Theorem fzofzim 10113

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 10037	. . . 4 |- ( K e. ( 0 ... M ) <-> ( K e. NN0 /\ M e. NN0 /\ K <_ M ) )
2		simpl1 989	. . . . . 6 |- ( ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) /\ K =/= M ) -> K e. NN0 )
3		necom 2418	. . . . . . . . 9 |- ( K =/= M <-> M =/= K )
4		nn0z 9202	. . . . . . . . . . . . 13 |- ( K e. NN0 -> K e. ZZ )
5		nn0z 9202	. . . . . . . . . . . . 13 |- ( M e. NN0 -> M e. ZZ )
6		zltlen 9260	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( K < M <-> ( K <_ M /\ M =/= K ) ) )
7	4, 5, 6	syl2an 287	. . . . . . . . . . . 12 |- ( ( K e. NN0 /\ M e. NN0 ) -> ( K < M <-> ( K <_ M /\ M =/= K ) ) )
8	7	bicomd 140	. . . . . . . . . . 11 |- ( ( K e. NN0 /\ M e. NN0 ) -> ( ( K <_ M /\ M =/= K ) <-> K < M ) )
9		elnn0z 9195	. . . . . . . . . . . . 13 |- ( K e. NN0 <-> ( K e. ZZ /\ 0 <_ K ) )
10		0red 7891	. . . . . . . . . . . . . . . . 17 |- ( ( K e. ZZ /\ M e. NN0 ) -> 0 e. RR )
11		zre 9186	. . . . . . . . . . . . . . . . . 18 |- ( K e. ZZ -> K e. RR )
12	11	adantr 274	. . . . . . . . . . . . . . . . 17 |- ( ( K e. ZZ /\ M e. NN0 ) -> K e. RR )
13		nn0re 9114	. . . . . . . . . . . . . . . . . 18 |- ( M e. NN0 -> M e. RR )
14	13	adantl 275	. . . . . . . . . . . . . . . . 17 |- ( ( K e. ZZ /\ M e. NN0 ) -> M e. RR )
15		lelttr 7978	. . . . . . . . . . . . . . . . 17 |- ( ( 0 e. RR /\ K e. RR /\ M e. RR ) -> ( ( 0 <_ K /\ K < M ) -> 0 < M ) )
16	10, 12, 14, 15	syl3anc 1227	. . . . . . . . . . . . . . . 16 |- ( ( K e. ZZ /\ M e. NN0 ) -> ( ( 0 <_ K /\ K < M ) -> 0 < M ) )
17		elnnz 9192	. . . . . . . . . . . . . . . . . . 19 |- ( M e. NN <-> ( M e. ZZ /\ 0 < M ) )
18	17	simplbi2 383	. . . . . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . 14 |- ( ( K e. ZZ /\ M e. NN0 ) -> ( 0 <_ K -> ( K < M -> M e. NN ) ) )
23	22	impancom 258	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ 0 <_ K ) -> ( M e. NN0 -> ( K < M -> M e. NN ) ) )
24	9, 23	sylbi 120	. . . . . . . . . . . 12 |- ( K e. NN0 -> ( M e. NN0 -> ( K < M -> M e. NN ) ) )
25	24	imp 123	. . . . . . . . . . 11 |- ( ( K e. NN0 /\ M e. NN0 ) -> ( K < M -> M e. NN ) )
26	8, 25	sylbid 149	. . . . . . . . . 10 |- ( ( K e. NN0 /\ M e. NN0 ) -> ( ( K <_ M /\ M =/= K ) -> M e. NN ) )
27	26	expd 256	. . . . . . . . 9 |- ( ( K e. NN0 /\ M e. NN0 ) -> ( K <_ M -> ( M =/= K -> M e. NN ) ) )
28	3, 27	syl7bi 164	. . . . . . . 8 |- ( ( K e. NN0 /\ M e. NN0 ) -> ( K <_ M -> ( K =/= M -> M e. NN ) ) )
29	28	3impia 1189	. . . . . . 7 |- ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) -> ( K =/= M -> M e. NN ) )
30	29	imp 123	. . . . . 6 |- ( ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) /\ K =/= M ) -> M e. NN )
31	8	biimpd 143	. . . . . . . . . 10 |- ( ( K e. NN0 /\ M e. NN0 ) -> ( ( K <_ M /\ M =/= K ) -> K < M ) )
32	31	exp4b 365	. . . . . . . . 9 |- ( K e. NN0 -> ( M e. NN0 -> ( K <_ M -> ( M =/= K -> K < M ) ) ) )
33	32	3imp 1182	. . . . . . . 8 |- ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) -> ( M =/= K -> K < M ) )
34	3, 33	syl5bi 151	. . . . . . 7 |- ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) -> ( K =/= M -> K < M ) )
35	34	imp 123	. . . . . 6 |- ( ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) /\ K =/= M ) -> K < M )
36	2, 30, 35	3jca 1166	. . . . 5 |- ( ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) /\ K =/= M ) -> ( K e. NN0 /\ M e. NN /\ K < M ) )
37	36	ex 114	. . . 4 |- ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) -> ( K =/= M -> ( K e. NN0 /\ M e. NN /\ K < M ) ) )
38	1, 37	sylbi 120	. . 3 |- ( K e. ( 0 ... M ) -> ( K =/= M -> ( K e. NN0 /\ M e. NN /\ K < M ) ) )
39	38	impcom 124	. 2 |- ( ( K =/= M /\ K e. ( 0 ... M ) ) -> ( K e. NN0 /\ M e. NN /\ K < M ) )
40		elfzo0 10107	. 2 |- ( K e. ( 0..^ M ) <-> ( K e. NN0 /\ M e. NN /\ K < M ) )
41	39, 40	sylibr 133	1 |- ( ( K =/= M /\ K e. ( 0 ... M ) ) -> K e. ( 0..^ M ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 967   e. wcel 2135   =/= wne 2334   class class class wbr 3976  (class class class)co 5836   RRcr 7743   0cc0 7744   < clt 7924   <_ cle 7925   NNcn 8848   NN0cn0 9105   ZZcz 9182   ...cfz 9935  ..^cfzo 10067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-mulrcl 7843  ax-addcom 7844  ax-mulcom 7845  ax-addass 7846  ax-mulass 7847  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-1rid 7851  ax-0id 7852  ax-rnegex 7853  ax-precex 7854  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-apti 7859  ax-pre-ltadd 7860  ax-pre-mulgt0 7861

This theorem depends on definitions:  df-bi 116  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-fv 5190  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-reap 8464  df-ap 8471  df-inn 8849  df-n0 9106  df-z 9183  df-uz 9458  df-fz 9936  df-fzo 10068

This theorem is referenced by: (None)