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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 13576 | . . . 4 ⊢ (𝐴 ∈ (0..^𝐵) → 𝐴 ∈ (ℤ≥‘0)) | |
2 | elnn0uz 12808 | . . . 4 ⊢ (𝐴 ∈ ℕ0 ↔ 𝐴 ∈ (ℤ≥‘0)) | |
3 | 1, 2 | sylibr 233 | . . 3 ⊢ (𝐴 ∈ (0..^𝐵) → 𝐴 ∈ ℕ0) |
4 | elfzolt3b 13584 | . . . 4 ⊢ (𝐴 ∈ (0..^𝐵) → 0 ∈ (0..^𝐵)) | |
5 | lbfzo0 13612 | . . . 4 ⊢ (0 ∈ (0..^𝐵) ↔ 𝐵 ∈ ℕ) | |
6 | 4, 5 | sylib 217 | . . 3 ⊢ (𝐴 ∈ (0..^𝐵) → 𝐵 ∈ ℕ) |
7 | elfzolt2 13581 | . . 3 ⊢ (𝐴 ∈ (0..^𝐵) → 𝐴 < 𝐵) | |
8 | 3, 6, 7 | 3jca 1128 | . 2 ⊢ (𝐴 ∈ (0..^𝐵) → (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵)) |
9 | simp1 1136 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℕ0) | |
10 | 9, 2 | sylib 217 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐴 ∈ (ℤ≥‘0)) |
11 | nnz 12520 | . . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ) | |
12 | 11 | 3ad2ant2 1134 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℤ) |
13 | simp3 1138 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵) | |
14 | elfzo2 13575 | . . 3 ⊢ (𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ (ℤ≥‘0) ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵)) | |
15 | 10, 12, 13, 14 | syl3anbrc 1343 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐴 ∈ (0..^𝐵)) |
16 | 8, 15 | impbii 208 | 1 ⊢ (𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ w3a 1087 ∈ wcel 2106 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 0cc0 11051 < clt 11189 ℕcn 12153 ℕ0cn0 12413 ℤcz 12499 ℤ≥cuz 12763 ..^cfzo 13567 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-fzo 13568 |
This theorem is referenced by: elfzo0z 13614 nn0p1elfzo 13615 elfzo0le 13616 fzonmapblen 13618 fzofzim 13619 fzo1fzo0n0 13623 ubmelfzo 13637 elfzodifsumelfzo 13638 elfzonlteqm1 13648 fzonn0p1 13649 fzonn0p1p1 13651 elfzo0l 13662 ubmelm1fzo 13668 elfznelfzo 13677 subfzo0 13694 zmodidfzoimp 13806 modfzo0difsn 13848 modsumfzodifsn 13849 addmodlteq 13851 ccatalpha 14481 ccat2s1fvw 14526 swrdswrd 14593 swrdccatin1 14613 pfxccatin12lem3 14620 repswswrd 14672 cshwidxmod 14691 cshwidxmodr 14692 cshwidx0 14694 cshwidxm1 14695 cshf1 14698 2cshw 14701 cshweqrep 14709 cshw1 14710 cshco 14725 swrds2 14829 pfx2 14836 2swrd2eqwrdeq 14842 wwlktovf 14845 addmodlteqALT 16207 smueqlem 16370 hashgcdlem 16660 prmgaplem3 16925 cshwshashlem2 16969 psgnunilem5 19276 psgnunilem2 19277 psgnunilem3 19278 psgnunilem4 19279 usgr2pthlem 28711 uspgrn2crct 28753 crctcshwlkn0lem4 28758 crctcshwlkn0lem5 28759 crctcshwlkn0 28766 wwlksnredwwlkn 28840 clwlkclwwlklem2fv2 28940 clwlkclwwlklem2a4 28941 clwlkclwwlklem2a 28942 clwwisshclwwslemlem 28957 clwwlkel 28990 wwlksext2clwwlk 29001 umgr2cwwkdifex 29009 clwwlknonex2lem2 29052 upgr3v3e3cycl 29124 upgr4cycl4dv4e 29129 eucrctshift 29187 eucrct2eupth 29189 cshwrnid 31815 fiblem 32998 fib1 33000 fibp1 33001 signstfveq0 33189 lpadleft 33296 poimirlem17 36095 poimirlem20 36098 subsubelfzo0 45548 iccpartigtl 45605 lswn0 45626 bgoldbtbndlem4 45990 |
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