Theorem elfzp1b 9877

Description: An integer is a member of a 0-based finite set of sequential integers iff its successor is a member of the corresponding 1-based set. (Contributed by Paul Chapman, 22-Jun-2011.)

Assertion

Ref	Expression
elfzp1b	|- ( ( K e. ZZ /\ N e. ZZ ) -> ( K e. ( 0 ... ( N - 1 ) ) <-> ( K + 1 ) e. ( 1 ... N ) ) )

Proof of Theorem elfzp1b

Step	Hyp	Ref	Expression
1		peano2z 9090	. . . 4 |- ( K e. ZZ -> ( K + 1 ) e. ZZ )
2		1z 9080	. . . . 5 |- 1 e. ZZ
3		fzsubel 9840	. . . . . 6 |- ( ( ( 1 e. ZZ /\ N e. ZZ ) /\ ( ( K + 1 ) e. ZZ /\ 1 e. ZZ ) ) -> ( ( K + 1 ) e. ( 1 ... N ) <-> ( ( K + 1 ) - 1 ) e. ( ( 1 - 1 ) ... ( N - 1 ) ) ) )
4	2, 3	mpanl1 430	. . . . 5 |- ( ( N e. ZZ /\ ( ( K + 1 ) e. ZZ /\ 1 e. ZZ ) ) -> ( ( K + 1 ) e. ( 1 ... N ) <-> ( ( K + 1 ) - 1 ) e. ( ( 1 - 1 ) ... ( N - 1 ) ) ) )
5	2, 4	mpanr2 434	. . . 4 |- ( ( N e. ZZ /\ ( K + 1 ) e. ZZ ) -> ( ( K + 1 ) e. ( 1 ... N ) <-> ( ( K + 1 ) - 1 ) e. ( ( 1 - 1 ) ... ( N - 1 ) ) ) )
6	1, 5	sylan2 284	. . 3 |- ( ( N e. ZZ /\ K e. ZZ ) -> ( ( K + 1 ) e. ( 1 ... N ) <-> ( ( K + 1 ) - 1 ) e. ( ( 1 - 1 ) ... ( N - 1 ) ) ) )
7	6	ancoms 266	. 2 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( ( K + 1 ) e. ( 1 ... N ) <-> ( ( K + 1 ) - 1 ) e. ( ( 1 - 1 ) ... ( N - 1 ) ) ) )
8		zcn 9059	. . . . 5 |- ( K e. ZZ -> K e. CC )
9		ax-1cn 7713	. . . . 5 |- 1 e. CC
10		pncan 7968	. . . . 5 |- ( ( K e. CC /\ 1 e. CC ) -> ( ( K + 1 ) - 1 ) = K )
11	8, 9, 10	sylancl 409	. . . 4 |- ( K e. ZZ -> ( ( K + 1 ) - 1 ) = K )
12		1m1e0 8789	. . . . . 6 |- ( 1 - 1 ) = 0
13	12	oveq1i 5784	. . . . 5 |- ( ( 1 - 1 ) ... ( N - 1 ) ) = ( 0 ... ( N - 1 ) )
14	13	a1i 9	. . . 4 |- ( K e. ZZ -> ( ( 1 - 1 ) ... ( N - 1 ) ) = ( 0 ... ( N - 1 ) ) )
15	11, 14	eleq12d 2210	. . 3 |- ( K e. ZZ -> ( ( ( K + 1 ) - 1 ) e. ( ( 1 - 1 ) ... ( N - 1 ) ) <-> K e. ( 0 ... ( N - 1 ) ) ) )
16	15	adantr 274	. 2 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( ( ( K + 1 ) - 1 ) e. ( ( 1 - 1 ) ... ( N - 1 ) ) <-> K e. ( 0 ... ( N - 1 ) ) ) )
17	7, 16	bitr2d 188	1 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( K e. ( 0 ... ( N - 1 ) ) <-> ( K + 1 ) e. ( 1 ... N ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480  (class class class)co 5774   CCcc 7618   0cc0 7620   1c1 7621   + caddc 7623   - cmin 7933   ZZcz 9054   ...cfz 9790

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-ltadd 7736

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-inn 8721  df-n0 8978  df-z 9055  df-fz 9791

This theorem is referenced by: (None)