Theorem ubmelm1fzo 10003

Description: The result of subtracting 1 and an integer of a half-open range of nonnegative integers from the upper bound of this range is contained in this range. (Contributed by AV, 23-Mar-2018.) (Revised by AV, 30-Oct-2018.)

Assertion

Ref	Expression
ubmelm1fzo	|- ( K e. ( 0..^ N ) -> ( ( N - K ) - 1 ) e. ( 0..^ N ) )

Proof of Theorem ubmelm1fzo

Step	Hyp	Ref	Expression
1		elfzo0 9959	. 2 |- ( K e. ( 0..^ N ) <-> ( K e. NN0 /\ N e. NN /\ K < N ) )
2		nnz 9073	. . . . . . . . 9 |- ( N e. NN -> N e. ZZ )
3	2	adantr 274	. . . . . . . 8 |- ( ( N e. NN /\ K e. NN0 ) -> N e. ZZ )
4		nn0z 9074	. . . . . . . . 9 |- ( K e. NN0 -> K e. ZZ )
5	4	adantl 275	. . . . . . . 8 |- ( ( N e. NN /\ K e. NN0 ) -> K e. ZZ )
6	3, 5	zsubcld 9178	. . . . . . 7 |- ( ( N e. NN /\ K e. NN0 ) -> ( N - K ) e. ZZ )
7	6	ancoms 266	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> ( N - K ) e. ZZ )
8		peano2zm 9092	. . . . . 6 |- ( ( N - K ) e. ZZ -> ( ( N - K ) - 1 ) e. ZZ )
9	7, 8	syl 14	. . . . 5 |- ( ( K e. NN0 /\ N e. NN ) -> ( ( N - K ) - 1 ) e. ZZ )
10	9	3adant3 1001	. . . 4 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( ( N - K ) - 1 ) e. ZZ )
11		simp3 983	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> K < N )
12	4, 2	anim12i 336	. . . . . . . 8 |- ( ( K e. NN0 /\ N e. NN ) -> ( K e. ZZ /\ N e. ZZ ) )
13	12	3adant3 1001	. . . . . . 7 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( K e. ZZ /\ N e. ZZ ) )
14		znnsub 9105	. . . . . . 7 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( K < N <-> ( N - K ) e. NN ) )
15	13, 14	syl 14	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( K < N <-> ( N - K ) e. NN ) )
16	11, 15	mpbid 146	. . . . 5 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( N - K ) e. NN )
17		nnm1ge0 9137	. . . . 5 |- ( ( N - K ) e. NN -> 0 <_ ( ( N - K ) - 1 ) )
18	16, 17	syl 14	. . . 4 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> 0 <_ ( ( N - K ) - 1 ) )
19		elnn0z 9067	. . . 4 |- ( ( ( N - K ) - 1 ) e. NN0 <-> ( ( ( N - K ) - 1 ) e. ZZ /\ 0 <_ ( ( N - K ) - 1 ) ) )
20	10, 18, 19	sylanbrc 413	. . 3 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( ( N - K ) - 1 ) e. NN0 )
21		simp2 982	. . 3 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> N e. NN )
22		nncn 8728	. . . . . . 7 |- ( N e. NN -> N e. CC )
23	22	adantl 275	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> N e. CC )
24		nn0cn 8987	. . . . . . 7 |- ( K e. NN0 -> K e. CC )
25	24	adantr 274	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> K e. CC )
26		1cnd 7782	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> 1 e. CC )
27	23, 25, 26	subsub4d 8104	. . . . 5 |- ( ( K e. NN0 /\ N e. NN ) -> ( ( N - K ) - 1 ) = ( N - ( K + 1 ) ) )
28		nn0p1gt0 9006	. . . . . . 7 |- ( K e. NN0 -> 0 < ( K + 1 ) )
29	28	adantr 274	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> 0 < ( K + 1 ) )
30		nn0re 8986	. . . . . . . 8 |- ( K e. NN0 -> K e. RR )
31		peano2re 7898	. . . . . . . 8 |- ( K e. RR -> ( K + 1 ) e. RR )
32	30, 31	syl 14	. . . . . . 7 |- ( K e. NN0 -> ( K + 1 ) e. RR )
33		nnre 8727	. . . . . . 7 |- ( N e. NN -> N e. RR )
34		ltsubpos 8216	. . . . . . 7 |- ( ( ( K + 1 ) e. RR /\ N e. RR ) -> ( 0 < ( K + 1 ) <-> ( N - ( K + 1 ) ) < N ) )
35	32, 33, 34	syl2an 287	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> ( 0 < ( K + 1 ) <-> ( N - ( K + 1 ) ) < N ) )
36	29, 35	mpbid 146	. . . . 5 |- ( ( K e. NN0 /\ N e. NN ) -> ( N - ( K + 1 ) ) < N )
37	27, 36	eqbrtrd 3950	. . . 4 |- ( ( K e. NN0 /\ N e. NN ) -> ( ( N - K ) - 1 ) < N )
38	37	3adant3 1001	. . 3 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( ( N - K ) - 1 ) < N )
39		elfzo0 9959	. . 3 |- ( ( ( N - K ) - 1 ) e. ( 0..^ N ) <-> ( ( ( N - K ) - 1 ) e. NN0 /\ N e. NN /\ ( ( N - K ) - 1 ) < N ) )
40	20, 21, 38, 39	syl3anbrc 1165	. 2 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( ( N - K ) - 1 ) e. ( 0..^ N ) )
41	1, 40	sylbi 120	1 |- ( K e. ( 0..^ N ) -> ( ( N - K ) - 1 ) e. ( 0..^ N ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   e. wcel 1480   class class class wbr 3929  (class class class)co 5774   CCcc 7618   RRcr 7619   0cc0 7620   1c1 7621   + caddc 7623   < clt 7800   <_ cle 7801   - cmin 7933   NNcn 8720   NN0cn0 8977   ZZcz 9054  ..^cfzo 9919

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-csb 3004  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-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  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-uz 9327  df-fz 9791  df-fzo 9920

This theorem is referenced by: (None)