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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 13636 | . . . 4 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ ℤ) | |
2 | elfzoel1 13634 | . . . 4 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑀 ∈ ℤ) | |
3 | elfzoel2 13635 | . . . 4 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑁 ∈ ℤ) | |
4 | elfzo 13638 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀..^𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 < 𝑁))) | |
5 | 1, 2, 3, 4 | syl3anc 1369 | . . 3 ⊢ (𝐾 ∈ (𝑀..^𝑁) → (𝐾 ∈ (𝑀..^𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 < 𝑁))) |
6 | 5 | ibi 266 | . 2 ⊢ (𝐾 ∈ (𝑀..^𝑁) → (𝑀 ≤ 𝐾 ∧ 𝐾 < 𝑁)) |
7 | 6 | simprd 494 | 1 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 < 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2104 class class class wbr 5147 (class class class)co 7411 < clt 11252 ≤ cle 11253 ℤcz 12562 ..^cfzo 13631 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 |
This theorem is referenced by: elfzolt3 13646 elfzolt2b 13647 elfzop1le2 13649 fzonel 13650 elfzouz2 13651 fzonnsub 13661 fzospliti 13668 fzodisj 13670 fzouzdisj 13672 fzodisjsn 13674 elfzo0 13677 elfzo1 13686 fzoaddel 13689 elincfzoext 13694 ssfzo12 13729 elfznelfzob 13742 modaddmodlo 13904 ccatrn 14543 swrds2 14895 fzomaxdiflem 15293 fzo0dvdseq 16270 bitsfzolem 16379 bitsfzo 16380 sadcaddlem 16402 sadaddlem 16411 sadasslem 16415 sadeq 16417 smuval2 16427 smupvallem 16428 smueqlem 16435 crth 16715 eulerthlem2 16719 hashgcdlem 16725 prmgaplem6 16993 znf1o 21326 iundisj 25297 tgcgr4 28049 clwlkclwwlklem2fv1 29515 iundisjf 32087 iundisjfi 32274 fzone1 32278 ply1degltdimlem 32995 ply1degltdim 32996 smattl 33076 smattr 33077 smatbl 33078 signsplypnf 33859 breprexplemc 33942 poimirlem17 36808 poimirlem20 36811 frlmvscadiccat 41386 elfzfzo 44284 dvnmul 44957 iblspltprt 44987 itgspltprt 44993 stoweidlem3 45017 fourierdlem12 45133 fourierdlem50 45170 fourierdlem64 45184 fourierdlem79 45199 iccpartgt 46393 m1modmmod 47294 |
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