Theorem fzsubel 9995

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 9222	. . 3 |- ( K e. ZZ -> -u K e. ZZ )
2		fzaddel 9994	. . 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 403	. 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 9196	. . . 4 |- ( M e. ZZ -> M e. CC )
5		zcn 9196	. . . 4 |- ( N e. ZZ -> N e. CC )
6	4, 5	anim12i 336	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M e. CC /\ N e. CC ) )
7		zcn 9196	. . . 4 |- ( J e. ZZ -> J e. CC )
8		zcn 9196	. . . 4 |- ( K e. ZZ -> K e. CC )
9	7, 8	anim12i 336	. . 3 |- ( ( J e. ZZ /\ K e. ZZ ) -> ( J e. CC /\ K e. CC ) )
10		negsub 8146	. . . . 5 |- ( ( J e. CC /\ K e. CC ) -> ( J + -u K ) = ( J - K ) )
11	10	adantl 275	. . . 4 |- ( ( ( M e. CC /\ N e. CC ) /\ ( J e. CC /\ K e. CC ) ) -> ( J + -u K ) = ( J - K ) )
12		negsub 8146	. . . . . . 7 |- ( ( M e. CC /\ K e. CC ) -> ( M + -u K ) = ( M - K ) )
13		negsub 8146	. . . . . . 7 |- ( ( N e. CC /\ K e. CC ) -> ( N + -u K ) = ( N - K ) )
14	12, 13	oveqan12d 5861	. . . . . 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 583	. . . . 5 |- ( ( ( M e. CC /\ N e. CC ) /\ K e. CC ) -> ( ( M + -u K ) ... ( N + -u K ) ) = ( ( M - K ) ... ( N - K ) ) )
16	15	adantrl 470	. . . 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 2237	. . 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 287	. 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 187	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 103   <-> wb 104   = wceq 1343   e. wcel 2136  (class class class)co 5842   CCcc 7751   + caddc 7756   - cmin 8069   -ucneg 8070   ZZcz 9191   ...cfz 9944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-fz 9945

This theorem is referenced by:  elfzp1b  10032  elfzm1b  10033  fisum0diag2  11388