Theorem ubmelm1fzo 10474

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 10423	. 2 |- ( K e. ( 0..^ N ) <-> ( K e. NN0 /\ N e. NN /\ K < N ) )
2		nnz 9500	. . . . . . . . 9 |- ( N e. NN -> N e. ZZ )
3	2	adantr 276	. . . . . . . 8 |- ( ( N e. NN /\ K e. NN0 ) -> N e. ZZ )
4		nn0z 9501	. . . . . . . . 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 9609	. . . . . . 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 9519	. . . . . 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 1043	. . . 4 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( ( N - K ) - 1 ) e. ZZ )
11		simp3 1025	. . . . . 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 1043	. . . . . . 7 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( K e. ZZ /\ N e. ZZ ) )
14		znnsub 9533	. . . . . . 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 9568	. . . . 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 9494	. . . 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 1024	. . 3 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> N e. NN )
22		nncn 9153	. . . . . . 7 |- ( N e. NN -> N e. CC )
23	22	adantl 277	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> N e. CC )
24		nn0cn 9414	. . . . . . 7 |- ( K e. NN0 -> K e. CC )
25	24	adantr 276	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> K e. CC )
26		1cnd 8197	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> 1 e. CC )
27	23, 25, 26	subsub4d 8523	. . . . 5 |- ( ( K e. NN0 /\ N e. NN ) -> ( ( N - K ) - 1 ) = ( N - ( K + 1 ) ) )
28		nn0p1gt0 9433	. . . . . . 7 |- ( K e. NN0 -> 0 < ( K + 1 ) )
29	28	adantr 276	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> 0 < ( K + 1 ) )
30		nn0re 9413	. . . . . . . 8 |- ( K e. NN0 -> K e. RR )
31		peano2re 8317	. . . . . . . 8 |- ( K e. RR -> ( K + 1 ) e. RR )
32	30, 31	syl 14	. . . . . . 7 |- ( K e. NN0 -> ( K + 1 ) e. RR )
33		nnre 9152	. . . . . . 7 |- ( N e. NN -> N e. RR )
34		ltsubpos 8636	. . . . . . 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 4109	. . . 4 |- ( ( K e. NN0 /\ N e. NN ) -> ( ( N - K ) - 1 ) < N )
38	37	3adant3 1043	. . 3 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( ( N - K ) - 1 ) < N )
39		elfzo0 10423	. . 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 1207	. 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 1004   e. wcel 2201   class class class wbr 4087  (class class class)co 6020   CCcc 8032   RRcr 8033   0cc0 8034   1c1 8035   + caddc 8037   < clt 8216   <_ cle 8217   - cmin 8352   NNcn 9145   NN0cn0 9404   ZZcz 9481  ..^cfzo 10379

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4206  ax-pow 4263  ax-pr 4298  ax-un 4529  ax-setind 4634  ax-cnex 8125  ax-resscn 8126  ax-1cn 8127  ax-1re 8128  ax-icn 8129  ax-addcl 8130  ax-addrcl 8131  ax-mulcl 8132  ax-addcom 8134  ax-addass 8136  ax-distr 8138  ax-i2m1 8139  ax-0lt1 8140  ax-0id 8142  ax-rnegex 8143  ax-cnre 8145  ax-pre-ltirr 8146  ax-pre-ltwlin 8147  ax-pre-lttrn 8148  ax-pre-ltadd 8150

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-pw 3653  df-sn 3674  df-pr 3675  df-op 3677  df-uni 3893  df-int 3928  df-iun 3971  df-br 4088  df-opab 4150  df-mpt 4151  df-id 4389  df-xp 4730  df-rel 4731  df-cnv 4732  df-co 4733  df-dm 4734  df-rn 4735  df-res 4736  df-ima 4737  df-iota 5285  df-fun 5327  df-fn 5328  df-f 5329  df-fv 5333  df-riota 5973  df-ov 6023  df-oprab 6024  df-mpo 6025  df-1st 6305  df-2nd 6306  df-pnf 8218  df-mnf 8219  df-xr 8220  df-ltxr 8221  df-le 8222  df-sub 8354  df-neg 8355  df-inn 9146  df-n0 9405  df-z 9482  df-uz 9758  df-fz 10246  df-fzo 10380

This theorem is referenced by: (None)