Theorem fzofzim 12514

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 12431	. . . 4 |- ( K e. ( 0 ... M ) <-> ( K e. NN0 /\ M e. NN0 /\ K <_ M ) )
2		simpl1 1064	. . . . . 6 |- ( ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) /\ K =/= M ) -> K e. NN0 )
3		necom 2847	. . . . . . . . 9 |- ( K =/= M <-> M =/= K )
4		nn0re 11301	. . . . . . . . . . . . 13 |- ( K e. NN0 -> K e. RR )
5		nn0re 11301	. . . . . . . . . . . . 13 |- ( M e. NN0 -> M e. RR )
6		ltlen 10138	. . . . . . . . . . . . 13 |- ( ( K e. RR /\ M e. RR ) -> ( K < M <-> ( K <_ M /\ M =/= K ) ) )
7	4, 5, 6	syl2an 494	. . . . . . . . . . . 12 |- ( ( K e. NN0 /\ M e. NN0 ) -> ( K < M <-> ( K <_ M /\ M =/= K ) ) )
8	7	bicomd 213	. . . . . . . . . . 11 |- ( ( K e. NN0 /\ M e. NN0 ) -> ( ( K <_ M /\ M =/= K ) <-> K < M ) )
9		elnn0z 11390	. . . . . . . . . . . . 13 |- ( K e. NN0 <-> ( K e. ZZ /\ 0 <_ K ) )
10		0red 10041	. . . . . . . . . . . . . . . . 17 |- ( ( K e. ZZ /\ M e. NN0 ) -> 0 e. RR )
11		zre 11381	. . . . . . . . . . . . . . . . . 18 |- ( K e. ZZ -> K e. RR )
12	11	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( K e. ZZ /\ M e. NN0 ) -> K e. RR )
13	5	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( K e. ZZ /\ M e. NN0 ) -> M e. RR )
14		lelttr 10128	. . . . . . . . . . . . . . . . 17 |- ( ( 0 e. RR /\ K e. RR /\ M e. RR ) -> ( ( 0 <_ K /\ K < M ) -> 0 < M ) )
15	10, 12, 13, 14	syl3anc 1326	. . . . . . . . . . . . . . . 16 |- ( ( K e. ZZ /\ M e. NN0 ) -> ( ( 0 <_ K /\ K < M ) -> 0 < M ) )
16		nn0z 11400	. . . . . . . . . . . . . . . . . 18 |- ( M e. NN0 -> M e. ZZ )
17		elnnz 11387	. . . . . . . . . . . . . . . . . . 19 |- ( M e. NN <-> ( M e. ZZ /\ 0 < M ) )
18	17	simplbi2 655	. . . . . . . . . . . . . . . . . 18 |- ( M e. ZZ -> ( 0 < M -> M e. NN ) )
19	16, 18	syl 17	. . . . . . . . . . . . . . . . 17 |- ( M e. NN0 -> ( 0 < M -> M e. NN ) )
20	19	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( K e. ZZ /\ M e. NN0 ) -> ( 0 < M -> M e. NN ) )
21	15, 20	syld 47	. . . . . . . . . . . . . . 15 |- ( ( K e. ZZ /\ M e. NN0 ) -> ( ( 0 <_ K /\ K < M ) -> M e. NN ) )
22	21	expd 452	. . . . . . . . . . . . . 14 |- ( ( K e. ZZ /\ M e. NN0 ) -> ( 0 <_ K -> ( K < M -> M e. NN ) ) )
23	22	impancom 456	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ 0 <_ K ) -> ( M e. NN0 -> ( K < M -> M e. NN ) ) )
24	9, 23	sylbi 207	. . . . . . . . . . . 12 |- ( K e. NN0 -> ( M e. NN0 -> ( K < M -> M e. NN ) ) )
25	24	imp 445	. . . . . . . . . . 11 |- ( ( K e. NN0 /\ M e. NN0 ) -> ( K < M -> M e. NN ) )
26	8, 25	sylbid 230	. . . . . . . . . 10 |- ( ( K e. NN0 /\ M e. NN0 ) -> ( ( K <_ M /\ M =/= K ) -> M e. NN ) )
27	26	expd 452	. . . . . . . . 9 |- ( ( K e. NN0 /\ M e. NN0 ) -> ( K <_ M -> ( M =/= K -> M e. NN ) ) )
28	3, 27	syl7bi 245	. . . . . . . 8 |- ( ( K e. NN0 /\ M e. NN0 ) -> ( K <_ M -> ( K =/= M -> M e. NN ) ) )
29	28	3impia 1261	. . . . . . 7 |- ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) -> ( K =/= M -> M e. NN ) )
30	29	imp 445	. . . . . 6 |- ( ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) /\ K =/= M ) -> M e. NN )
31	8	biimpd 219	. . . . . . . . . 10 |- ( ( K e. NN0 /\ M e. NN0 ) -> ( ( K <_ M /\ M =/= K ) -> K < M ) )
32	31	exp4b 632	. . . . . . . . 9 |- ( K e. NN0 -> ( M e. NN0 -> ( K <_ M -> ( M =/= K -> K < M ) ) ) )
33	32	3imp 1256	. . . . . . . 8 |- ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) -> ( M =/= K -> K < M ) )
34	3, 33	syl5bi 232	. . . . . . 7 |- ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) -> ( K =/= M -> K < M ) )
35	34	imp 445	. . . . . 6 |- ( ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) /\ K =/= M ) -> K < M )
36	2, 30, 35	3jca 1242	. . . . 5 |- ( ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) /\ K =/= M ) -> ( K e. NN0 /\ M e. NN /\ K < M ) )
37	36	ex 450	. . . 4 |- ( ( K e. NN0 /\ M e. NN0 /\ K <_ M ) -> ( K =/= M -> ( K e. NN0 /\ M e. NN /\ K < M ) ) )
38	1, 37	sylbi 207	. . 3 |- ( K e. ( 0 ... M ) -> ( K =/= M -> ( K e. NN0 /\ M e. NN /\ K < M ) ) )
39	38	impcom 446	. 2 |- ( ( K =/= M /\ K e. ( 0 ... M ) ) -> ( K e. NN0 /\ M e. NN /\ K < M ) )
40		elfzo0 12508	. 2 |- ( K e. ( 0..^ M ) <-> ( K e. NN0 /\ M e. NN /\ K < M ) )
41	39, 40	sylibr 224	1 |- ( ( K =/= M /\ K e. ( 0 ... M ) ) -> K e. ( 0..^ M ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   =/= wne 2794   class class class wbr 4653  (class class class)co 6650   RRcr 9935   0cc0 9936   < clt 10074   <_ cle 10075   NNcn 11020   NN0cn0 11292   ZZcz 11377   ...cfz 12326  ..^cfzo 12465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466

This theorem is referenced by:  cshwshashlem1  15802  clwwisshclwwsn  26929