Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > rebtwnz | Structured version Visualization version GIF version |
Description: There is a unique greatest integer less than or equal to a real number. Exercise 4 of [Apostol] p. 28. (Contributed by NM, 15-Nov-2004.) |
Ref | Expression |
---|---|
rebtwnz | ⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | renegcl 10943 | . . 3 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
2 | zbtwnre 12340 | . . 3 ⊢ (-𝐴 ∈ ℝ → ∃!𝑦 ∈ ℤ (-𝐴 ≤ 𝑦 ∧ 𝑦 < (-𝐴 + 1))) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ → ∃!𝑦 ∈ ℤ (-𝐴 ≤ 𝑦 ∧ 𝑦 < (-𝐴 + 1))) |
4 | znegcl 12011 | . . . 4 ⊢ (𝑥 ∈ ℤ → -𝑥 ∈ ℤ) | |
5 | znegcl 12011 | . . . . 5 ⊢ (𝑦 ∈ ℤ → -𝑦 ∈ ℤ) | |
6 | zcn 11980 | . . . . . 6 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ) | |
7 | zcn 11980 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
8 | negcon2 10933 | . . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑦 = -𝑥 ↔ 𝑥 = -𝑦)) | |
9 | 6, 7, 8 | syl2an 597 | . . . . 5 ⊢ ((𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ) → (𝑦 = -𝑥 ↔ 𝑥 = -𝑦)) |
10 | 5, 9 | reuhyp 5312 | . . . 4 ⊢ (𝑦 ∈ ℤ → ∃!𝑥 ∈ ℤ 𝑦 = -𝑥) |
11 | breq2 5062 | . . . . 5 ⊢ (𝑦 = -𝑥 → (-𝐴 ≤ 𝑦 ↔ -𝐴 ≤ -𝑥)) | |
12 | breq1 5061 | . . . . 5 ⊢ (𝑦 = -𝑥 → (𝑦 < (-𝐴 + 1) ↔ -𝑥 < (-𝐴 + 1))) | |
13 | 11, 12 | anbi12d 632 | . . . 4 ⊢ (𝑦 = -𝑥 → ((-𝐴 ≤ 𝑦 ∧ 𝑦 < (-𝐴 + 1)) ↔ (-𝐴 ≤ -𝑥 ∧ -𝑥 < (-𝐴 + 1)))) |
14 | 4, 10, 13 | reuxfr1 3742 | . . 3 ⊢ (∃!𝑦 ∈ ℤ (-𝐴 ≤ 𝑦 ∧ 𝑦 < (-𝐴 + 1)) ↔ ∃!𝑥 ∈ ℤ (-𝐴 ≤ -𝑥 ∧ -𝑥 < (-𝐴 + 1))) |
15 | zre 11979 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ) | |
16 | leneg 11137 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑥 ≤ 𝐴 ↔ -𝐴 ≤ -𝑥)) | |
17 | 16 | ancoms 461 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑥 ≤ 𝐴 ↔ -𝐴 ≤ -𝑥)) |
18 | peano2rem 10947 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℝ) | |
19 | ltneg 11134 | . . . . . . . . 9 ⊢ (((𝐴 − 1) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝐴 − 1) < 𝑥 ↔ -𝑥 < -(𝐴 − 1))) | |
20 | 18, 19 | sylan 582 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝐴 − 1) < 𝑥 ↔ -𝑥 < -(𝐴 − 1))) |
21 | 1re 10635 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
22 | ltsubadd 11104 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝐴 − 1) < 𝑥 ↔ 𝐴 < (𝑥 + 1))) | |
23 | 21, 22 | mp3an2 1445 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝐴 − 1) < 𝑥 ↔ 𝐴 < (𝑥 + 1))) |
24 | recn 10621 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
25 | ax-1cn 10589 | . . . . . . . . . . 11 ⊢ 1 ∈ ℂ | |
26 | negsubdi 10936 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝐴 − 1) = (-𝐴 + 1)) | |
27 | 24, 25, 26 | sylancl 588 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → -(𝐴 − 1) = (-𝐴 + 1)) |
28 | 27 | adantr 483 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → -(𝐴 − 1) = (-𝐴 + 1)) |
29 | 28 | breq2d 5070 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (-𝑥 < -(𝐴 − 1) ↔ -𝑥 < (-𝐴 + 1))) |
30 | 20, 23, 29 | 3bitr3d 311 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝐴 < (𝑥 + 1) ↔ -𝑥 < (-𝐴 + 1))) |
31 | 17, 30 | anbi12d 632 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ↔ (-𝐴 ≤ -𝑥 ∧ -𝑥 < (-𝐴 + 1)))) |
32 | 15, 31 | sylan2 594 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℤ) → ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ↔ (-𝐴 ≤ -𝑥 ∧ -𝑥 < (-𝐴 + 1)))) |
33 | 32 | bicomd 225 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℤ) → ((-𝐴 ≤ -𝑥 ∧ -𝑥 < (-𝐴 + 1)) ↔ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))) |
34 | 33 | reubidva 3388 | . . 3 ⊢ (𝐴 ∈ ℝ → (∃!𝑥 ∈ ℤ (-𝐴 ≤ -𝑥 ∧ -𝑥 < (-𝐴 + 1)) ↔ ∃!𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))) |
35 | 14, 34 | syl5bb 285 | . 2 ⊢ (𝐴 ∈ ℝ → (∃!𝑦 ∈ ℤ (-𝐴 ≤ 𝑦 ∧ 𝑦 < (-𝐴 + 1)) ↔ ∃!𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))) |
36 | 3, 35 | mpbid 234 | 1 ⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃!wreu 3140 class class class wbr 5058 (class class class)co 7150 ℂcc 10529 ℝcr 10530 1c1 10532 + caddc 10534 < clt 10669 ≤ cle 10670 − cmin 10864 -cneg 10865 ℤcz 11975 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 |
This theorem is referenced by: flcl 13159 fllelt 13161 flflp1 13171 flbi 13180 ltflcei 34874 |
Copyright terms: Public domain | W3C validator |