Theorem fzofzim 10174

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 10098	. . . 4 |- ( K e. ( 0 ... M ) <-> ( K e. NN0 /\ M e. NN0 /\ K <_ M ) )
2		simpl1 1000	. . . . . 6 |- ( ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) /\ K =/= M ) -> K e. NN0 )
3		necom 2431	. . . . . . . . 9 |- ( K =/= M <-> M =/= K )
4		nn0z 9262	. . . . . . . . . . . . 13 |- ( K e. NN0 -> K e. ZZ )
5		nn0z 9262	. . . . . . . . . . . . 13 |- ( M e. NN0 -> M e. ZZ )
6		zltlen 9320	. . . . . . . . . . . . 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 9255	. . . . . . . . . . . . 13 |- ( K e. NN0 <-> ( K e. ZZ /\ 0 <_ K ) )
10		0red 7949	. . . . . . . . . . . . . . . . 17 |- ( ( K e. ZZ /\ M e. NN0 ) -> 0 e. RR )
11		zre 9246	. . . . . . . . . . . . . . . . . 18 |- ( K e. ZZ -> K e. RR )
12	11	adantr 276	. . . . . . . . . . . . . . . . 17 |- ( ( K e. ZZ /\ M e. NN0 ) -> K e. RR )
13		nn0re 9174	. . . . . . . . . . . . . . . . . 18 |- ( M e. NN0 -> M e. RR )
14	13	adantl 277	. . . . . . . . . . . . . . . . 17 |- ( ( K e. ZZ /\ M e. NN0 ) -> M e. RR )
15		lelttr 8036	. . . . . . . . . . . . . . . . 17 |- ( ( 0 e. RR /\ K e. RR /\ M e. RR ) -> ( ( 0 <_ K /\ K < M ) -> 0 < M ) )
16	10, 12, 14, 15	syl3anc 1238	. . . . . . . . . . . . . . . 16 |- ( ( K e. ZZ /\ M e. NN0 ) -> ( ( 0 <_ K /\ K < M ) -> 0 < M ) )
17		elnnz 9252	. . . . . . . . . . . . . . . . . . 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 1200	. . . . . . 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 1193	. . . . . . . 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 1177	. . . . 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 10168	. 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 978   e. wcel 2148   =/= wne 2347   class class class wbr 4000  (class class class)co 5869   RRcr 7801   0cc0 7802   < clt 7982   <_ cle 7983   NNcn 8908   NN0cn0 9165   ZZcz 9242   ...cfz 9995  ..^cfzo 10128

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 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919

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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518  df-fz 9996  df-fzo 10129

This theorem is referenced by: (None)