Theorem elfzm1b 10323

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

Assertion

Ref	Expression
elfzm1b	|- ( ( K e. ZZ /\ N e. ZZ ) -> ( K e. ( 1 ... N ) <-> ( K - 1 ) e. ( 0 ... ( N - 1 ) ) ) )

Proof of Theorem elfzm1b

Step	Hyp	Ref	Expression
1		1z 9495	. . . 4 |- 1 e. ZZ
2		fzsubel 10285	. . . . 5 |- ( ( ( 1 e. ZZ /\ N e. ZZ ) /\ ( K e. ZZ /\ 1 e. ZZ ) ) -> ( K e. ( 1 ... N ) <-> ( K - 1 ) e. ( ( 1 - 1 ) ... ( N - 1 ) ) ) )
3	1, 2	mpanl1 434	. . . 4 |- ( ( N e. ZZ /\ ( K e. ZZ /\ 1 e. ZZ ) ) -> ( K e. ( 1 ... N ) <-> ( K - 1 ) e. ( ( 1 - 1 ) ... ( N - 1 ) ) ) )
4	1, 3	mpanr2 438	. . 3 |- ( ( N e. ZZ /\ K e. ZZ ) -> ( K e. ( 1 ... N ) <-> ( K - 1 ) e. ( ( 1 - 1 ) ... ( N - 1 ) ) ) )
5		1m1e0 9202	. . . . 5 |- ( 1 - 1 ) = 0
6	5	oveq1i 6023	. . . 4 |- ( ( 1 - 1 ) ... ( N - 1 ) ) = ( 0 ... ( N - 1 ) )
7	6	eleq2i 2296	. . 3 |- ( ( K - 1 ) e. ( ( 1 - 1 ) ... ( N - 1 ) ) <-> ( K - 1 ) e. ( 0 ... ( N - 1 ) ) )
8	4, 7	bitrdi 196	. 2 |- ( ( N e. ZZ /\ K e. ZZ ) -> ( K e. ( 1 ... N ) <-> ( K - 1 ) e. ( 0 ... ( N - 1 ) ) ) )
9	8	ancoms 268	1 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( K e. ( 1 ... N ) <-> ( K - 1 ) e. ( 0 ... ( N - 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2200  (class class class)co 6013   0cc0 8022   1c1 8023   - cmin 8340   ZZcz 9469   ...cfz 10233

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-n0 9393  df-z 9470  df-fz 10234

This theorem is referenced by:  elfzom1b  10464  bcpasc  11018