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Mirrors > Home > MPE Home > Th. List > elfzolt2 | Structured version Visualization version GIF version |
Description: A member in a half-open integer interval is less than the upper bound. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
Ref | Expression |
---|---|
elfzolt2 | ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 < 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzoelz 13632 | . . . 4 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ ℤ) | |
2 | elfzoel1 13630 | . . . 4 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑀 ∈ ℤ) | |
3 | elfzoel2 13631 | . . . 4 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑁 ∈ ℤ) | |
4 | elfzo 13634 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀..^𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 < 𝑁))) | |
5 | 1, 2, 3, 4 | syl3anc 1372 | . . 3 ⊢ (𝐾 ∈ (𝑀..^𝑁) → (𝐾 ∈ (𝑀..^𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 < 𝑁))) |
6 | 5 | ibi 267 | . 2 ⊢ (𝐾 ∈ (𝑀..^𝑁) → (𝑀 ≤ 𝐾 ∧ 𝐾 < 𝑁)) |
7 | 6 | simprd 497 | 1 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 < 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 class class class wbr 5149 (class class class)co 7409 < clt 11248 ≤ cle 11249 ℤcz 12558 ..^cfzo 13627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 |
This theorem is referenced by: elfzolt3 13642 elfzolt2b 13643 elfzop1le2 13645 fzonel 13646 elfzouz2 13647 fzonnsub 13657 fzospliti 13664 fzodisj 13666 fzouzdisj 13668 fzodisjsn 13670 elfzo0 13673 elfzo1 13682 fzoaddel 13685 elincfzoext 13690 ssfzo12 13725 elfznelfzob 13738 modaddmodlo 13900 ccatrn 14539 swrds2 14891 fzomaxdiflem 15289 fzo0dvdseq 16266 bitsfzolem 16375 bitsfzo 16376 sadcaddlem 16398 sadaddlem 16407 sadasslem 16411 sadeq 16413 smuval2 16423 smupvallem 16424 smueqlem 16431 crth 16711 eulerthlem2 16715 hashgcdlem 16721 prmgaplem6 16989 znf1o 21107 iundisj 25065 tgcgr4 27782 clwlkclwwlklem2fv1 29248 iundisjf 31820 iundisjfi 32007 fzone1 32011 ply1degltdimlem 32707 ply1degltdim 32708 smattl 32778 smattr 32779 smatbl 32780 signsplypnf 33561 breprexplemc 33644 poimirlem17 36505 poimirlem20 36508 frlmvscadiccat 41080 elfzfzo 43986 dvnmul 44659 iblspltprt 44689 itgspltprt 44695 stoweidlem3 44719 fourierdlem12 44835 fourierdlem50 44872 fourierdlem64 44886 fourierdlem79 44901 iccpartgt 46095 m1modmmod 47207 |
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