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Mirrors > Home > MPE Home > Th. List > elfzo0 | Structured version Visualization version GIF version |
Description: Membership in a half-open integer range based at 0. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.) |
Ref | Expression |
---|---|
elfzo0 | ⊢ (𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzouz 13037 | . . . 4 ⊢ (𝐴 ∈ (0..^𝐵) → 𝐴 ∈ (ℤ≥‘0)) | |
2 | elnn0uz 12271 | . . . 4 ⊢ (𝐴 ∈ ℕ0 ↔ 𝐴 ∈ (ℤ≥‘0)) | |
3 | 1, 2 | sylibr 237 | . . 3 ⊢ (𝐴 ∈ (0..^𝐵) → 𝐴 ∈ ℕ0) |
4 | elfzolt3b 13045 | . . . 4 ⊢ (𝐴 ∈ (0..^𝐵) → 0 ∈ (0..^𝐵)) | |
5 | lbfzo0 13072 | . . . 4 ⊢ (0 ∈ (0..^𝐵) ↔ 𝐵 ∈ ℕ) | |
6 | 4, 5 | sylib 221 | . . 3 ⊢ (𝐴 ∈ (0..^𝐵) → 𝐵 ∈ ℕ) |
7 | elfzolt2 13042 | . . 3 ⊢ (𝐴 ∈ (0..^𝐵) → 𝐴 < 𝐵) | |
8 | 3, 6, 7 | 3jca 1125 | . 2 ⊢ (𝐴 ∈ (0..^𝐵) → (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵)) |
9 | simp1 1133 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℕ0) | |
10 | 9, 2 | sylib 221 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐴 ∈ (ℤ≥‘0)) |
11 | nnz 11992 | . . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ) | |
12 | 11 | 3ad2ant2 1131 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℤ) |
13 | simp3 1135 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵) | |
14 | elfzo2 13036 | . . 3 ⊢ (𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ (ℤ≥‘0) ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵)) | |
15 | 10, 12, 13, 14 | syl3anbrc 1340 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐴 ∈ (0..^𝐵)) |
16 | 8, 15 | impbii 212 | 1 ⊢ (𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ w3a 1084 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 0cc0 10526 < clt 10664 ℕcn 11625 ℕ0cn0 11885 ℤcz 11969 ℤ≥cuz 12231 ..^cfzo 13028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 |
This theorem is referenced by: elfzo0z 13074 nn0p1elfzo 13075 elfzo0le 13076 fzonmapblen 13078 fzofzim 13079 fzo1fzo0n0 13083 ubmelfzo 13097 elfzodifsumelfzo 13098 elfzonlteqm1 13108 fzonn0p1 13109 fzonn0p1p1 13111 elfzo0l 13122 ubmelm1fzo 13128 elfznelfzo 13137 subfzo0 13154 zmodidfzoimp 13264 modfzo0difsn 13306 modsumfzodifsn 13307 addmodlteq 13309 ccatalpha 13938 ccat2s1fvw 13989 ccat2s1fvwOLD 13990 swrdswrd 14058 swrdccatin1 14078 pfxccatin12lem3 14085 repswswrd 14137 cshwidxmod 14156 cshwidxmodr 14157 cshwidx0 14159 cshwidxm1 14160 cshf1 14163 2cshw 14166 cshweqrep 14174 cshw1 14175 cshco 14189 swrds2 14293 pfx2 14300 2swrd2eqwrdeq 14306 wwlktovf 14311 addmodlteqALT 15667 smueqlem 15829 hashgcdlem 16115 prmgaplem3 16379 cshwshashlem2 16422 psgnunilem5 18614 psgnunilem2 18615 psgnunilem3 18616 psgnunilem4 18617 usgr2pthlem 27552 uspgrn2crct 27594 crctcshwlkn0lem4 27599 crctcshwlkn0lem5 27600 crctcshwlkn0 27607 wwlksnredwwlkn 27681 clwlkclwwlklem2fv2 27781 clwlkclwwlklem2a4 27782 clwlkclwwlklem2a 27783 clwwisshclwwslemlem 27798 clwwlkel 27831 wwlksext2clwwlk 27842 umgr2cwwkdifex 27850 clwwlknonex2lem2 27893 upgr3v3e3cycl 27965 upgr4cycl4dv4e 27970 eucrctshift 28028 eucrct2eupth 28030 cshwrnid 30661 fiblem 31766 fib1 31768 fibp1 31769 signstfveq0 31957 lpadleft 32064 poimirlem17 35074 poimirlem20 35077 subsubelfzo0 43883 iccpartigtl 43940 lswn0 43961 bgoldbtbndlem4 44326 |
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