Theorem ubmelm1fzo 10229

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 10185	. 2 |- ( K e. ( 0..^ N ) <-> ( K e. NN0 /\ N e. NN /\ K < N ) )
2		nnz 9275	. . . . . . . . 9 |- ( N e. NN -> N e. ZZ )
3	2	adantr 276	. . . . . . . 8 |- ( ( N e. NN /\ K e. NN0 ) -> N e. ZZ )
4		nn0z 9276	. . . . . . . . 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 9383	. . . . . . 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 9294	. . . . . 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 1017	. . . 4 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( ( N - K ) - 1 ) e. ZZ )
11		simp3 999	. . . . . 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 1017	. . . . . . 7 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( K e. ZZ /\ N e. ZZ ) )
14		znnsub 9307	. . . . . . 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 9342	. . . . 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 9269	. . . 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 998	. . 3 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> N e. NN )
22		nncn 8930	. . . . . . 7 |- ( N e. NN -> N e. CC )
23	22	adantl 277	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> N e. CC )
24		nn0cn 9189	. . . . . . 7 |- ( K e. NN0 -> K e. CC )
25	24	adantr 276	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> K e. CC )
26		1cnd 7976	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> 1 e. CC )
27	23, 25, 26	subsub4d 8302	. . . . 5 |- ( ( K e. NN0 /\ N e. NN ) -> ( ( N - K ) - 1 ) = ( N - ( K + 1 ) ) )
28		nn0p1gt0 9208	. . . . . . 7 |- ( K e. NN0 -> 0 < ( K + 1 ) )
29	28	adantr 276	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> 0 < ( K + 1 ) )
30		nn0re 9188	. . . . . . . 8 |- ( K e. NN0 -> K e. RR )
31		peano2re 8096	. . . . . . . 8 |- ( K e. RR -> ( K + 1 ) e. RR )
32	30, 31	syl 14	. . . . . . 7 |- ( K e. NN0 -> ( K + 1 ) e. RR )
33		nnre 8929	. . . . . . 7 |- ( N e. NN -> N e. RR )
34		ltsubpos 8414	. . . . . . 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 4027	. . . 4 |- ( ( K e. NN0 /\ N e. NN ) -> ( ( N - K ) - 1 ) < N )
38	37	3adant3 1017	. . 3 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( ( N - K ) - 1 ) < N )
39		elfzo0 10185	. . 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 1181	. 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 978   e. wcel 2148   class class class wbr 4005  (class class class)co 5878   CCcc 7812   RRcr 7813   0cc0 7814   1c1 7815   + caddc 7817   < clt 7995   <_ cle 7996   - cmin 8131   NNcn 8922   NN0cn0 9179   ZZcz 9256  ..^cfzo 10145

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-addcom 7914  ax-addass 7916  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-0id 7922  ax-rnegex 7923  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-ltadd 7930

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-inn 8923  df-n0 9180  df-z 9257  df-uz 9532  df-fz 10012  df-fzo 10146

This theorem is referenced by: (None)