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Mirrors > Home > ILE Home > Th. List > elfzo0 | 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 9921 | . . . 4 ⊢ (𝐴 ∈ (0..^𝐵) → 𝐴 ∈ (ℤ≥‘0)) | |
2 | elnn0uz 9356 | . . . 4 ⊢ (𝐴 ∈ ℕ0 ↔ 𝐴 ∈ (ℤ≥‘0)) | |
3 | 1, 2 | sylibr 133 | . . 3 ⊢ (𝐴 ∈ (0..^𝐵) → 𝐴 ∈ ℕ0) |
4 | elfzolt3b 9929 | . . . 4 ⊢ (𝐴 ∈ (0..^𝐵) → 0 ∈ (0..^𝐵)) | |
5 | lbfzo0 9951 | . . . 4 ⊢ (0 ∈ (0..^𝐵) ↔ 𝐵 ∈ ℕ) | |
6 | 4, 5 | sylib 121 | . . 3 ⊢ (𝐴 ∈ (0..^𝐵) → 𝐵 ∈ ℕ) |
7 | elfzolt2 9926 | . . 3 ⊢ (𝐴 ∈ (0..^𝐵) → 𝐴 < 𝐵) | |
8 | 3, 6, 7 | 3jca 1161 | . 2 ⊢ (𝐴 ∈ (0..^𝐵) → (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵)) |
9 | simp1 981 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℕ0) | |
10 | 9, 2 | sylib 121 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐴 ∈ (ℤ≥‘0)) |
11 | nnz 9066 | . . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ) | |
12 | 11 | 3ad2ant2 1003 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℤ) |
13 | simp3 983 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵) | |
14 | elfzo2 9920 | . . 3 ⊢ (𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ (ℤ≥‘0) ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵)) | |
15 | 10, 12, 13, 14 | syl3anbrc 1165 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐴 ∈ (0..^𝐵)) |
16 | 8, 15 | impbii 125 | 1 ⊢ (𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∧ w3a 962 ∈ wcel 1480 class class class wbr 3924 ‘cfv 5118 (class class class)co 5767 0cc0 7613 < clt 7793 ℕcn 8713 ℕ0cn0 8970 ℤcz 9047 ℤ≥cuz 9319 ..^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-uz 9320 df-fz 9784 df-fzo 9913 |
This theorem is referenced by: fzo1fzo0n0 9953 elfzo0z 9954 elfzo0le 9955 fzonmapblen 9957 fzofzim 9958 ubmelfzo 9970 elfzodifsumelfzo 9971 elfzonlteqm1 9980 fzonn0p1 9981 fzonn0p1p1 9983 elfzom1p1elfzo 9984 ubmelm1fzo 9996 subfzo0 10012 zmodidfzoimp 10120 modfzo0difsn 10161 modsumfzodifsn 10162 addmodlteq 10164 addmodlteqALT 11546 hashgcdlem 11892 |
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