Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fzofi | Structured version Visualization version GIF version |
Description: Half-open integer sets are finite. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
Ref | Expression |
---|---|
fzofi | ⊢ (𝑀..^𝑁) ∈ Fin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzoval 13040 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) | |
2 | 1 | adantl 484 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
3 | fzfi 13341 | . . 3 ⊢ (𝑀...(𝑁 − 1)) ∈ Fin | |
4 | 2, 3 | eqeltrdi 2921 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) ∈ Fin) |
5 | fzof 13036 | . . . . 5 ⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ | |
6 | 5 | fdmi 6524 | . . . 4 ⊢ dom ..^ = (ℤ × ℤ) |
7 | 6 | ndmov 7332 | . . 3 ⊢ (¬ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) = ∅) |
8 | 0fin 8746 | . . 3 ⊢ ∅ ∈ Fin | |
9 | 7, 8 | eqeltrdi 2921 | . 2 ⊢ (¬ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) ∈ Fin) |
10 | 4, 9 | pm2.61i 184 | 1 ⊢ (𝑀..^𝑁) ∈ Fin |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∅c0 4291 𝒫 cpw 4539 × cxp 5553 (class class class)co 7156 Fincfn 8509 1c1 10538 − cmin 10870 ℤcz 11982 ...cfz 12893 ..^cfzo 13034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 |
This theorem is referenced by: uzindi 13351 fnfzo0hashnn0 13810 wrdfin 13882 hashwrdn 13898 ccatalpha 13947 telfsumo 15157 fsumparts 15161 geoserg 15221 pwdif 15223 bitsfi 15786 bitsinv1 15791 bitsinvp1 15798 sadcaddlem 15806 sadadd2lem 15808 sadadd3 15810 sadaddlem 15815 sadasslem 15819 sadeq 15821 crth 16115 phimullem 16116 eulerthlem2 16119 eulerth 16120 phisum 16127 prmgaplem3 16389 cshwshashnsame 16437 ablfaclem3 19209 ablfac2 19211 iunmbl 24154 volsup 24157 dvfsumle 24618 dvfsumge 24619 dvfsumabs 24620 advlogexp 25238 dchrisumlem1 26065 dchrisumlem2 26066 dchrisum 26068 vdegp1bi 27319 eupthfi 27984 trlsegvdeglem6 28004 fz1nnct 30526 cycpmconjslem2 30797 sigapildsys 31421 carsgclctunlem3 31578 ccatmulgnn0dir 31812 ofcccat 31813 signsplypnf 31820 signsvvf 31849 prodfzo03 31874 fsum2dsub 31878 reprle 31885 reprsuc 31886 reprfi 31887 reprlt 31890 hashreprin 31891 reprgt 31892 reprinfz1 31893 reprpmtf1o 31897 breprexplema 31901 breprexplemc 31903 breprexpnat 31905 circlemeth 31911 circlemethnat 31912 circlevma 31913 circlemethhgt 31914 hgt750lema 31928 lpadlem2 31951 mvrsfpw 32753 poimirlem26 34933 poimirlem27 34934 poimirlem28 34935 poimirlem30 34937 frlmfzowrdb 39163 frlmvscadiccat 39165 fltnltalem 39294 amgm2d 40571 amgm3d 40572 amgm4d 40573 fourierdlem25 42437 fourierdlem70 42481 fourierdlem71 42482 fourierdlem73 42484 fourierdlem79 42490 fourierdlem80 42491 meaiunlelem 42770 2pwp1prm 43771 nn0sumshdiglemA 44699 nn0sumshdiglemB 44700 nn0mullong 44705 amgmw2d 44925 |
Copyright terms: Public domain | W3C validator |