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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 13700 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) | |
| 2 | 1 | adantl 481 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
| 3 | fzfi 14013 | . . 3 ⊢ (𝑀...(𝑁 − 1)) ∈ Fin | |
| 4 | 2, 3 | eqeltrdi 2849 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) ∈ Fin) |
| 5 | fzof 13696 | . . . . 5 ⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ | |
| 6 | 5 | fdmi 6747 | . . . 4 ⊢ dom ..^ = (ℤ × ℤ) |
| 7 | 6 | ndmov 7617 | . . 3 ⊢ (¬ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) = ∅) |
| 8 | 0fi 9082 | . . 3 ⊢ ∅ ∈ Fin | |
| 9 | 7, 8 | eqeltrdi 2849 | . 2 ⊢ (¬ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) ∈ Fin) |
| 10 | 4, 9 | pm2.61i 182 | 1 ⊢ (𝑀..^𝑁) ∈ Fin |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∅c0 4333 𝒫 cpw 4600 × cxp 5683 (class class class)co 7431 Fincfn 8985 1c1 11156 − cmin 11492 ℤcz 12613 ...cfz 13547 ..^cfzo 13694 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 |
| This theorem is referenced by: uzindi 14023 fnfzo0hashnn0 14490 tpf1o 14540 wrdfin 14570 hashwrdn 14585 ccatalpha 14631 s7f1o 15005 telfsumo 15838 fsumparts 15842 geoserg 15902 pwdif 15904 bitsfi 16474 bitsinv1 16479 bitsinvp1 16486 sadcaddlem 16494 sadadd2lem 16496 sadadd3 16498 sadaddlem 16503 sadasslem 16507 sadeq 16509 crth 16815 phimullem 16816 eulerthlem2 16819 eulerth 16820 phisum 16828 prmgaplem3 17091 cshwshashnsame 17141 ablfaclem3 20107 ablfac2 20109 iunmbl 25588 volsup 25591 dvfsumle 26060 dvfsumleOLD 26061 dvfsumge 26062 dvfsumabs 26063 advlogexp 26697 dchrisumlem1 27533 dchrisumlem2 27534 dchrisum 27536 vdegp1bi 29555 eupthfi 30224 trlsegvdeglem6 30244 fz1nnct 32805 wrdfsupp 32921 cycpmconjslem2 33175 evl1deg2 33602 evl1deg3 33603 gsummoncoe1fzo 33618 ply1degltdimlem 33673 sigapildsys 34163 carsgclctunlem3 34322 ccatmulgnn0dir 34557 ofcccat 34558 signsplypnf 34565 signsvvf 34594 prodfzo03 34618 fsum2dsub 34622 reprle 34629 reprsuc 34630 reprfi 34631 reprlt 34634 hashreprin 34635 reprgt 34636 reprinfz1 34637 reprpmtf1o 34641 breprexplema 34645 breprexplemc 34647 breprexpnat 34649 circlemeth 34655 circlemethnat 34656 circlevma 34657 circlemethhgt 34658 hgt750lema 34672 lpadlem2 34695 mvrsfpw 35511 poimirlem26 37653 poimirlem27 37654 poimirlem28 37655 poimirlem30 37657 frlmfzowrdb 42514 frlmvscadiccat 42516 fltnltalem 42672 amgm2d 44211 amgm3d 44212 amgm4d 44213 fourierdlem25 46147 fourierdlem70 46191 fourierdlem71 46192 fourierdlem73 46194 fourierdlem79 46200 fourierdlem80 46201 meaiunlelem 46483 2pwp1prm 47576 gpgorder 48013 nn0sumshdiglemA 48540 nn0sumshdiglemB 48541 nn0mullong 48546 amgmw2d 49323 |
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