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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 13699 | . . . 4 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ ℤ) | |
| 2 | elfzoel1 13697 | . . . 4 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑀 ∈ ℤ) | |
| 3 | elfzoel2 13698 | . . . 4 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑁 ∈ ℤ) | |
| 4 | elfzo 13701 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀..^𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 < 𝑁))) | |
| 5 | 1, 2, 3, 4 | syl3anc 1373 | . . 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 495 | 1 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 < 𝑁) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 < clt 11295 ≤ cle 11296 ℤcz 12613 ..^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-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 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: elfzolt3 13709 elfzolt2b 13710 elfzop1le2 13712 fzonel 13713 elfzouz2 13714 fzonnsub 13724 fzospliti 13731 fzodisj 13733 fzouzdisj 13735 fzodisjsn 13737 elfzo0 13740 elfzo1 13752 fzoaddel 13756 elincfzoext 13762 ssfzo12 13798 elfznelfzob 13812 modaddmodlo 13976 ccatrn 14627 swrds2 14979 fzomaxdiflem 15381 fzo0dvdseq 16360 bitsfzolem 16471 bitsfzo 16472 sadcaddlem 16494 sadaddlem 16503 sadasslem 16507 sadeq 16509 smuval2 16519 smupvallem 16520 smueqlem 16527 crth 16815 eulerthlem2 16819 hashgcdlem 16825 prmgaplem6 17094 znf1o 21570 iundisj 25583 tgcgr4 28539 clwlkclwwlklem2fv1 30014 iundisjf 32602 iundisjfi 32798 fzone1 32802 ply1degltdimlem 33673 ply1degltdim 33674 smattl 33797 smattr 33798 smatbl 33799 signsplypnf 34565 breprexplemc 34647 poimirlem17 37644 poimirlem20 37647 frlmvscadiccat 42516 elfzfzo 45288 dvnmul 45958 iblspltprt 45988 itgspltprt 45994 stoweidlem3 46018 fourierdlem12 46134 fourierdlem50 46171 fourierdlem64 46185 fourierdlem79 46200 iccpartgt 47414 gpgedgvtx1 48020 m1modmmod 48442 |
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