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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 13312 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 13312 | . 2 ⊢ (𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵)) | |
2 | nnz 12228 | . . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ) | |
3 | 2 | 3anim2i 1155 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵)) |
4 | simp1 1138 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℕ0) | |
5 | elnn0z 12218 | . . . . . 6 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℤ ∧ 0 ≤ 𝐴)) | |
6 | 0red 10865 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 0 ∈ ℝ) | |
7 | zre 12209 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
8 | 7 | adantr 484 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℝ) |
9 | zre 12209 | . . . . . . . . . . 11 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ) | |
10 | 9 | adantl 485 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℝ) |
11 | lelttr 10952 | . . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 ≤ 𝐴 ∧ 𝐴 < 𝐵) → 0 < 𝐵)) | |
12 | 6, 8, 10, 11 | syl3anc 1373 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((0 ≤ 𝐴 ∧ 𝐴 < 𝐵) → 0 < 𝐵)) |
13 | elnnz 12215 | . . . . . . . . . . 11 ⊢ (𝐵 ∈ ℕ ↔ (𝐵 ∈ ℤ ∧ 0 < 𝐵)) | |
14 | 13 | simplbi2 504 | . . . . . . . . . 10 ⊢ (𝐵 ∈ ℤ → (0 < 𝐵 → 𝐵 ∈ ℕ)) |
15 | 14 | adantl 485 | . . . . . . . . 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 419 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (0 ≤ 𝐴 → (𝐴 < 𝐵 → 𝐵 ∈ ℕ))) |
18 | 17 | impancom 455 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 0 ≤ 𝐴) → (𝐵 ∈ ℤ → (𝐴 < 𝐵 → 𝐵 ∈ ℕ))) |
19 | 5, 18 | sylbi 220 | . . . . 5 ⊢ (𝐴 ∈ ℕ0 → (𝐵 ∈ ℤ → (𝐴 < 𝐵 → 𝐵 ∈ ℕ))) |
20 | 19 | 3imp 1113 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℕ) |
21 | simp3 1140 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵) | |
22 | 4, 20, 21 | 3jca 1130 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵)) |
23 | 3, 22 | impbii 212 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) ↔ (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵)) |
24 | 1, 23 | bitri 278 | 1 ⊢ (𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 ∈ wcel 2112 class class class wbr 5069 (class class class)co 7234 ℝcr 10757 0cc0 10758 < clt 10896 ≤ cle 10897 ℕcn 11859 ℕ0cn0 12119 ℤcz 12205 ..^cfzo 13267 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 ax-cnex 10814 ax-resscn 10815 ax-1cn 10816 ax-icn 10817 ax-addcl 10818 ax-addrcl 10819 ax-mulcl 10820 ax-mulrcl 10821 ax-mulcom 10822 ax-addass 10823 ax-mulass 10824 ax-distr 10825 ax-i2m1 10826 ax-1ne0 10827 ax-1rid 10828 ax-rnegex 10829 ax-rrecex 10830 ax-cnre 10831 ax-pre-lttri 10832 ax-pre-lttrn 10833 ax-pre-ltadd 10834 ax-pre-mulgt0 10835 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4836 df-iun 4922 df-br 5070 df-opab 5132 df-mpt 5152 df-tr 5178 df-id 5471 df-eprel 5477 df-po 5485 df-so 5486 df-fr 5526 df-we 5528 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-pred 6178 df-ord 6236 df-on 6237 df-lim 6238 df-suc 6239 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-f1 6405 df-fo 6406 df-f1o 6407 df-fv 6408 df-riota 7191 df-ov 7237 df-oprab 7238 df-mpo 7239 df-om 7666 df-1st 7782 df-2nd 7783 df-wrecs 8070 df-recs 8131 df-rdg 8169 df-er 8414 df-en 8650 df-dom 8651 df-sdom 8652 df-pnf 10898 df-mnf 10899 df-xr 10900 df-ltxr 10901 df-le 10902 df-sub 11093 df-neg 11094 df-nn 11860 df-n0 12120 df-z 12206 df-uz 12468 df-fz 13125 df-fzo 13268 |
This theorem is referenced by: ccat2s1fvwALT 14553 ccat2s1fvwALTOLD 14554 clwwlkel 28160 |
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