Theorem eluzgtdifelfzo 10593

Description: Membership of the difference of integers in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 17-Sep-2018.)

Assertion

Ref	Expression
eluzgtdifelfzo	|- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( N e. ( ZZ>= ` A ) /\ B < A ) -> ( N - A ) e. ( 0..^ ( N - B ) ) ) )

Proof of Theorem eluzgtdifelfzo

Step	Hyp	Ref	Expression
1		simpl 109	. . . . 5 |- ( ( N e. ( ZZ>= ` A ) /\ B < A ) -> N e. ( ZZ>= ` A ) )
2	1	adantl 277	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( N e. ( ZZ>= ` A ) /\ B < A ) ) -> N e. ( ZZ>= ` A ) )
3		simpl 109	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> A e. ZZ )
4	3	adantr 276	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( N e. ( ZZ>= ` A ) /\ B < A ) ) -> A e. ZZ )
5		eluzelz 9910	. . . . . . . 8 |- ( N e. ( ZZ>= ` A ) -> N e. ZZ )
6	5	ad2antrr 492	. . . . . . 7 |- ( ( ( N e. ( ZZ>= ` A ) /\ B < A ) /\ ( A e. ZZ /\ B e. ZZ ) ) -> N e. ZZ )
7		simprr 537	. . . . . . 7 |- ( ( ( N e. ( ZZ>= ` A ) /\ B < A ) /\ ( A e. ZZ /\ B e. ZZ ) ) -> B e. ZZ )
8	6, 7	zsubcld 9752	. . . . . 6 |- ( ( ( N e. ( ZZ>= ` A ) /\ B < A ) /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( N - B ) e. ZZ )
9	8	ancoms 268	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( N e. ( ZZ>= ` A ) /\ B < A ) ) -> ( N - B ) e. ZZ )
10	4, 9	zaddcld 9751	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( N e. ( ZZ>= ` A ) /\ B < A ) ) -> ( A + ( N - B ) ) e. ZZ )
11		zre 9627	. . . . . . . . 9 |- ( B e. ZZ -> B e. RR )
12		zre 9627	. . . . . . . . 9 |- ( A e. ZZ -> A e. RR )
13		posdif 8773	. . . . . . . . . 10 |- ( ( B e. RR /\ A e. RR ) -> ( B < A <-> 0 < ( A - B ) ) )
14	13	biimpd 144	. . . . . . . . 9 |- ( ( B e. RR /\ A e. RR ) -> ( B < A -> 0 < ( A - B ) ) )
15	11, 12, 14	syl2anr 290	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( B < A -> 0 < ( A - B ) ) )
16	15	adantld 278	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( N e. ( ZZ>= ` A ) /\ B < A ) -> 0 < ( A - B ) ) )
17	16	imp 124	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( N e. ( ZZ>= ` A ) /\ B < A ) ) -> 0 < ( A - B ) )
18		resubcl 8580	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR )
19	12, 11, 18	syl2an 289	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A - B ) e. RR )
20	19	adantr 276	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( N e. ( ZZ>= ` A ) /\ B < A ) ) -> ( A - B ) e. RR )
21		eluzelre 9911	. . . . . . . 8 |- ( N e. ( ZZ>= ` A ) -> N e. RR )
22	21	ad2antrl 494	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( N e. ( ZZ>= ` A ) /\ B < A ) ) -> N e. RR )
23	20, 22	ltaddposd 8847	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( N e. ( ZZ>= ` A ) /\ B < A ) ) -> ( 0 < ( A - B ) <-> N < ( N + ( A - B ) ) ) )
24	17, 23	mpbid 147	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( N e. ( ZZ>= ` A ) /\ B < A ) ) -> N < ( N + ( A - B ) ) )
25		zcn 9628	. . . . . . 7 |- ( A e. ZZ -> A e. CC )
26	25	ad2antrr 492	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( N e. ( ZZ>= ` A ) /\ B < A ) ) -> A e. CC )
27		eluzelcn 9912	. . . . . . 7 |- ( N e. ( ZZ>= ` A ) -> N e. CC )
28	27	ad2antrl 494	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( N e. ( ZZ>= ` A ) /\ B < A ) ) -> N e. CC )
29		zcn 9628	. . . . . . . 8 |- ( B e. ZZ -> B e. CC )
30	29	adantl 277	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> B e. CC )
31	30	adantr 276	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( N e. ( ZZ>= ` A ) /\ B < A ) ) -> B e. CC )
32		addsub12 8529	. . . . . . 7 |- ( ( A e. CC /\ N e. CC /\ B e. CC ) -> ( A + ( N - B ) ) = ( N + ( A - B ) ) )
33	32	breq2d 4137	. . . . . 6 |- ( ( A e. CC /\ N e. CC /\ B e. CC ) -> ( N < ( A + ( N - B ) ) <-> N < ( N + ( A - B ) ) ) )
34	26, 28, 31, 33	syl3anc 1278	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( N e. ( ZZ>= ` A ) /\ B < A ) ) -> ( N < ( A + ( N - B ) ) <-> N < ( N + ( A - B ) ) ) )
35	24, 34	mpbird 167	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( N e. ( ZZ>= ` A ) /\ B < A ) ) -> N < ( A + ( N - B ) ) )
36		elfzo2 10535	. . . 4 |- ( N e. ( A..^ ( A + ( N - B ) ) ) <-> ( N e. ( ZZ>= ` A ) /\ ( A + ( N - B ) ) e. ZZ /\ N < ( A + ( N - B ) ) ) )
37	2, 10, 35, 36	syl3anbrc 1212	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( N e. ( ZZ>= ` A ) /\ B < A ) ) -> N e. ( A..^ ( A + ( N - B ) ) ) )
38		fzosubel3 10592	. . 3 |- ( ( N e. ( A..^ ( A + ( N - B ) ) ) /\ ( N - B ) e. ZZ ) -> ( N - A ) e. ( 0..^ ( N - B ) ) )
39	37, 9, 38	syl2anc 415	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( N e. ( ZZ>= ` A ) /\ B < A ) ) -> ( N - A ) e. ( 0..^ ( N - B ) ) )
40	39	ex 115	1 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( N e. ( ZZ>= ` A ) /\ B < A ) -> ( N - A ) e. ( 0..^ ( N - B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   e. wcel 2209   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   CCcc 8167   RRcr 8168   0cc0 8169   + caddc 8172   < clt 8350   - cmin 8487   ZZcz 9623   ZZ>=cuz 9900  ..^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:  ige2m2fzo  10594