Theorem fzo1fzo0n0 10595

Description: An integer between 1 and an upper bound of a half-open integer range is not 0 and between 0 and the upper bound of the half-open integer range. (Contributed by Alexander van der Vekens, 21-Mar-2018.)

Assertion

Ref	Expression
fzo1fzo0n0	|- ( K e. ( 1..^ N ) <-> ( K e. ( 0..^ N ) /\ K =/= 0 ) )

Proof of Theorem fzo1fzo0n0

Step	Hyp	Ref	Expression
1		elfzo2 10557	. . 3 |- ( K e. ( 1..^ N ) <-> ( K e. ( ZZ>= ` 1 ) /\ N e. ZZ /\ K < N ) )
2		elnnuz 9959	. . . . . . 7 |- ( K e. NN <-> K e. ( ZZ>= ` 1 ) )
3		nnnn0 9570	. . . . . . . . . . 11 |- ( K e. NN -> K e. NN0 )
4	3	adantr 276	. . . . . . . . . 10 |- ( ( K e. NN /\ N e. ZZ ) -> K e. NN0 )
5	4	adantr 276	. . . . . . . . 9 |- ( ( ( K e. NN /\ N e. ZZ ) /\ K < N ) -> K e. NN0 )
6		nngt0 9329	. . . . . . . . . . 11 |- ( K e. NN -> 0 < K )
7		0red 8327	. . . . . . . . . . . . . . 15 |- ( ( N e. ZZ /\ K e. NN ) -> 0 e. RR )
8		nnre 9311	. . . . . . . . . . . . . . . 16 |- ( K e. NN -> K e. RR )
9	8	adantl 277	. . . . . . . . . . . . . . 15 |- ( ( N e. ZZ /\ K e. NN ) -> K e. RR )
10		zre 9648	. . . . . . . . . . . . . . . 16 |- ( N e. ZZ -> N e. RR )
11	10	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( N e. ZZ /\ K e. NN ) -> N e. RR )
12		lttr 8399	. . . . . . . . . . . . . . 15 |- ( ( 0 e. RR /\ K e. RR /\ N e. RR ) -> ( ( 0 < K /\ K < N ) -> 0 < N ) )
13	7, 9, 11, 12	syl3anc 1278	. . . . . . . . . . . . . 14 |- ( ( N e. ZZ /\ K e. NN ) -> ( ( 0 < K /\ K < N ) -> 0 < N ) )
14		elnnz 9654	. . . . . . . . . . . . . . . 16 |- ( N e. NN <-> ( N e. ZZ /\ 0 < N ) )
15	14	simplbi2 385	. . . . . . . . . . . . . . 15 |- ( N e. ZZ -> ( 0 < N -> N e. NN ) )
16	15	adantr 276	. . . . . . . . . . . . . 14 |- ( ( N e. ZZ /\ K e. NN ) -> ( 0 < N -> N e. NN ) )
17	13, 16	syld 45	. . . . . . . . . . . . 13 |- ( ( N e. ZZ /\ K e. NN ) -> ( ( 0 < K /\ K < N ) -> N e. NN ) )
18	17	exp4b 367	. . . . . . . . . . . 12 |- ( N e. ZZ -> ( K e. NN -> ( 0 < K -> ( K < N -> N e. NN ) ) ) )
19	18	com13 80	. . . . . . . . . . 11 |- ( 0 < K -> ( K e. NN -> ( N e. ZZ -> ( K < N -> N e. NN ) ) ) )
20	6, 19	mpcom 36	. . . . . . . . . 10 |- ( K e. NN -> ( N e. ZZ -> ( K < N -> N e. NN ) ) )
21	20	imp31 256	. . . . . . . . 9 |- ( ( ( K e. NN /\ N e. ZZ ) /\ K < N ) -> N e. NN )
22		simpr 110	. . . . . . . . 9 |- ( ( ( K e. NN /\ N e. ZZ ) /\ K < N ) -> K < N )
23	5, 21, 22	3jca 1208	. . . . . . . 8 |- ( ( ( K e. NN /\ N e. ZZ ) /\ K < N ) -> ( K e. NN0 /\ N e. NN /\ K < N ) )
24	23	exp31 364	. . . . . . 7 |- ( K e. NN -> ( N e. ZZ -> ( K < N -> ( K e. NN0 /\ N e. NN /\ K < N ) ) ) )
25	2, 24	sylbir 135	. . . . . 6 |- ( K e. ( ZZ>= ` 1 ) -> ( N e. ZZ -> ( K < N -> ( K e. NN0 /\ N e. NN /\ K < N ) ) ) )
26	25	3imp 1224	. . . . 5 |- ( ( K e. ( ZZ>= ` 1 ) /\ N e. ZZ /\ K < N ) -> ( K e. NN0 /\ N e. NN /\ K < N ) )
27		elfzo0 10593	. . . . 5 |- ( K e. ( 0..^ N ) <-> ( K e. NN0 /\ N e. NN /\ K < N ) )
28	26, 27	sylibr 134	. . . 4 |- ( ( K e. ( ZZ>= ` 1 ) /\ N e. ZZ /\ K < N ) -> K e. ( 0..^ N ) )
29		nnne0 9332	. . . . . 6 |- ( K e. NN -> K =/= 0 )
30	2, 29	sylbir 135	. . . . 5 |- ( K e. ( ZZ>= ` 1 ) -> K =/= 0 )
31	30	3ad2ant1 1049	. . . 4 |- ( ( K e. ( ZZ>= ` 1 ) /\ N e. ZZ /\ K < N ) -> K =/= 0 )
32	28, 31	jca 306	. . 3 |- ( ( K e. ( ZZ>= ` 1 ) /\ N e. ZZ /\ K < N ) -> ( K e. ( 0..^ N ) /\ K =/= 0 ) )
33	1, 32	sylbi 121	. 2 |- ( K e. ( 1..^ N ) -> ( K e. ( 0..^ N ) /\ K =/= 0 ) )
34		elnnne0 9577	. . . . . 6 |- ( K e. NN <-> ( K e. NN0 /\ K =/= 0 ) )
35		nnge1 9327	. . . . . 6 |- ( K e. NN -> 1 <_ K )
36	34, 35	sylbir 135	. . . . 5 |- ( ( K e. NN0 /\ K =/= 0 ) -> 1 <_ K )
37	36	3ad2antl1 1190	. . . 4 |- ( ( ( K e. NN0 /\ N e. NN /\ K < N ) /\ K =/= 0 ) -> 1 <_ K )
38		simpl3 1033	. . . 4 |- ( ( ( K e. NN0 /\ N e. NN /\ K < N ) /\ K =/= 0 ) -> K < N )
39		nn0z 9664	. . . . . . . . 9 |- ( K e. NN0 -> K e. ZZ )
40	39	adantr 276	. . . . . . . 8 |- ( ( K e. NN0 /\ N e. NN ) -> K e. ZZ )
41		1zzd 9671	. . . . . . . 8 |- ( ( K e. NN0 /\ N e. NN ) -> 1 e. ZZ )
42		nnz 9663	. . . . . . . . 9 |- ( N e. NN -> N e. ZZ )
43	42	adantl 277	. . . . . . . 8 |- ( ( K e. NN0 /\ N e. NN ) -> N e. ZZ )
44	40, 41, 43	3jca 1208	. . . . . . 7 |- ( ( K e. NN0 /\ N e. NN ) -> ( K e. ZZ /\ 1 e. ZZ /\ N e. ZZ ) )
45	44	3adant3 1048	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN /\ K < N ) -> ( K e. ZZ /\ 1 e. ZZ /\ N e. ZZ ) )
46	45	adantr 276	. . . . 5 |- ( ( ( K e. NN0 /\ N e. NN /\ K < N ) /\ K =/= 0 ) -> ( K e. ZZ /\ 1 e. ZZ /\ N e. ZZ ) )
47		elfzo 10556	. . . . 5 |- ( ( K e. ZZ /\ 1 e. ZZ /\ N e. ZZ ) -> ( K e. ( 1..^ N ) <-> ( 1 <_ K /\ K < N ) ) )
48	46, 47	syl 14	. . . 4 |- ( ( ( K e. NN0 /\ N e. NN /\ K < N ) /\ K =/= 0 ) -> ( K e. ( 1..^ N ) <-> ( 1 <_ K /\ K < N ) ) )
49	37, 38, 48	mpbir2and 957	. . 3 |- ( ( ( K e. NN0 /\ N e. NN /\ K < N ) /\ K =/= 0 ) -> K e. ( 1..^ N ) )
50	27, 49	sylanb 284	. 2 |- ( ( K e. ( 0..^ N ) /\ K =/= 0 ) -> K e. ( 1..^ N ) )
51	33, 50	impbii 126	1 |- ( K e. ( 1..^ N ) <-> ( K e. ( 0..^ N ) /\ K =/= 0 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   e. wcel 2209   =/= wne 2420   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   RRcr 8178   0cc0 8179   1c1 8180   < clt 8360   <_ cle 8361   NNcn 9304   NN0cn0 9563   ZZcz 9644   ZZ>=cuz 9921  ..^cfzo 10549

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-fzo 10550

This theorem is used by:  modprmn0modprm0  13035