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Mirrors > Home > ILE Home > Th. List > elfzoelz | GIF version |
Description: Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
Ref | Expression |
---|---|
elfzoelz | ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐴 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzoel1 10048 | . . . 4 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐵 ∈ ℤ) | |
2 | elfzoel2 10049 | . . . 4 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐶 ∈ ℤ) | |
3 | fzof 10047 | . . . . 5 ⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ | |
4 | 3 | fovcl 5927 | . . . 4 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐵..^𝐶) ∈ 𝒫 ℤ) |
5 | 1, 2, 4 | syl2anc 409 | . . 3 ⊢ (𝐴 ∈ (𝐵..^𝐶) → (𝐵..^𝐶) ∈ 𝒫 ℤ) |
6 | 5 | elpwid 3554 | . 2 ⊢ (𝐴 ∈ (𝐵..^𝐶) → (𝐵..^𝐶) ⊆ ℤ) |
7 | id 19 | . 2 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐴 ∈ (𝐵..^𝐶)) | |
8 | 6, 7 | sseldd 3129 | 1 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐴 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2128 𝒫 cpw 3543 (class class class)co 5825 ℤcz 9168 ..^cfzo 10045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-addcom 7833 ax-addass 7835 ax-distr 7837 ax-i2m1 7838 ax-0lt1 7839 ax-0id 7841 ax-rnegex 7842 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-ltadd 7849 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-fv 5179 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-inn 8835 df-n0 9092 df-z 9169 df-fz 9914 df-fzo 10046 |
This theorem is referenced by: elfzo2 10053 elfzole1 10058 elfzolt2 10059 elfzolt3 10060 elfzolt2b 10061 elfzouz2 10064 fzonnsub 10072 fzospliti 10079 fzodisj 10081 fzonmapblen 10090 fzoaddel 10095 fzosubel 10097 modaddmodup 10290 modaddmodlo 10291 modfzo0difsn 10298 modsumfzodifsn 10299 addmodlteq 10301 iseqf1olemqk 10397 seq3f1olemp 10405 fzomaxdiflem 11016 fzomaxdif 11017 fzo0dvdseq 11753 fzocongeq 11754 addmodlteqALT 11755 crth 12103 phimullem 12104 eulerthlem1 12106 eulerthlemfi 12107 eulerthlemrprm 12108 hashgcdlem 12117 hashgcdeq 12118 phisum 12119 reumodprminv 12132 modprm0 12133 nnnn0modprm0 12134 modprmn0modprm0 12135 nninfdclemlt 12224 trilpolemeq1 13653 |
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