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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 12340 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) | |
2 | 1 | adantl 481 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
3 | fzfi 12633 | . . 3 ⊢ (𝑀...(𝑁 − 1)) ∈ Fin | |
4 | 2, 3 | syl6eqel 2696 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) ∈ Fin) |
5 | fzof 12336 | . . . . 5 ⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ | |
6 | 5 | fdmi 5965 | . . . 4 ⊢ dom ..^ = (ℤ × ℤ) |
7 | 6 | ndmov 6716 | . . 3 ⊢ (¬ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) = ∅) |
8 | 0fin 8073 | . . 3 ⊢ ∅ ∈ Fin | |
9 | 7, 8 | syl6eqel 2696 | . 2 ⊢ (¬ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) ∈ Fin) |
10 | 4, 9 | pm2.61i 175 | 1 ⊢ (𝑀..^𝑁) ∈ Fin |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∅c0 3874 𝒫 cpw 4108 × cxp 5036 (class class class)co 6549 Fincfn 7841 1c1 9816 − cmin 10145 ℤcz 11254 ...cfz 12197 ..^cfzo 12334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 |
This theorem is referenced by: uzindi 12643 fnfzo0hashnn0 13092 wrdfin 13178 hashwrdn 13192 ccatalpha 13228 telfsumo 14375 fsumparts 14379 geoserg 14437 bitsfi 14997 bitsinv1 15002 bitsinvp1 15009 sadcaddlem 15017 sadadd2lem 15019 sadadd3 15021 sadaddlem 15026 sadasslem 15030 sadeq 15032 crth 15321 phimullem 15322 eulerthlem2 15325 eulerth 15326 phisum 15333 prmgaplem3 15595 cshwshashnsame 15648 ablfaclem3 18309 ablfac2 18311 iunmbl 23128 volsup 23131 dvfsumle 23588 dvfsumge 23589 dvfsumabs 23590 advlogexp 24201 dchrisumlem1 24978 dchrisumlem2 24979 dchrisum 24981 vdegp1ai 26511 vdegp1bi 26512 fz1nnct 28947 sigapildsys 29552 carsgclctunlem3 29709 ccatmulgnn0dir 29945 ofcccat 29946 signsplypnf 29953 signsvvf 29982 mvrsfpw 30657 poimirlem26 32605 poimirlem27 32606 poimirlem28 32607 poimirlem30 32609 amgm2d 37523 amgm3d 37524 amgm4d 37525 fourierdlem25 39025 fourierdlem70 39069 fourierdlem71 39070 fourierdlem73 39072 fourierdlem79 39078 fourierdlem80 39079 meaiunlelem 39361 pwdif 40039 2pwp1prm 40041 vdegp1bi-av 40753 eupthfi 41373 trlsegvdeglem6 41393 nn0sumshdiglemA 42211 nn0sumshdiglemB 42212 nn0mullong 42217 amgmw2d 42359 |
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