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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 13579 | . . . 4 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ ℤ) | |
2 | elfzoel1 13577 | . . . 4 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑀 ∈ ℤ) | |
3 | elfzoel2 13578 | . . . 4 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑁 ∈ ℤ) | |
4 | elfzo 13581 | . . . 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 5110 (class class class)co 7362 < clt 11196 ≤ cle 11197 ℤcz 12506 ..^cfzo 13574 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-n0 12421 df-z 12507 df-uz 12771 df-fz 13432 df-fzo 13575 |
This theorem is referenced by: elfzolt3 13589 elfzolt2b 13590 elfzop1le2 13592 fzonel 13593 elfzouz2 13594 fzonnsub 13604 fzospliti 13611 fzodisj 13613 fzouzdisj 13615 fzodisjsn 13617 elfzo0 13620 elfzo1 13629 fzoaddel 13632 elincfzoext 13637 ssfzo12 13672 elfznelfzob 13685 modaddmodlo 13847 ccatrn 14484 swrds2 14836 fzomaxdiflem 15234 fzo0dvdseq 16212 bitsfzolem 16321 bitsfzo 16322 sadcaddlem 16344 sadaddlem 16353 sadasslem 16357 sadeq 16359 smuval2 16369 smupvallem 16370 smueqlem 16377 crth 16657 eulerthlem2 16661 hashgcdlem 16667 prmgaplem6 16935 znf1o 20974 iundisj 24928 tgcgr4 27515 clwlkclwwlklem2fv1 28981 iundisjf 31549 iundisjfi 31741 fzone1 31745 smattl 32419 smattr 32420 smatbl 32421 signsplypnf 33202 breprexplemc 33285 poimirlem17 36124 poimirlem20 36127 frlmvscadiccat 40710 elfzfzo 43584 dvnmul 44258 iblspltprt 44288 itgspltprt 44294 stoweidlem3 44318 fourierdlem12 44434 fourierdlem50 44471 fourierdlem64 44485 fourierdlem79 44500 iccpartgt 45693 m1modmmod 46681 |
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