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Mirrors > Home > ILE Home > Th. List > hashfzo | Unicode version |
Description: Cardinality of a half-open set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
Ref | Expression |
---|---|
hashfzo | ♯..^ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 9331 | . . . . . 6 | |
2 | 1 | zcnd 9174 | . . . . 5 |
3 | 2 | subidd 8061 | . . . 4 |
4 | fzo0 9945 | . . . . . 6 ..^ | |
5 | 4 | fveq2i 5424 | . . . . 5 ♯..^ ♯ |
6 | hash0 10543 | . . . . 5 ♯ | |
7 | 5, 6 | eqtri 2160 | . . . 4 ♯..^ |
8 | 3, 7 | syl6reqr 2191 | . . 3 ♯..^ |
9 | oveq2 5782 | . . . . 5 ..^ ..^ | |
10 | 9 | fveq2d 5425 | . . . 4 ♯..^ ♯..^ |
11 | oveq1 5781 | . . . 4 | |
12 | 10, 11 | eqeq12d 2154 | . . 3 ♯..^ ♯..^ |
13 | 8, 12 | syl5ibrcom 156 | . 2 ♯..^ |
14 | eluzelz 9335 | . . . . . . 7 | |
15 | fzoval 9925 | . . . . . . 7 ..^ | |
16 | 14, 15 | syl 14 | . . . . . 6 ..^ |
17 | 16 | fveq2d 5425 | . . . . 5 ♯..^ ♯ |
18 | 17 | adantr 274 | . . . 4 ♯..^ ♯ |
19 | hashfz 10567 | . . . . 5 ♯ | |
20 | 14 | zcnd 9174 | . . . . . . . 8 |
21 | 1cnd 7782 | . . . . . . . 8 | |
22 | 20, 21, 2 | sub32d 8105 | . . . . . . 7 |
23 | 22 | oveq1d 5789 | . . . . . 6 |
24 | 20, 2 | subcld 8073 | . . . . . . 7 |
25 | ax-1cn 7713 | . . . . . . 7 | |
26 | npcan 7971 | . . . . . . 7 | |
27 | 24, 25, 26 | sylancl 409 | . . . . . 6 |
28 | 23, 27 | eqtrd 2172 | . . . . 5 |
29 | 19, 28 | sylan9eqr 2194 | . . . 4 ♯ |
30 | 18, 29 | eqtrd 2172 | . . 3 ♯..^ |
31 | 30 | ex 114 | . 2 ♯..^ |
32 | uzm1 9356 | . 2 | |
33 | 13, 31, 32 | mpjaod 707 | 1 ♯..^ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 c0 3363 cfv 5123 (class class class)co 5774 cc 7618 cc0 7620 c1 7621 caddc 7623 cmin 7933 cz 9054 cuz 9326 cfz 9790 ..^cfzo 9919 ♯chash 10521 |
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-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 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-apti 7735 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-dc 820 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-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 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-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-1o 6313 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 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 df-ihash 10522 |
This theorem is referenced by: hashfzo0 10569 |
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