Theorem elfzp1b 10053

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 9248	. . . 4 |- ( K e. ZZ -> ( K + 1 ) e. ZZ )
2		1z 9238	. . . . 5 |- 1 e. ZZ
3		fzsubel 10016	. . . . . 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 432	. . . . 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 436	. . . 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 9217	. . . . 5 |- ( K e. ZZ -> K e. CC )
9		ax-1cn 7867	. . . . 5 |- 1 e. CC
10		pncan 8125	. . . . 5 |- ( ( K e. CC /\ 1 e. CC ) -> ( ( K + 1 ) - 1 ) = K )
11	8, 9, 10	sylancl 411	. . . 4 |- ( K e. ZZ -> ( ( K + 1 ) - 1 ) = K )
12		1m1e0 8947	. . . . . 6 |- ( 1 - 1 ) = 0
13	12	oveq1i 5863	. . . . 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 2241	. . 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 1348   e. wcel 2141  (class class class)co 5853   CCcc 7772   0cc0 7774   1c1 7775   + caddc 7777   - cmin 8090   ZZcz 9212   ...cfz 9965

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 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

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 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-fz 9966

This theorem is referenced by: (None)