Theorem elfzp1b 10430

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 9612	. . . 4 |- ( K e. ZZ -> ( K + 1 ) e. ZZ )
2		1z 9602	. . . . 5 |- 1 e. ZZ
3		fzsubel 10393	. . . . . 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 434	. . . . 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 438	. . . 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 286	. . 3 |- ( ( N e. ZZ /\ K e. ZZ ) -> ( ( K + 1 ) e. ( 1 ... N ) <-> ( ( K + 1 ) - 1 ) e. ( ( 1 - 1 ) ... ( N - 1 ) ) ) )
7	6	ancoms 268	. 2 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( ( K + 1 ) e. ( 1 ... N ) <-> ( ( K + 1 ) - 1 ) e. ( ( 1 - 1 ) ... ( N - 1 ) ) ) )
8		zcn 9581	. . . . 5 |- ( K e. ZZ -> K e. CC )
9		ax-1cn 8219	. . . . 5 |- 1 e. CC
10		pncan 8478	. . . . 5 |- ( ( K e. CC /\ 1 e. CC ) -> ( ( K + 1 ) - 1 ) = K )
11	8, 9, 10	sylancl 413	. . . 4 |- ( K e. ZZ -> ( ( K + 1 ) - 1 ) = K )
12		1m1e0 9305	. . . . . 6 |- ( 1 - 1 ) = 0
13	12	oveq1i 6059	. . . . 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 2303	. . 3 |- ( K e. ZZ -> ( ( ( K + 1 ) - 1 ) e. ( ( 1 - 1 ) ... ( N - 1 ) ) <-> K e. ( 0 ... ( N - 1 ) ) ) )
16	15	adantr 276	. 2 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( ( ( K + 1 ) - 1 ) e. ( ( 1 - 1 ) ... ( N - 1 ) ) <-> K e. ( 0 ... ( N - 1 ) ) ) )
17	7, 16	bitr2d 189	1 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( K e. ( 0 ... ( N - 1 ) ) <-> ( K + 1 ) e. ( 1 ... N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203  (class class class)co 6049   CCcc 8124   0cc0 8126   1c1 8127   + caddc 8129   - cmin 8443   ZZcz 9576   ...cfz 10341

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 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-addass 8228  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-0id 8234  ax-rnegex 8235  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-ltadd 8242

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 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-iota 5311  df-fun 5353  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-inn 9237  df-n0 9496  df-z 9577  df-fz 10342

This theorem is referenced by: (None)