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| Mirrors > Home > MPE Home > Th. List > btwnz | Structured version Visualization version GIF version | ||
| Description: Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28. (Contributed by NM, 10-Nov-2004.) |
| Ref | Expression |
|---|---|
| btwnz | ⊢ (𝐴 ∈ ℝ → (∃𝑥 ∈ ℤ 𝑥 < 𝐴 ∧ ∃𝑦 ∈ ℤ 𝐴 < 𝑦)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | renegcl 11539 | . . . 4 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
| 2 | arch 12491 | . . . 4 ⊢ (-𝐴 ∈ ℝ → ∃𝑧 ∈ ℕ -𝐴 < 𝑧) | |
| 3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℝ → ∃𝑧 ∈ ℕ -𝐴 < 𝑧) |
| 4 | nnre 12240 | . . . . . . . 8 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℝ) | |
| 5 | ltnegcon1 11731 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (-𝐴 < 𝑧 ↔ -𝑧 < 𝐴)) | |
| 6 | 5 | ex 412 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝑧 ∈ ℝ → (-𝐴 < 𝑧 ↔ -𝑧 < 𝐴))) |
| 7 | 4, 6 | syl5 34 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝑧 ∈ ℕ → (-𝐴 < 𝑧 ↔ -𝑧 < 𝐴))) |
| 8 | 7 | pm5.32d 577 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → ((𝑧 ∈ ℕ ∧ -𝐴 < 𝑧) ↔ (𝑧 ∈ ℕ ∧ -𝑧 < 𝐴))) |
| 9 | nnnegz 12584 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → -𝑧 ∈ ℤ) | |
| 10 | breq1 5120 | . . . . . . . 8 ⊢ (𝑥 = -𝑧 → (𝑥 < 𝐴 ↔ -𝑧 < 𝐴)) | |
| 11 | 10 | rspcev 3599 | . . . . . . 7 ⊢ ((-𝑧 ∈ ℤ ∧ -𝑧 < 𝐴) → ∃𝑥 ∈ ℤ 𝑥 < 𝐴) |
| 12 | 9, 11 | sylan 580 | . . . . . 6 ⊢ ((𝑧 ∈ ℕ ∧ -𝑧 < 𝐴) → ∃𝑥 ∈ ℤ 𝑥 < 𝐴) |
| 13 | 8, 12 | biimtrdi 253 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((𝑧 ∈ ℕ ∧ -𝐴 < 𝑧) → ∃𝑥 ∈ ℤ 𝑥 < 𝐴)) |
| 14 | 13 | expd 415 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝑧 ∈ ℕ → (-𝐴 < 𝑧 → ∃𝑥 ∈ ℤ 𝑥 < 𝐴))) |
| 15 | 14 | rexlimdv 3137 | . . 3 ⊢ (𝐴 ∈ ℝ → (∃𝑧 ∈ ℕ -𝐴 < 𝑧 → ∃𝑥 ∈ ℤ 𝑥 < 𝐴)) |
| 16 | 3, 15 | mpd 15 | . 2 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℤ 𝑥 < 𝐴) |
| 17 | arch 12491 | . . 3 ⊢ (𝐴 ∈ ℝ → ∃𝑦 ∈ ℕ 𝐴 < 𝑦) | |
| 18 | nnz 12602 | . . . . 5 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ) | |
| 19 | 18 | anim1i 615 | . . . 4 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 < 𝑦) → (𝑦 ∈ ℤ ∧ 𝐴 < 𝑦)) |
| 20 | 19 | reximi2 3068 | . . 3 ⊢ (∃𝑦 ∈ ℕ 𝐴 < 𝑦 → ∃𝑦 ∈ ℤ 𝐴 < 𝑦) |
| 21 | 17, 20 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ → ∃𝑦 ∈ ℤ 𝐴 < 𝑦) |
| 22 | 16, 21 | jca 511 | 1 ⊢ (𝐴 ∈ ℝ → (∃𝑥 ∈ ℤ 𝑥 < 𝐴 ∧ ∃𝑦 ∈ ℤ 𝐴 < 𝑦)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2107 ∃wrex 3059 class class class wbr 5117 ℝcr 11121 < clt 11262 -cneg 11460 ℕcn 12233 ℤcz 12581 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5264 ax-nul 5274 ax-pow 5333 ax-pr 5400 ax-un 7724 ax-resscn 11179 ax-1cn 11180 ax-icn 11181 ax-addcl 11182 ax-addrcl 11183 ax-mulcl 11184 ax-mulrcl 11185 ax-mulcom 11186 ax-addass 11187 ax-mulass 11188 ax-distr 11189 ax-i2m1 11190 ax-1ne0 11191 ax-1rid 11192 ax-rnegex 11193 ax-rrecex 11194 ax-cnre 11195 ax-pre-lttri 11196 ax-pre-lttrn 11197 ax-pre-ltadd 11198 ax-pre-mulgt0 11199 ax-pre-sup 11200 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3358 df-rab 3414 df-v 3459 df-sbc 3764 df-csb 3873 df-dif 3927 df-un 3929 df-in 3931 df-ss 3941 df-pss 3944 df-nul 4307 df-if 4499 df-pw 4575 df-sn 4600 df-pr 4602 df-op 4606 df-uni 4882 df-iun 4967 df-br 5118 df-opab 5180 df-mpt 5200 df-tr 5228 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6288 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7857 df-2nd 7984 df-frecs 8275 df-wrecs 8306 df-recs 8380 df-rdg 8419 df-er 8714 df-en 8955 df-dom 8956 df-sdom 8957 df-pnf 11264 df-mnf 11265 df-xr 11266 df-ltxr 11267 df-le 11268 df-sub 11461 df-neg 11462 df-nn 12234 df-z 12582 |
| This theorem is referenced by: lbzbi 12945 rpnnen1lem2 12986 rpnnen1lem1 12987 rpnnen1lem3 12988 rpnnen1lem5 12990 fourierdlem64 46135 |
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