Theorem ubmelm1fzo 10245

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 10201	. 2 |- ( K e. ( 0..^ N ) <-> ( K e. NN0 /\ N e. NN /\ K < N ) )
2		nnz 9291	. . . . . . . . 9 |- ( N e. NN -> N e. ZZ )
3	2	adantr 276	. . . . . . . 8 |- ( ( N e. NN /\ K e. NN0 ) -> N e. ZZ )
4		nn0z 9292	. . . . . . . . 9 |- ( K e. NN0 -> K e. ZZ )
5	4	adantl 277	. . . . . . . 8 |- ( ( N e. NN /\ K e. NN0 ) -> K e. ZZ )
6	3, 5	zsubcld 9399	. . . . . . 7 |- ( ( N e. NN /\ K e. NN0 ) -> ( N - K ) e. ZZ )
7	6	ancoms 268	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> ( N - K ) e. ZZ )
8		peano2zm 9310	. . . . . 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 1019	. . . 4 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( ( N - K ) - 1 ) e. ZZ )
11		simp3 1001	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> K < N )
12	4, 2	anim12i 338	. . . . . . . 8 |- ( ( K e. NN0 /\ N e. NN ) -> ( K e. ZZ /\ N e. ZZ ) )
13	12	3adant3 1019	. . . . . . 7 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( K e. ZZ /\ N e. ZZ ) )
14		znnsub 9323	. . . . . . 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 147	. . . . 5 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( N - K ) e. NN )
17		nnm1ge0 9358	. . . . 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 9285	. . . 4 |- ( ( ( N - K ) - 1 ) e. NN0 <-> ( ( ( N - K ) - 1 ) e. ZZ /\ 0 <_ ( ( N - K ) - 1 ) ) )
20	10, 18, 19	sylanbrc 417	. . 3 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( ( N - K ) - 1 ) e. NN0 )
21		simp2 1000	. . 3 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> N e. NN )
22		nncn 8946	. . . . . . 7 |- ( N e. NN -> N e. CC )
23	22	adantl 277	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> N e. CC )
24		nn0cn 9205	. . . . . . 7 |- ( K e. NN0 -> K e. CC )
25	24	adantr 276	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> K e. CC )
26		1cnd 7992	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> 1 e. CC )
27	23, 25, 26	subsub4d 8318	. . . . 5 |- ( ( K e. NN0 /\ N e. NN ) -> ( ( N - K ) - 1 ) = ( N - ( K + 1 ) ) )
28		nn0p1gt0 9224	. . . . . . 7 |- ( K e. NN0 -> 0 < ( K + 1 ) )
29	28	adantr 276	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> 0 < ( K + 1 ) )
30		nn0re 9204	. . . . . . . 8 |- ( K e. NN0 -> K e. RR )
31		peano2re 8112	. . . . . . . 8 |- ( K e. RR -> ( K + 1 ) e. RR )
32	30, 31	syl 14	. . . . . . 7 |- ( K e. NN0 -> ( K + 1 ) e. RR )
33		nnre 8945	. . . . . . 7 |- ( N e. NN -> N e. RR )
34		ltsubpos 8430	. . . . . . 7 |- ( ( ( K + 1 ) e. RR /\ N e. RR ) -> ( 0 < ( K + 1 ) <-> ( N - ( K + 1 ) ) < N ) )
35	32, 33, 34	syl2an 289	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> ( 0 < ( K + 1 ) <-> ( N - ( K + 1 ) ) < N ) )
36	29, 35	mpbid 147	. . . . 5 |- ( ( K e. NN0 /\ N e. NN ) -> ( N - ( K + 1 ) ) < N )
37	27, 36	eqbrtrd 4040	. . . 4 |- ( ( K e. NN0 /\ N e. NN ) -> ( ( N - K ) - 1 ) < N )
38	37	3adant3 1019	. . 3 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( ( N - K ) - 1 ) < N )
39		elfzo0 10201	. . 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 1183	. 2 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( ( N - K ) - 1 ) e. ( 0..^ N ) )
41	1, 40	sylbi 121	1 |- ( K e. ( 0..^ N ) -> ( ( N - K ) - 1 ) e. ( 0..^ N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   e. wcel 2160   class class class wbr 4018  (class class class)co 5891   CCcc 7828   RRcr 7829   0cc0 7830   1c1 7831   + caddc 7833   < clt 8011   <_ cle 8012   - cmin 8147   NNcn 8938   NN0cn0 9195   ZZcz 9272  ..^cfzo 10161

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-addcom 7930  ax-addass 7932  ax-distr 7934  ax-i2m1 7935  ax-0lt1 7936  ax-0id 7938  ax-rnegex 7939  ax-cnre 7941  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-ltadd 7946

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-sub 8149  df-neg 8150  df-inn 8939  df-n0 9196  df-z 9273  df-uz 9548  df-fz 10028  df-fzo 10162

This theorem is referenced by: (None)