Theorem fzofzim 10264

Description: If a nonnegative integer in a finite interval of integers is not the upper bound of the interval, it is contained in the corresponding half-open integer range. (Contributed by Alexander van der Vekens, 15-Jun-2018.)

Assertion

Ref	Expression
fzofzim	|- ( ( K =/= M /\ K e. ( 0 ... M ) ) -> K e. ( 0..^ M ) )

Proof of Theorem fzofzim

Step	Hyp	Ref	Expression
1		elfz2nn0 10187	. . . 4 |- ( K e. ( 0 ... M ) <-> ( K e. NN0 /\ M e. NN0 /\ K <_ M ) )
2		simpl1 1002	. . . . . 6 |- ( ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) /\ K =/= M ) -> K e. NN0 )
3		necom 2451	. . . . . . . . 9 |- ( K =/= M <-> M =/= K )
4		nn0z 9346	. . . . . . . . . . . . 13 |- ( K e. NN0 -> K e. ZZ )
5		nn0z 9346	. . . . . . . . . . . . 13 |- ( M e. NN0 -> M e. ZZ )
6		zltlen 9404	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( K < M <-> ( K <_ M /\ M =/= K ) ) )
7	4, 5, 6	syl2an 289	. . . . . . . . . . . 12 |- ( ( K e. NN0 /\ M e. NN0 ) -> ( K < M <-> ( K <_ M /\ M =/= K ) ) )
8	7	bicomd 141	. . . . . . . . . . 11 |- ( ( K e. NN0 /\ M e. NN0 ) -> ( ( K <_ M /\ M =/= K ) <-> K < M ) )
9		elnn0z 9339	. . . . . . . . . . . . 13 |- ( K e. NN0 <-> ( K e. ZZ /\ 0 <_ K ) )
10		0red 8027	. . . . . . . . . . . . . . . . 17 |- ( ( K e. ZZ /\ M e. NN0 ) -> 0 e. RR )
11		zre 9330	. . . . . . . . . . . . . . . . . 18 |- ( K e. ZZ -> K e. RR )
12	11	adantr 276	. . . . . . . . . . . . . . . . 17 |- ( ( K e. ZZ /\ M e. NN0 ) -> K e. RR )
13		nn0re 9258	. . . . . . . . . . . . . . . . . 18 |- ( M e. NN0 -> M e. RR )
14	13	adantl 277	. . . . . . . . . . . . . . . . 17 |- ( ( K e. ZZ /\ M e. NN0 ) -> M e. RR )
15		lelttr 8115	. . . . . . . . . . . . . . . . 17 |- ( ( 0 e. RR /\ K e. RR /\ M e. RR ) -> ( ( 0 <_ K /\ K < M ) -> 0 < M ) )
16	10, 12, 14, 15	syl3anc 1249	. . . . . . . . . . . . . . . 16 |- ( ( K e. ZZ /\ M e. NN0 ) -> ( ( 0 <_ K /\ K < M ) -> 0 < M ) )
17		elnnz 9336	. . . . . . . . . . . . . . . . . . 19 |- ( M e. NN <-> ( M e. ZZ /\ 0 < M ) )
18	17	simplbi2 385	. . . . . . . . . . . . . . . . . 18 |- ( M e. ZZ -> ( 0 < M -> M e. NN ) )
19	5, 18	syl 14	. . . . . . . . . . . . . . . . 17 |- ( M e. NN0 -> ( 0 < M -> M e. NN ) )
20	19	adantl 277	. . . . . . . . . . . . . . . 16 |- ( ( K e. ZZ /\ M e. NN0 ) -> ( 0 < M -> M e. NN ) )
21	16, 20	syld 45	. . . . . . . . . . . . . . 15 |- ( ( K e. ZZ /\ M e. NN0 ) -> ( ( 0 <_ K /\ K < M ) -> M e. NN ) )
22	21	expd 258	. . . . . . . . . . . . . 14 |- ( ( K e. ZZ /\ M e. NN0 ) -> ( 0 <_ K -> ( K < M -> M e. NN ) ) )
23	22	impancom 260	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ 0 <_ K ) -> ( M e. NN0 -> ( K < M -> M e. NN ) ) )
24	9, 23	sylbi 121	. . . . . . . . . . . 12 |- ( K e. NN0 -> ( M e. NN0 -> ( K < M -> M e. NN ) ) )
25	24	imp 124	. . . . . . . . . . 11 |- ( ( K e. NN0 /\ M e. NN0 ) -> ( K < M -> M e. NN ) )
26	8, 25	sylbid 150	. . . . . . . . . 10 |- ( ( K e. NN0 /\ M e. NN0 ) -> ( ( K <_ M /\ M =/= K ) -> M e. NN ) )
27	26	expd 258	. . . . . . . . 9 |- ( ( K e. NN0 /\ M e. NN0 ) -> ( K <_ M -> ( M =/= K -> M e. NN ) ) )
28	3, 27	syl7bi 165	. . . . . . . 8 |- ( ( K e. NN0 /\ M e. NN0 ) -> ( K <_ M -> ( K =/= M -> M e. NN ) ) )
29	28	3impia 1202	. . . . . . 7 |- ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) -> ( K =/= M -> M e. NN ) )
30	29	imp 124	. . . . . 6 |- ( ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) /\ K =/= M ) -> M e. NN )
31	8	biimpd 144	. . . . . . . . . 10 |- ( ( K e. NN0 /\ M e. NN0 ) -> ( ( K <_ M /\ M =/= K ) -> K < M ) )
32	31	exp4b 367	. . . . . . . . 9 |- ( K e. NN0 -> ( M e. NN0 -> ( K <_ M -> ( M =/= K -> K < M ) ) ) )
33	32	3imp 1195	. . . . . . . 8 |- ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) -> ( M =/= K -> K < M ) )
34	3, 33	biimtrid 152	. . . . . . 7 |- ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) -> ( K =/= M -> K < M ) )
35	34	imp 124	. . . . . 6 |- ( ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) /\ K =/= M ) -> K < M )
36	2, 30, 35	3jca 1179	. . . . 5 |- ( ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) /\ K =/= M ) -> ( K e. NN0 /\ M e. NN /\ K < M ) )
37	36	ex 115	. . . 4 |- ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) -> ( K =/= M -> ( K e. NN0 /\ M e. NN /\ K < M ) ) )
38	1, 37	sylbi 121	. . 3 |- ( K e. ( 0 ... M ) -> ( K =/= M -> ( K e. NN0 /\ M e. NN /\ K < M ) ) )
39	38	impcom 125	. 2 |- ( ( K =/= M /\ K e. ( 0 ... M ) ) -> ( K e. NN0 /\ M e. NN /\ K < M ) )
40		elfzo0 10258	. 2 |- ( K e. ( 0..^ M ) <-> ( K e. NN0 /\ M e. NN /\ K < M ) )
41	39, 40	sylibr 134	1 |- ( ( K =/= M /\ K e. ( 0 ... M ) ) -> K e. ( 0..^ M ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   e. wcel 2167   =/= wne 2367   class class class wbr 4033  (class class class)co 5922   RRcr 7878   0cc0 7879   < clt 8061   <_ cle 8062   NNcn 8990   NN0cn0 9249   ZZcz 9326   ...cfz 10083  ..^cfzo 10217

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602  df-fz 10084  df-fzo 10218

This theorem is referenced by: (None)