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Mirrors > Home > MPE Home > Th. List > elfzo0z | Structured version Visualization version GIF version |
Description: Membership in a half-open range of nonnegative integers, generalization of elfzo0 13474 requiring the upper bound to be an integer only. (Contributed by Alexander van der Vekens, 23-Sep-2018.) |
Ref | Expression |
---|---|
elfzo0z | ⊢ (𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzo0 13474 | . 2 ⊢ (𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵)) | |
2 | nnz 12388 | . . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ) | |
3 | 2 | 3anim2i 1153 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵)) |
4 | simp1 1136 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℕ0) | |
5 | elnn0z 12378 | . . . . . 6 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℤ ∧ 0 ≤ 𝐴)) | |
6 | 0red 11024 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 0 ∈ ℝ) | |
7 | zre 12369 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
8 | 7 | adantr 482 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℝ) |
9 | zre 12369 | . . . . . . . . . . 11 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ) | |
10 | 9 | adantl 483 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℝ) |
11 | lelttr 11111 | . . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 ≤ 𝐴 ∧ 𝐴 < 𝐵) → 0 < 𝐵)) | |
12 | 6, 8, 10, 11 | syl3anc 1371 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((0 ≤ 𝐴 ∧ 𝐴 < 𝐵) → 0 < 𝐵)) |
13 | elnnz 12375 | . . . . . . . . . . 11 ⊢ (𝐵 ∈ ℕ ↔ (𝐵 ∈ ℤ ∧ 0 < 𝐵)) | |
14 | 13 | simplbi2 502 | . . . . . . . . . 10 ⊢ (𝐵 ∈ ℤ → (0 < 𝐵 → 𝐵 ∈ ℕ)) |
15 | 14 | adantl 483 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (0 < 𝐵 → 𝐵 ∈ ℕ)) |
16 | 12, 15 | syld 47 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((0 ≤ 𝐴 ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℕ)) |
17 | 16 | expd 417 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (0 ≤ 𝐴 → (𝐴 < 𝐵 → 𝐵 ∈ ℕ))) |
18 | 17 | impancom 453 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 0 ≤ 𝐴) → (𝐵 ∈ ℤ → (𝐴 < 𝐵 → 𝐵 ∈ ℕ))) |
19 | 5, 18 | sylbi 216 | . . . . 5 ⊢ (𝐴 ∈ ℕ0 → (𝐵 ∈ ℤ → (𝐴 < 𝐵 → 𝐵 ∈ ℕ))) |
20 | 19 | 3imp 1111 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℕ) |
21 | simp3 1138 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵) | |
22 | 4, 20, 21 | 3jca 1128 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵)) |
23 | 3, 22 | impbii 208 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) ↔ (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵)) |
24 | 1, 23 | bitri 275 | 1 ⊢ (𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1087 ∈ wcel 2104 class class class wbr 5081 (class class class)co 7307 ℝcr 10916 0cc0 10917 < clt 11055 ≤ cle 11056 ℕcn 12019 ℕ0cn0 12279 ℤcz 12365 ..^cfzo 13428 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-er 8529 df-en 8765 df-dom 8766 df-sdom 8767 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-nn 12020 df-n0 12280 df-z 12366 df-uz 12629 df-fz 13286 df-fzo 13429 |
This theorem is referenced by: ccat2s1fvwALT 14714 ccat2s1fvwALTOLD 14715 clwwlkel 28455 |
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