel 2148   class class class wbr 4000  (class class class)co 5869   CCcc 7800   RRcr 7801   0cc0 7802   + caddc 7805   < clt 7982   <_ cle 7983   - cmin 8118   -ucneg 8119   NNcn 8908   NN0cn0 9165  ..^cfzo 10128

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 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-ltadd 7918

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-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518  df-fz 9996  df-fzo 10129

This theorem is referenced by:  addmodlteq  10384