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Mirrors > Home > ILE Home > Th. List > hashfzp1 | Unicode version |
Description: Value of the numeric cardinality of a (possibly empty) integer range. (Contributed by AV, 19-Jun-2021.) |
Ref | Expression |
---|---|
hashfzp1 | ♯ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 9445 | . . . 4 | |
2 | eluzelz 9449 | . . . 4 | |
3 | zdceq 9240 | . . . 4 DECID | |
4 | 1, 2, 3 | syl2anc 409 | . . 3 DECID |
5 | exmiddc 822 | . . 3 DECID | |
6 | 4, 5 | syl 14 | . 2 |
7 | hash0 10675 | . . . . 5 ♯ | |
8 | eluzelre 9450 | . . . . . . . 8 | |
9 | 8 | ltp1d 8802 | . . . . . . 7 |
10 | peano2z 9204 | . . . . . . . . 9 | |
11 | 10 | ancri 322 | . . . . . . . 8 |
12 | fzn 9945 | . . . . . . . 8 | |
13 | 2, 11, 12 | 3syl 17 | . . . . . . 7 |
14 | 9, 13 | mpbid 146 | . . . . . 6 |
15 | 14 | fveq2d 5473 | . . . . 5 ♯ ♯ |
16 | 2 | zcnd 9288 | . . . . . 6 |
17 | 16 | subidd 8175 | . . . . 5 |
18 | 7, 15, 17 | 3eqtr4a 2216 | . . . 4 ♯ |
19 | oveq1 5832 | . . . . . . 7 | |
20 | 19 | oveq1d 5840 | . . . . . 6 |
21 | 20 | fveq2d 5473 | . . . . 5 ♯ ♯ |
22 | oveq2 5833 | . . . . 5 | |
23 | 21, 22 | eqeq12d 2172 | . . . 4 ♯ ♯ |
24 | 18, 23 | syl5ibr 155 | . . 3 ♯ |
25 | uzp1 9473 | . . . . . . . 8 | |
26 | pm2.24 611 | . . . . . . . . . 10 | |
27 | 26 | eqcoms 2160 | . . . . . . . . 9 |
28 | ax-1 6 | . . . . . . . . 9 | |
29 | 27, 28 | jaoi 706 | . . . . . . . 8 |
30 | 25, 29 | syl 14 | . . . . . . 7 |
31 | 30 | impcom 124 | . . . . . 6 |
32 | hashfz 10699 | . . . . . 6 ♯ | |
33 | 31, 32 | syl 14 | . . . . 5 ♯ |
34 | 1 | zcnd 9288 | . . . . . . 7 |
35 | 1cnd 7895 | . . . . . . 7 | |
36 | 16, 34, 35 | nppcan2d 8213 | . . . . . 6 |
37 | 36 | adantl 275 | . . . . 5 |
38 | 33, 37 | eqtrd 2190 | . . . 4 ♯ |
39 | 38 | ex 114 | . . 3 ♯ |
40 | 24, 39 | jaoi 706 | . 2 ♯ |
41 | 6, 40 | mpcom 36 | 1 ♯ |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 DECID wdc 820 wceq 1335 wcel 2128 c0 3394 class class class wbr 3966 cfv 5171 (class class class)co 5825 cc0 7733 c1 7734 caddc 7736 clt 7913 cmin 8047 cz 9168 cuz 9440 cfz 9913 ♯chash 10653 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-addcom 7833 ax-addass 7835 ax-distr 7837 ax-i2m1 7838 ax-0lt1 7839 ax-0id 7841 ax-rnegex 7842 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-apti 7848 ax-pre-ltadd 7849 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-iord 4327 df-on 4329 df-ilim 4330 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-recs 6253 df-frec 6339 df-1o 6364 df-er 6481 df-en 6687 df-dom 6688 df-fin 6689 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-inn 8835 df-n0 9092 df-z 9169 df-uz 9441 df-fz 9914 df-ihash 10654 |
This theorem is referenced by: (None) |
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