Theorem eluzgtdifelfzo 10432

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 9755	. . . . . . . 8 |- ( N e. ( ZZ>= ` A ) -> N e. ZZ )
6	5	ad2antrr 488	. . . . . . 7 |- ( ( ( N e. ( ZZ>= ` A ) /\ B < A ) /\ ( A e. ZZ /\ B e. ZZ ) ) -> N e. ZZ )
7		simprr 531	. . . . . . 7 |- ( ( ( N e. ( ZZ>= ` A ) /\ B < A ) /\ ( A e. ZZ /\ B e. ZZ ) ) -> B e. ZZ )
8	6, 7	zsubcld 9597	. . . . . 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 9596	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( N e. ( ZZ>= ` A ) /\ B < A ) ) -> ( A + ( N - B ) ) e. ZZ )
11		zre 9473	. . . . . . . . 9 |- ( B e. ZZ -> B e. RR )
12		zre 9473	. . . . . . . . 9 |- ( A e. ZZ -> A e. RR )
13		posdif 8625	. . . . . . . . . 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 8433	. . . . . . . . 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 9756	. . . . . . . 8 |- ( N e. ( ZZ>= ` A ) -> N e. RR )
22	21	ad2antrl 490	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( N e. ( ZZ>= ` A ) /\ B < A ) ) -> N e. RR )
23	20, 22	ltaddposd 8699	. . . . . 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 9474	. . . . . . 7 |- ( A e. ZZ -> A e. CC )
26	25	ad2antrr 488	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( N e. ( ZZ>= ` A ) /\ B < A ) ) -> A e. CC )
27		eluzelcn 9757	. . . . . . 7 |- ( N e. ( ZZ>= ` A ) -> N e. CC )
28	27	ad2antrl 490	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( N e. ( ZZ>= ` A ) /\ B < A ) ) -> N e. CC )
29		zcn 9474	. . . . . . . 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 8382	. . . . . . 7 |- ( ( A e. CC /\ N e. CC /\ B e. CC ) -> ( A + ( N - B ) ) = ( N + ( A - B ) ) )
33	32	breq2d 4098	. . . . . 6 |- ( ( A e. CC /\ N e. CC /\ B e. CC ) -> ( N < ( A + ( N - B ) ) <-> N < ( N + ( A - B ) ) ) )
34	26, 28, 31, 33	syl3anc 1271	. . . . 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 10375	. . . 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 1205	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( N e. ( ZZ>= ` A ) /\ B < A ) ) -> N e. ( A..^ ( A + ( N - B ) ) ) )
38		fzosubel3 10431	. . 3 |- ( ( N e. ( A..^ ( A + ( N - B ) ) ) /\ ( N - B ) e. ZZ ) -> ( N - A ) e. ( 0..^ ( N - B ) ) )
39	37, 9, 38	syl2anc 411	. 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 1002   e. wcel 2200   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   CCcc 8020   RRcr 8021   0cc0 8022   + caddc 8025   < clt 8204   - cmin 8340   ZZcz 9469   ZZ>=cuz 9745  ..^cfzo 10367

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-n0 9393  df-z 9470  df-uz 9746  df-fz 10234  df-fzo 10368

This theorem is referenced by:  ige2m2fzo  10433