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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 9915 | . . . 4 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐵 ∈ ℤ) | |
2 | elfzoel2 9916 | . . . 4 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐶 ∈ ℤ) | |
3 | fzof 9914 | . . . . 5 ⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ | |
4 | 3 | fovcl 5869 | . . . 4 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐵..^𝐶) ∈ 𝒫 ℤ) |
5 | 1, 2, 4 | syl2anc 408 | . . 3 ⊢ (𝐴 ∈ (𝐵..^𝐶) → (𝐵..^𝐶) ∈ 𝒫 ℤ) |
6 | 5 | elpwid 3516 | . 2 ⊢ (𝐴 ∈ (𝐵..^𝐶) → (𝐵..^𝐶) ⊆ ℤ) |
7 | id 19 | . 2 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐴 ∈ (𝐵..^𝐶)) | |
8 | 6, 7 | sseldd 3093 | 1 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐴 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 𝒫 cpw 3505 (class class class)co 5767 ℤcz 9047 ..^cfzo 9912 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 df-fz 9784 df-fzo 9913 |
This theorem is referenced by: elfzo2 9920 elfzole1 9925 elfzolt2 9926 elfzolt3 9927 elfzolt2b 9928 elfzouz2 9931 fzonnsub 9939 fzospliti 9946 fzodisj 9948 fzonmapblen 9957 fzoaddel 9962 fzosubel 9964 modaddmodup 10153 modaddmodlo 10154 modfzo0difsn 10161 modsumfzodifsn 10162 addmodlteq 10164 iseqf1olemqk 10260 seq3f1olemp 10268 fzomaxdiflem 10877 fzomaxdif 10878 fzo0dvdseq 11544 fzocongeq 11545 addmodlteqALT 11546 crth 11889 phimullem 11890 hashgcdlem 11892 hashgcdeq 11893 trilpolemeq1 13222 |
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