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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 13572 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) | |
2 | 1 | adantl 482 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
3 | fzfi 13876 | . . 3 ⊢ (𝑀...(𝑁 − 1)) ∈ Fin | |
4 | 2, 3 | eqeltrdi 2846 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) ∈ Fin) |
5 | fzof 13568 | . . . . 5 ⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ | |
6 | 5 | fdmi 6680 | . . . 4 ⊢ dom ..^ = (ℤ × ℤ) |
7 | 6 | ndmov 7537 | . . 3 ⊢ (¬ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) = ∅) |
8 | 0fin 9114 | . . 3 ⊢ ∅ ∈ Fin | |
9 | 7, 8 | eqeltrdi 2846 | . 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 396 = wceq 1541 ∈ wcel 2106 ∅c0 4282 𝒫 cpw 4560 × cxp 5631 (class class class)co 7356 Fincfn 8882 1c1 11051 − cmin 11384 ℤcz 12498 ...cfz 13423 ..^cfzo 13566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-n0 12413 df-z 12499 df-uz 12763 df-fz 13424 df-fzo 13567 |
This theorem is referenced by: uzindi 13886 fnfzo0hashnn0 14347 wrdfin 14419 hashwrdn 14434 ccatalpha 14480 telfsumo 15686 fsumparts 15690 geoserg 15750 pwdif 15752 bitsfi 16316 bitsinv1 16321 bitsinvp1 16328 sadcaddlem 16336 sadadd2lem 16338 sadadd3 16340 sadaddlem 16345 sadasslem 16349 sadeq 16351 crth 16649 phimullem 16650 eulerthlem2 16653 eulerth 16654 phisum 16661 prmgaplem3 16924 cshwshashnsame 16975 ablfaclem3 19864 ablfac2 19866 iunmbl 24915 volsup 24918 dvfsumle 25383 dvfsumge 25384 dvfsumabs 25385 advlogexp 26008 dchrisumlem1 26835 dchrisumlem2 26836 dchrisum 26838 vdegp1bi 28432 eupthfi 29096 trlsegvdeglem6 29116 fz1nnct 31650 cycpmconjslem2 31948 sigapildsys 32701 carsgclctunlem3 32860 ccatmulgnn0dir 33094 ofcccat 33095 signsplypnf 33102 signsvvf 33131 prodfzo03 33156 fsum2dsub 33160 reprle 33167 reprsuc 33168 reprfi 33169 reprlt 33172 hashreprin 33173 reprgt 33174 reprinfz1 33175 reprpmtf1o 33179 breprexplema 33183 breprexplemc 33185 breprexpnat 33187 circlemeth 33193 circlemethnat 33194 circlevma 33195 circlemethhgt 33196 hgt750lema 33210 lpadlem2 33233 mvrsfpw 34040 poimirlem26 36094 poimirlem27 36095 poimirlem28 36096 poimirlem30 36098 frlmfzowrdb 40668 frlmvscadiccat 40670 fltnltalem 40977 amgm2d 42452 amgm3d 42453 amgm4d 42454 fourierdlem25 44344 fourierdlem70 44388 fourierdlem71 44389 fourierdlem73 44391 fourierdlem79 44397 fourierdlem80 44398 meaiunlelem 44680 2pwp1prm 45752 nn0sumshdiglemA 46676 nn0sumshdiglemB 46677 nn0mullong 46682 amgmw2d 47222 |
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