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Mirrors > Home > MPE Home > Th. List > fzossfz | Structured version Visualization version GIF version |
Description: A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.) |
Ref | Expression |
---|---|
fzossfz | ⊢ (𝐴..^𝐵) ⊆ (𝐴...𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzofz 12692 | . 2 ⊢ (𝑥 ∈ (𝐴..^𝐵) → 𝑥 ∈ (𝐴...𝐵)) | |
2 | 1 | ssriv 3756 | 1 ⊢ (𝐴..^𝐵) ⊆ (𝐴...𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3723 (class class class)co 6792 ...cfz 12532 ..^cfzo 12672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-1st 7314 df-2nd 7315 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-er 7895 df-en 8109 df-dom 8110 df-sdom 8111 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-nn 11222 df-n0 11494 df-z 11579 df-uz 11888 df-fz 12533 df-fzo 12673 |
This theorem is referenced by: fzossnn0 12706 fzossnn 12724 elfzom1elp1fzo 12742 injresinjlem 12795 injresinj 12796 zmodfzp1 12901 uzindi 12988 wrdind 13684 wrd2ind 13685 scshwfzeqfzo 13780 telfsumo 14740 dfphi2 15685 cshwshashlem1 16008 psgnunilem5 18120 psgnunilem2 18121 efgredlemf 18360 efgredlemd 18363 efgredlemc 18364 uspgr2wlkeq 26776 wlkres 26801 redwlklem 26802 pthdivtx 26859 eucrct2eupth 27424 signstfvn 30983 signsvtn0 30984 breprexplemc 31047 fzossuz 40110 fzossz 40111 fourierdlem20 40857 fourierdlem25 40862 fourierdlem37 40874 fourierdlem64 40900 fourierdlem79 40915 fourierdlem89 40925 fourierdlem91 40927 fourierdlem101 40937 iccpartres 41878 iccpartipre 41881 iccpartleu 41888 bgoldbtbndlem2 42218 |
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