Theorem elfzp1b 10487

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 9663	. . . 4 |- ( K e. ZZ -> ( K + 1 ) e. ZZ )
2		1z 9653	. . . . 5 |- 1 e. ZZ
3		fzsubel 10449	. . . . . 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 438	. . . . 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 442	. . . 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 9632	. . . . 5 |- ( K e. ZZ -> K e. CC )
9		ax-1cn 8266	. . . . 5 |- 1 e. CC
10		pncan 8526	. . . . 5 |- ( ( K e. CC /\ 1 e. CC ) -> ( ( K + 1 ) - 1 ) = K )
11	8, 9, 10	sylancl 417	. . . 4 |- ( K e. ZZ -> ( ( K + 1 ) - 1 ) = K )
12		1m1e0 9356	. . . . . 6 |- ( 1 - 1 ) = 0
13	12	oveq1i 6089	. . . . 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 2309	. . 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 1402   e. wcel 2209  (class class class)co 6079   CCcc 8171   0cc0 8173   1c1 8174   + caddc 8176   - cmin 8491   ZZcz 9627   ...cfz 10394

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-iota 5335  df-fun 5377  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-inn 9288  df-n0 9547  df-z 9628  df-fz 10395

This theorem is referenced by: (None)