Theorem fzsubel 10394

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 9608	. . 3 |- ( K e. ZZ -> -u K e. ZZ )
2		fzaddel 10393	. . 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 9582	. . . 4 |- ( M e. ZZ -> M e. CC )
5		zcn 9582	. . . 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 9582	. . . 4 |- ( J e. ZZ -> J e. CC )
8		zcn 9582	. . . 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 8521	. . . . 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 8521	. . . . . . 7 |- ( ( M e. CC /\ K e. CC ) -> ( M + -u K ) = ( M - K ) )
13		negsub 8521	. . . . . . 7 |- ( ( N e. CC /\ K e. CC ) -> ( N + -u K ) = ( N - K ) )
14	12, 13	oveqan12d 6069	. . . . . 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 597	. . . . 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 2303	. . 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 1398   e. wcel 2203  (class class class)co 6050   CCcc 8125   + caddc 8130   - cmin 8444   -ucneg 8445   ZZcz 9577   ...cfz 10342

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-fz 10343

This theorem is referenced by:  elfzp1b  10431  elfzm1b  10432  fisum0diag2  12133