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Mirrors > Home > MPE Home > Th. List > zmin | Structured version Visualization version GIF version |
Description: There is a unique smallest integer greater than or equal to a given real number. (Contributed by NM, 12-Nov-2004.) (Revised by Mario Carneiro, 13-Jun-2014.) |
Ref | Expression |
---|---|
zmin | ⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ ℤ (𝐴 ≤ 𝑥 ∧ ∀𝑦 ∈ ℤ (𝐴 ≤ 𝑦 → 𝑥 ≤ 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnssz 12633 | . . . . . 6 ⊢ ℕ ⊆ ℤ | |
2 | arch 12521 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → ∃𝑧 ∈ ℕ 𝐴 < 𝑧) | |
3 | ssrexv 4065 | . . . . . 6 ⊢ (ℕ ⊆ ℤ → (∃𝑧 ∈ ℕ 𝐴 < 𝑧 → ∃𝑧 ∈ ℤ 𝐴 < 𝑧)) | |
4 | 1, 2, 3 | mpsyl 68 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ∃𝑧 ∈ ℤ 𝐴 < 𝑧) |
5 | zre 12615 | . . . . . . 7 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℝ) | |
6 | ltle 11347 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝐴 < 𝑧 → 𝐴 ≤ 𝑧)) | |
7 | 5, 6 | sylan2 593 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑧 ∈ ℤ) → (𝐴 < 𝑧 → 𝐴 ≤ 𝑧)) |
8 | 7 | reximdva 3166 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (∃𝑧 ∈ ℤ 𝐴 < 𝑧 → ∃𝑧 ∈ ℤ 𝐴 ≤ 𝑧)) |
9 | 4, 8 | mpd 15 | . . . 4 ⊢ (𝐴 ∈ ℝ → ∃𝑧 ∈ ℤ 𝐴 ≤ 𝑧) |
10 | rabn0 4395 | . . . 4 ⊢ ({𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧} ≠ ∅ ↔ ∃𝑧 ∈ ℤ 𝐴 ≤ 𝑧) | |
11 | 9, 10 | sylibr 234 | . . 3 ⊢ (𝐴 ∈ ℝ → {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧} ≠ ∅) |
12 | breq2 5152 | . . . . . 6 ⊢ (𝑧 = 𝑛 → (𝐴 ≤ 𝑧 ↔ 𝐴 ≤ 𝑛)) | |
13 | 12 | cbvrabv 3444 | . . . . 5 ⊢ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧} = {𝑛 ∈ ℤ ∣ 𝐴 ≤ 𝑛} |
14 | 13 | eqimssi 4056 | . . . 4 ⊢ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧} ⊆ {𝑛 ∈ ℤ ∣ 𝐴 ≤ 𝑛} |
15 | uzwo3 12983 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ ({𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧} ⊆ {𝑛 ∈ ℤ ∣ 𝐴 ≤ 𝑛} ∧ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧} ≠ ∅)) → ∃!𝑥 ∈ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧}∀𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧}𝑥 ≤ 𝑦) | |
16 | 14, 15 | mpanr1 703 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧} ≠ ∅) → ∃!𝑥 ∈ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧}∀𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧}𝑥 ≤ 𝑦) |
17 | 11, 16 | mpdan 687 | . 2 ⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧}∀𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧}𝑥 ≤ 𝑦) |
18 | breq2 5152 | . . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝐴 ≤ 𝑧 ↔ 𝐴 ≤ 𝑥)) | |
19 | 18 | elrab 3695 | . . . . . 6 ⊢ (𝑥 ∈ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧} ↔ (𝑥 ∈ ℤ ∧ 𝐴 ≤ 𝑥)) |
20 | breq2 5152 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝐴 ≤ 𝑧 ↔ 𝐴 ≤ 𝑦)) | |
21 | 20 | ralrab 3702 | . . . . . 6 ⊢ (∀𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧}𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ ℤ (𝐴 ≤ 𝑦 → 𝑥 ≤ 𝑦)) |
22 | 19, 21 | anbi12i 628 | . . . . 5 ⊢ ((𝑥 ∈ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧} ∧ ∀𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧}𝑥 ≤ 𝑦) ↔ ((𝑥 ∈ ℤ ∧ 𝐴 ≤ 𝑥) ∧ ∀𝑦 ∈ ℤ (𝐴 ≤ 𝑦 → 𝑥 ≤ 𝑦))) |
23 | anass 468 | . . . . 5 ⊢ (((𝑥 ∈ ℤ ∧ 𝐴 ≤ 𝑥) ∧ ∀𝑦 ∈ ℤ (𝐴 ≤ 𝑦 → 𝑥 ≤ 𝑦)) ↔ (𝑥 ∈ ℤ ∧ (𝐴 ≤ 𝑥 ∧ ∀𝑦 ∈ ℤ (𝐴 ≤ 𝑦 → 𝑥 ≤ 𝑦)))) | |
24 | 22, 23 | bitri 275 | . . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧} ∧ ∀𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧}𝑥 ≤ 𝑦) ↔ (𝑥 ∈ ℤ ∧ (𝐴 ≤ 𝑥 ∧ ∀𝑦 ∈ ℤ (𝐴 ≤ 𝑦 → 𝑥 ≤ 𝑦)))) |
25 | 24 | eubii 2583 | . . 3 ⊢ (∃!𝑥(𝑥 ∈ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧} ∧ ∀𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧}𝑥 ≤ 𝑦) ↔ ∃!𝑥(𝑥 ∈ ℤ ∧ (𝐴 ≤ 𝑥 ∧ ∀𝑦 ∈ ℤ (𝐴 ≤ 𝑦 → 𝑥 ≤ 𝑦)))) |
26 | df-reu 3379 | . . 3 ⊢ (∃!𝑥 ∈ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧}∀𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧}𝑥 ≤ 𝑦 ↔ ∃!𝑥(𝑥 ∈ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧} ∧ ∀𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧}𝑥 ≤ 𝑦)) | |
27 | df-reu 3379 | . . 3 ⊢ (∃!𝑥 ∈ ℤ (𝐴 ≤ 𝑥 ∧ ∀𝑦 ∈ ℤ (𝐴 ≤ 𝑦 → 𝑥 ≤ 𝑦)) ↔ ∃!𝑥(𝑥 ∈ ℤ ∧ (𝐴 ≤ 𝑥 ∧ ∀𝑦 ∈ ℤ (𝐴 ≤ 𝑦 → 𝑥 ≤ 𝑦)))) | |
28 | 25, 26, 27 | 3bitr4i 303 | . 2 ⊢ (∃!𝑥 ∈ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧}∀𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝐴 ≤ 𝑧}𝑥 ≤ 𝑦 ↔ ∃!𝑥 ∈ ℤ (𝐴 ≤ 𝑥 ∧ ∀𝑦 ∈ ℤ (𝐴 ≤ 𝑦 → 𝑥 ≤ 𝑦))) |
29 | 17, 28 | sylib 218 | 1 ⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ ℤ (𝐴 ≤ 𝑥 ∧ ∀𝑦 ∈ ℤ (𝐴 ≤ 𝑦 → 𝑥 ≤ 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ∃!weu 2566 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 ∃!wreu 3376 {crab 3433 ⊆ wss 3963 ∅c0 4339 class class class wbr 5148 ℝcr 11152 < clt 11293 ≤ cle 11294 ℕcn 12264 ℤcz 12611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 |
This theorem is referenced by: zmax 12985 zbtwnre 12986 |
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