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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 13043 | . . . 4 ⊢ (𝐴 ∈ (0..^𝐵) → 𝐴 ∈ (ℤ≥‘0)) | |
2 | elnn0uz 12284 | . . . 4 ⊢ (𝐴 ∈ ℕ0 ↔ 𝐴 ∈ (ℤ≥‘0)) | |
3 | 1, 2 | sylibr 236 | . . 3 ⊢ (𝐴 ∈ (0..^𝐵) → 𝐴 ∈ ℕ0) |
4 | elfzolt3b 13051 | . . . 4 ⊢ (𝐴 ∈ (0..^𝐵) → 0 ∈ (0..^𝐵)) | |
5 | lbfzo0 13078 | . . . 4 ⊢ (0 ∈ (0..^𝐵) ↔ 𝐵 ∈ ℕ) | |
6 | 4, 5 | sylib 220 | . . 3 ⊢ (𝐴 ∈ (0..^𝐵) → 𝐵 ∈ ℕ) |
7 | elfzolt2 13048 | . . 3 ⊢ (𝐴 ∈ (0..^𝐵) → 𝐴 < 𝐵) | |
8 | 3, 6, 7 | 3jca 1124 | . 2 ⊢ (𝐴 ∈ (0..^𝐵) → (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵)) |
9 | simp1 1132 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℕ0) | |
10 | 9, 2 | sylib 220 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐴 ∈ (ℤ≥‘0)) |
11 | nnz 12005 | . . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ) | |
12 | 11 | 3ad2ant2 1130 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℤ) |
13 | simp3 1134 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵) | |
14 | elfzo2 13042 | . . 3 ⊢ (𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ (ℤ≥‘0) ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵)) | |
15 | 10, 12, 13, 14 | syl3anbrc 1339 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐴 ∈ (0..^𝐵)) |
16 | 8, 15 | impbii 211 | 1 ⊢ (𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ w3a 1083 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 0cc0 10537 < clt 10675 ℕcn 11638 ℕ0cn0 11898 ℤcz 11982 ℤ≥cuz 12244 ..^cfzo 13034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 |
This theorem is referenced by: elfzo0z 13080 nn0p1elfzo 13081 elfzo0le 13082 fzonmapblen 13084 fzofzim 13085 fzo1fzo0n0 13089 ubmelfzo 13103 elfzodifsumelfzo 13104 elfzonlteqm1 13114 fzonn0p1 13115 fzonn0p1p1 13117 elfzo0l 13128 ubmelm1fzo 13134 elfznelfzo 13143 subfzo0 13160 zmodidfzoimp 13270 modfzo0difsn 13312 modsumfzodifsn 13313 addmodlteq 13315 ccatalpha 13947 ccat2s1fvw 13998 ccat2s1fvwOLD 13999 swrdswrd 14067 swrdccatin1 14087 pfxccatin12lem3 14094 repswswrd 14146 cshwidxmod 14165 cshwidxmodr 14166 cshwidx0 14168 cshwidxm1 14169 cshf1 14172 2cshw 14175 cshweqrep 14183 cshw1 14184 cshco 14198 swrds2 14302 pfx2 14309 2swrd2eqwrdeq 14315 wwlktovf 14320 addmodlteqALT 15675 smueqlem 15839 hashgcdlem 16125 prmgaplem3 16389 cshwshashlem2 16430 psgnunilem5 18622 psgnunilem2 18623 psgnunilem3 18624 psgnunilem4 18625 usgr2pthlem 27544 uspgrn2crct 27586 crctcshwlkn0lem4 27591 crctcshwlkn0lem5 27592 crctcshwlkn0 27599 wwlksnredwwlkn 27673 clwlkclwwlklem2fv2 27774 clwlkclwwlklem2a4 27775 clwlkclwwlklem2a 27776 clwwisshclwwslemlem 27791 clwwlkel 27825 wwlksext2clwwlk 27836 umgr2cwwkdifex 27844 clwwlknonex2lem2 27887 upgr3v3e3cycl 27959 upgr4cycl4dv4e 27964 eucrctshift 28022 eucrct2eupth 28024 cshwrnid 30635 fiblem 31656 fib1 31658 fibp1 31659 signstfveq0 31847 lpadleft 31954 poimirlem17 34924 poimirlem20 34927 subsubelfzo0 43546 iccpartigtl 43603 lswn0 43624 bgoldbtbndlem4 43993 |
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