Theorem fzsubel 10009

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 9236	. . 3 |- ( K e. ZZ -> -u K e. ZZ )
2		fzaddel 10008	. . 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 9210	. . . 4 |- ( M e. ZZ -> M e. CC )
5		zcn 9210	. . . 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 9210	. . . 4 |- ( J e. ZZ -> J e. CC )
8		zcn 9210	. . . 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 8160	. . . . 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 8160	. . . . . . 7 |- ( ( M e. CC /\ K e. CC ) -> ( M + -u K ) = ( M - K ) )
13		negsub 8160	. . . . . . 7 |- ( ( N e. CC /\ K e. CC ) -> ( N + -u K ) = ( N - K ) )
14	12, 13	oveqan12d 5870	. . . . . 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 588	. . . . 5 |- ( ( ( M e. CC /\ N e. CC ) /\ K e. CC ) -> ( ( M + -u K ) ... ( N + -u K ) ) = ( ( M - K ) ... ( N - K ) ) )
16	15	adantrl 475	. . . 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 2241	. . 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 1348   e. wcel 2141  (class class class)co 5851   CCcc 7765   + caddc 7770   - cmin 8083   -ucneg 8084   ZZcz 9205   ...cfz 9958

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7858  ax-resscn 7859  ax-1cn 7860  ax-1re 7861  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-addcom 7867  ax-addass 7869  ax-distr 7871  ax-i2m1 7872  ax-0lt1 7873  ax-0id 7875  ax-rnegex 7876  ax-cnre 7878  ax-pre-ltirr 7879  ax-pre-ltwlin 7880  ax-pre-lttrn 7881  ax-pre-ltadd 7883

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-pnf 7949  df-mnf 7950  df-xr 7951  df-ltxr 7952  df-le 7953  df-sub 8085  df-neg 8086  df-inn 8872  df-n0 9129  df-z 9206  df-fz 9959

This theorem is referenced by:  elfzp1b  10046  elfzm1b  10047  fisum0diag2  11403