Theorem fzofzim 10549

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 10468	. . . 4 |- ( K e. ( 0 ... M ) <-> ( K e. NN0 /\ M e. NN0 /\ K <_ M ) )
2		simpl1 1027	. . . . . 6 |- ( ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) /\ K =/= M ) -> K e. NN0 )
3		necom 2498	. . . . . . . . 9 |- ( K =/= M <-> M =/= K )
4		nn0z 9614	. . . . . . . . . . . . 13 |- ( K e. NN0 -> K e. ZZ )
5		nn0z 9614	. . . . . . . . . . . . 13 |- ( M e. NN0 -> M e. ZZ )
6		zltlen 9674	. . . . . . . . . . . . 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 9607	. . . . . . . . . . . . 13 |- ( K e. NN0 <-> ( K e. ZZ /\ 0 <_ K ) )
10		0red 8291	. . . . . . . . . . . . . . . . 17 |- ( ( K e. ZZ /\ M e. NN0 ) -> 0 e. RR )
11		zre 9598	. . . . . . . . . . . . . . . . . 18 |- ( K e. ZZ -> K e. RR )
12	11	adantr 276	. . . . . . . . . . . . . . . . 17 |- ( ( K e. ZZ /\ M e. NN0 ) -> K e. RR )
13		nn0re 9522	. . . . . . . . . . . . . . . . . 18 |- ( M e. NN0 -> M e. RR )
14	13	adantl 277	. . . . . . . . . . . . . . . . 17 |- ( ( K e. ZZ /\ M e. NN0 ) -> M e. RR )
15		lelttr 8378	. . . . . . . . . . . . . . . . 17 |- ( ( 0 e. RR /\ K e. RR /\ M e. RR ) -> ( ( 0 <_ K /\ K < M ) -> 0 < M ) )
16	10, 12, 14, 15	syl3anc 1274	. . . . . . . . . . . . . . . 16 |- ( ( K e. ZZ /\ M e. NN0 ) -> ( ( 0 <_ K /\ K < M ) -> 0 < M ) )
17		elnnz 9604	. . . . . . . . . . . . . . . . . . 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 1227	. . . . . . 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 1220	. . . . . . . 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 1204	. . . . 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 10542	. 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 1005   e. wcel 2205   =/= wne 2414   class class class wbr 4114  (class class class)co 6058   RRcr 8142   0cc0 8143   < clt 8324   <_ cle 8325   NNcn 9254   NN0cn0 9513   ZZcz 9594   ...cfz 10361  ..^cfzo 10498

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499

This theorem is referenced by: (None)