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Mirrors > Home > MPE Home > Th. List > flval3 | Structured version Visualization version GIF version |
Description: An alternate way to define the floor (greatest integer) function, as the supremum of all integers less than or equal to its argument. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Mario Carneiro, 6-Sep-2014.) |
Ref | Expression |
---|---|
flval3 | ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) = sup({𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴}, ℝ, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3977 | . . . . 5 ⊢ {𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴} ⊆ ℤ | |
2 | zssre 11836 | . . . . 5 ⊢ ℤ ⊆ ℝ | |
3 | 1, 2 | sstri 3898 | . . . 4 ⊢ {𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴} ⊆ ℝ |
4 | 3 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℝ → {𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴} ⊆ ℝ) |
5 | breq1 4965 | . . . . 5 ⊢ (𝑥 = (⌊‘𝐴) → (𝑥 ≤ 𝐴 ↔ (⌊‘𝐴) ≤ 𝐴)) | |
6 | flcl 13015 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℤ) | |
7 | flle 13019 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴) | |
8 | 5, 6, 7 | elrabd 3620 | . . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ {𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴}) |
9 | 8 | ne0d 4221 | . . 3 ⊢ (𝐴 ∈ ℝ → {𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴} ≠ ∅) |
10 | reflcl 13016 | . . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ) | |
11 | breq1 4965 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 ≤ 𝐴 ↔ 𝑧 ≤ 𝐴)) | |
12 | 11 | elrab 3618 | . . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴} ↔ (𝑧 ∈ ℤ ∧ 𝑧 ≤ 𝐴)) |
13 | flge 13025 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑧 ∈ ℤ) → (𝑧 ≤ 𝐴 ↔ 𝑧 ≤ (⌊‘𝐴))) | |
14 | 13 | biimpd 230 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑧 ∈ ℤ) → (𝑧 ≤ 𝐴 → 𝑧 ≤ (⌊‘𝐴))) |
15 | 14 | expimpd 454 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → ((𝑧 ∈ ℤ ∧ 𝑧 ≤ 𝐴) → 𝑧 ≤ (⌊‘𝐴))) |
16 | 12, 15 | syl5bi 243 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝑧 ∈ {𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴} → 𝑧 ≤ (⌊‘𝐴))) |
17 | 16 | ralrimiv 3148 | . . . 4 ⊢ (𝐴 ∈ ℝ → ∀𝑧 ∈ {𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴}𝑧 ≤ (⌊‘𝐴)) |
18 | brralrspcev 5022 | . . . 4 ⊢ (((⌊‘𝐴) ∈ ℝ ∧ ∀𝑧 ∈ {𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴}𝑧 ≤ (⌊‘𝐴)) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ {𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴}𝑧 ≤ 𝑦) | |
19 | 10, 17, 18 | syl2anc 584 | . . 3 ⊢ (𝐴 ∈ ℝ → ∃𝑦 ∈ ℝ ∀𝑧 ∈ {𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴}𝑧 ≤ 𝑦) |
20 | 4, 9, 19, 8 | suprubd 11451 | . 2 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ sup({𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴}, ℝ, < )) |
21 | suprleub 11455 | . . . 4 ⊢ ((({𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴} ⊆ ℝ ∧ {𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴} ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ {𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴}𝑧 ≤ 𝑦) ∧ (⌊‘𝐴) ∈ ℝ) → (sup({𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴}, ℝ, < ) ≤ (⌊‘𝐴) ↔ ∀𝑧 ∈ {𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴}𝑧 ≤ (⌊‘𝐴))) | |
22 | 4, 9, 19, 10, 21 | syl31anc 1366 | . . 3 ⊢ (𝐴 ∈ ℝ → (sup({𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴}, ℝ, < ) ≤ (⌊‘𝐴) ↔ ∀𝑧 ∈ {𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴}𝑧 ≤ (⌊‘𝐴))) |
23 | 17, 22 | mpbird 258 | . 2 ⊢ (𝐴 ∈ ℝ → sup({𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴}, ℝ, < ) ≤ (⌊‘𝐴)) |
24 | 4, 9, 19 | suprcld 11452 | . . 3 ⊢ (𝐴 ∈ ℝ → sup({𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴}, ℝ, < ) ∈ ℝ) |
25 | 10, 24 | letri3d 10629 | . 2 ⊢ (𝐴 ∈ ℝ → ((⌊‘𝐴) = sup({𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴}, ℝ, < ) ↔ ((⌊‘𝐴) ≤ sup({𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴}, ℝ, < ) ∧ sup({𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴}, ℝ, < ) ≤ (⌊‘𝐴)))) |
26 | 20, 23, 25 | mpbir2and 709 | 1 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) = sup({𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴}, ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ∀wral 3105 ∃wrex 3106 {crab 3109 ⊆ wss 3859 ∅c0 4211 class class class wbr 4962 ‘cfv 6225 supcsup 8750 ℝcr 10382 < clt 10521 ≤ cle 10522 ℤcz 11829 ⌊cfl 13010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-sup 8752 df-inf 8753 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-n0 11746 df-z 11830 df-uz 12094 df-fl 13012 |
This theorem is referenced by: (None) |
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