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Mirrors > Home > ILE Home > Th. List > fzo1fzo0n0 | Unicode version |
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.) |
Ref | Expression |
---|---|
fzo1fzo0n0 | ..^ ..^ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzo2 9927 | . . 3 ..^ | |
2 | elnnuz 9362 | . . . . . . 7 | |
3 | nnnn0 8984 | . . . . . . . . . . 11 | |
4 | 3 | adantr 274 | . . . . . . . . . 10 |
5 | 4 | adantr 274 | . . . . . . . . 9 |
6 | nngt0 8745 | . . . . . . . . . . 11 | |
7 | 0red 7767 | . . . . . . . . . . . . . . 15 | |
8 | nnre 8727 | . . . . . . . . . . . . . . . 16 | |
9 | 8 | adantl 275 | . . . . . . . . . . . . . . 15 |
10 | zre 9058 | . . . . . . . . . . . . . . . 16 | |
11 | 10 | adantr 274 | . . . . . . . . . . . . . . 15 |
12 | lttr 7838 | . . . . . . . . . . . . . . 15 | |
13 | 7, 9, 11, 12 | syl3anc 1216 | . . . . . . . . . . . . . 14 |
14 | elnnz 9064 | . . . . . . . . . . . . . . . 16 | |
15 | 14 | simplbi2 382 | . . . . . . . . . . . . . . 15 |
16 | 15 | adantr 274 | . . . . . . . . . . . . . 14 |
17 | 13, 16 | syld 45 | . . . . . . . . . . . . 13 |
18 | 17 | exp4b 364 | . . . . . . . . . . . 12 |
19 | 18 | com13 80 | . . . . . . . . . . 11 |
20 | 6, 19 | mpcom 36 | . . . . . . . . . 10 |
21 | 20 | imp31 254 | . . . . . . . . 9 |
22 | simpr 109 | . . . . . . . . 9 | |
23 | 5, 21, 22 | 3jca 1161 | . . . . . . . 8 |
24 | 23 | exp31 361 | . . . . . . 7 |
25 | 2, 24 | sylbir 134 | . . . . . 6 |
26 | 25 | 3imp 1175 | . . . . 5 |
27 | elfzo0 9959 | . . . . 5 ..^ | |
28 | 26, 27 | sylibr 133 | . . . 4 ..^ |
29 | nnne0 8748 | . . . . . 6 | |
30 | 2, 29 | sylbir 134 | . . . . 5 |
31 | 30 | 3ad2ant1 1002 | . . . 4 |
32 | 28, 31 | jca 304 | . . 3 ..^ |
33 | 1, 32 | sylbi 120 | . 2 ..^ ..^ |
34 | elnnne0 8991 | . . . . . 6 | |
35 | nnge1 8743 | . . . . . 6 | |
36 | 34, 35 | sylbir 134 | . . . . 5 |
37 | 36 | 3ad2antl1 1143 | . . . 4 |
38 | simpl3 986 | . . . 4 | |
39 | nn0z 9074 | . . . . . . . . 9 | |
40 | 39 | adantr 274 | . . . . . . . 8 |
41 | 1zzd 9081 | . . . . . . . 8 | |
42 | nnz 9073 | . . . . . . . . 9 | |
43 | 42 | adantl 275 | . . . . . . . 8 |
44 | 40, 41, 43 | 3jca 1161 | . . . . . . 7 |
45 | 44 | 3adant3 1001 | . . . . . 6 |
46 | 45 | adantr 274 | . . . . 5 |
47 | elfzo 9926 | . . . . 5 ..^ | |
48 | 46, 47 | syl 14 | . . . 4 ..^ |
49 | 37, 38, 48 | mpbir2and 928 | . . 3 ..^ |
50 | 27, 49 | sylanb 282 | . 2 ..^ ..^ |
51 | 33, 50 | impbii 125 | 1 ..^ ..^ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wcel 1480 wne 2308 class class class wbr 3929 cfv 5123 (class class class)co 5774 cr 7619 cc0 7620 c1 7621 clt 7800 cle 7801 cn 8720 cn0 8977 cz 9054 cuz 9326 ..^cfzo 9919 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-fz 9791 df-fzo 9920 |
This theorem is referenced by: (None) |
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