Theorem fzsubel 10062

Description: Membership of a difference in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)

Assertion

Ref	Expression
fzsubel	|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( J e. ZZ /\ K e. ZZ ) ) -> ( J e. ( M ... N ) <-> ( J - K ) e. ( ( M - K ) ... ( N - K ) ) ) )

Proof of Theorem fzsubel

Step	Hyp	Ref	Expression
1		znegcl 9286	. . 3 |- ( K e. ZZ -> -u K e. ZZ )
2		fzaddel 10061	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( J e. ZZ /\ -u K e. ZZ ) ) -> ( J e. ( M ... N ) <-> ( J + -u K ) e. ( ( M + -u K ) ... ( N + -u K ) ) ) )
3	1, 2	sylanr2 405	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( J e. ZZ /\ K e. ZZ ) ) -> ( J e. ( M ... N ) <-> ( J + -u K ) e. ( ( M + -u K ) ... ( N + -u K ) ) ) )
4		zcn 9260	. . . 4 |- ( M e. ZZ -> M e. CC )
5		zcn 9260	. . . 4 |- ( N e. ZZ -> N e. CC )
6	4, 5	anim12i 338	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M e. CC /\ N e. CC ) )
7		zcn 9260	. . . 4 |- ( J e. ZZ -> J e. CC )
8		zcn 9260	. . . 4 |- ( K e. ZZ -> K e. CC )
9	7, 8	anim12i 338	. . 3 |- ( ( J e. ZZ /\ K e. ZZ ) -> ( J e. CC /\ K e. CC ) )
10		negsub 8207	. . . . 5 |- ( ( J e. CC /\ K e. CC ) -> ( J + -u K ) = ( J - K ) )
11	10	adantl 277	. . . 4 |- ( ( ( M e. CC /\ N e. CC ) /\ ( J e. CC /\ K e. CC ) ) -> ( J + -u K ) = ( J - K ) )
12		negsub 8207	. . . . . . 7 |- ( ( M e. CC /\ K e. CC ) -> ( M + -u K ) = ( M - K ) )
13		negsub 8207	. . . . . . 7 |- ( ( N e. CC /\ K e. CC ) -> ( N + -u K ) = ( N - K ) )
14	12, 13	oveqan12d 5896	. . . . . 6 |- ( ( ( M e. CC /\ K e. CC ) /\ ( N e. CC /\ K e. CC ) ) -> ( ( M + -u K ) ... ( N + -u K ) ) = ( ( M - K ) ... ( N - K ) ) )
15	14	anandirs 593	. . . . 5 |- ( ( ( M e. CC /\ N e. CC ) /\ K e. CC ) -> ( ( M + -u K ) ... ( N + -u K ) ) = ( ( M - K ) ... ( N - K ) ) )
16	15	adantrl 478	. . . 4 |- ( ( ( M e. CC /\ N e. CC ) /\ ( J e. CC /\ K e. CC ) ) -> ( ( M + -u K ) ... ( N + -u K ) ) = ( ( M - K ) ... ( N - K ) ) )
17	11, 16	eleq12d 2248	. . 3 |- ( ( ( M e. CC /\ N e. CC ) /\ ( J e. CC /\ K e. CC ) ) -> ( ( J + -u K ) e. ( ( M + -u K ) ... ( N + -u K ) ) <-> ( J - K ) e. ( ( M - K ) ... ( N - K ) ) ) )
18	6, 9, 17	syl2an 289	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( J e. ZZ /\ K e. ZZ ) ) -> ( ( J + -u K ) e. ( ( M + -u K ) ... ( N + -u K ) ) <-> ( J - K ) e. ( ( M - K ) ... ( N - K ) ) ) )
19	3, 18	bitrd 188	1 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( J e. ZZ /\ K e. ZZ ) ) -> ( J e. ( M ... N ) <-> ( J - K ) e. ( ( M - K ) ... ( N - K ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148  (class class class)co 5877   CCcc 7811   + caddc 7816   - cmin 8130   -ucneg 8131   ZZcz 9255   ...cfz 10010

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 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-ltadd 7929

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 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-z 9256  df-fz 10011

This theorem is referenced by:  elfzp1b  10099  elfzm1b  10100  fisum0diag2  11457