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Mirrors > Home > MPE Home > Th. List > Mathboxes > flsqrt5 | Structured version Visualization version GIF version |
Description: The floor of the square root of a nonnegative number is 5 iff the number is between 25 and 35. (Contributed by AV, 17-Aug-2021.) |
Ref | Expression |
---|---|
flsqrt5 | ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → ((;25 ≤ 𝑋 ∧ 𝑋 < ;36) ↔ (⌊‘(√‘𝑋)) = 5)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 5nn0 12488 | . . 3 ⊢ 5 ∈ ℕ0 | |
2 | flsqrt 46196 | . . 3 ⊢ (((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ 5 ∈ ℕ0) → ((⌊‘(√‘𝑋)) = 5 ↔ ((5↑2) ≤ 𝑋 ∧ 𝑋 < ((5 + 1)↑2)))) | |
3 | 1, 2 | mpan2 690 | . 2 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → ((⌊‘(√‘𝑋)) = 5 ↔ ((5↑2) ≤ 𝑋 ∧ 𝑋 < ((5 + 1)↑2)))) |
4 | 5cn 12296 | . . . . . . 7 ⊢ 5 ∈ ℂ | |
5 | 4 | sqvali 14140 | . . . . . 6 ⊢ (5↑2) = (5 · 5) |
6 | 5t5e25 12776 | . . . . . 6 ⊢ (5 · 5) = ;25 | |
7 | 5, 6 | eqtri 2761 | . . . . 5 ⊢ (5↑2) = ;25 |
8 | 7 | breq1i 5154 | . . . 4 ⊢ ((5↑2) ≤ 𝑋 ↔ ;25 ≤ 𝑋) |
9 | 8 | a1i 11 | . . 3 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → ((5↑2) ≤ 𝑋 ↔ ;25 ≤ 𝑋)) |
10 | 5p1e6 12355 | . . . . . . 7 ⊢ (5 + 1) = 6 | |
11 | 10 | oveq1i 7414 | . . . . . 6 ⊢ ((5 + 1)↑2) = (6↑2) |
12 | 6cn 12299 | . . . . . . . 8 ⊢ 6 ∈ ℂ | |
13 | 12 | sqvali 14140 | . . . . . . 7 ⊢ (6↑2) = (6 · 6) |
14 | 6t6e36 12781 | . . . . . . 7 ⊢ (6 · 6) = ;36 | |
15 | 13, 14 | eqtri 2761 | . . . . . 6 ⊢ (6↑2) = ;36 |
16 | 11, 15 | eqtri 2761 | . . . . 5 ⊢ ((5 + 1)↑2) = ;36 |
17 | 16 | breq2i 5155 | . . . 4 ⊢ (𝑋 < ((5 + 1)↑2) ↔ 𝑋 < ;36) |
18 | 17 | a1i 11 | . . 3 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → (𝑋 < ((5 + 1)↑2) ↔ 𝑋 < ;36)) |
19 | 9, 18 | anbi12d 632 | . 2 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → (((5↑2) ≤ 𝑋 ∧ 𝑋 < ((5 + 1)↑2)) ↔ (;25 ≤ 𝑋 ∧ 𝑋 < ;36))) |
20 | 3, 19 | bitr2d 280 | 1 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → ((;25 ≤ 𝑋 ∧ 𝑋 < ;36) ↔ (⌊‘(√‘𝑋)) = 5)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5147 ‘cfv 6540 (class class class)co 7404 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 < clt 11244 ≤ cle 11245 2c2 12263 3c3 12264 5c5 12266 6c6 12267 ℕ0cn0 12468 ;cdc 12673 ⌊cfl 13751 ↑cexp 14023 √csqrt 15176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-rp 12971 df-fl 13753 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 |
This theorem is referenced by: 31prm 46200 |
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