Theorem subfzo0 10639

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 10571	. . 3 |- ( I e. ( 0..^ N ) <-> ( I e. NN0 /\ N e. NN /\ I < N ) )
2		elfzo0 10571	. . . . 5 |- ( J e. ( 0..^ N ) <-> ( J e. NN0 /\ N e. NN /\ J < N ) )
3		nn0re 9551	. . . . . . . . . . . 12 |- ( I e. NN0 -> I e. RR )
4	3	adantr 276	. . . . . . . . . . 11 |- ( ( I e. NN0 /\ I < N ) -> I e. RR )
5		nnre 9290	. . . . . . . . . . . . . 14 |- ( N e. NN -> N e. RR )
6		nn0re 9551	. . . . . . . . . . . . . 14 |- ( J e. NN0 -> J e. RR )
7		resubcl 8580	. . . . . . . . . . . . . 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 1048	. . . . . . . . . . 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 9567	. . . . . . . . . . . 12 |- ( I e. NN0 -> 0 <_ I )
13	12	adantr 276	. . . . . . . . . . 11 |- ( ( I e. NN0 /\ I < N ) -> 0 <_ I )
14		posdif 8773	. . . . . . . . . . . . 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 1386	. . . . . . . . . . 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 8767	. . . . . . . . . 10 |- ( ( ( I e. RR /\ ( N - J ) e. RR ) /\ ( 0 <_ I /\ 0 < ( N - J ) ) ) -> 0 < ( I + ( N - J ) ) )
19	11, 17, 18	syl2anc 415	. . . . . . . . 9 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> 0 < ( I + ( N - J ) ) )
20		nn0cn 9552	. . . . . . . . . . . 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 9552	. . . . . . . . . . . 12 |- ( J e. NN0 -> J e. CC )
24	23	3ad2ant1 1049	. . . . . . . . . . 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 9291	. . . . . . . . . . . 12 |- ( N e. NN -> N e. CC )
27	26	3ad2ant2 1050	. . . . . . . . . . 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 8649	. . . . . . . . 9 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( ( I - J ) + N ) = ( I + ( N - J ) ) )
30	19, 29	breqtrrd 4153	. . . . . . . 8 |- ( ( ( I e. NN0 /\ I < N ) /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> 0 < ( ( I - J ) + N ) )
31	6	3ad2ant1 1049	. . . . . . . . . 10 |- ( ( J e. NN0 /\ N e. NN /\ J < N ) -> J e. RR )
32		resubcl 8580	. . . . . . . . . 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 1050	. . . . . . . . . 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 8887	. . . . . . . 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 8295	. . . . . . . . . . . . . . 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 1048	. . . . . . . . . . . . 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 1208	. . . . . . . . . . 11 |- ( ( I e. NN0 /\ ( J e. NN0 /\ N e. NN /\ J < N ) ) -> ( I e. RR /\ N e. RR /\ ( J + N ) e. RR ) )
45		nn0ge0 9567	. . . . . . . . . . . . . 14 |- ( J e. NN0 -> 0 <_ J )
46	45	3ad2ant1 1049	. . . . . . . . . . . . 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 1048	. . . . . . . . . . . . . 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 8791	. . . . . . . . . . . . 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 8744	. . . . . . . . . 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 8861	. . . . . . . 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 1047	. . 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 1009   e. wcel 2209   class class class wbr 4125  (class class class)co 6075   CCcc 8167   RRcr 8168   0cc0 8169   + caddc 8172   < clt 8350   <_ cle 8351   - cmin 8487   -ucneg 8488   NNcn 9283   NN0cn0 9542  ..^cfzo 10527

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528

This theorem is referenced by:  addmodlteq  10813