![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
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 11602 | . . 3 ⊢ 5 ∈ ℕ0 | |
2 | flsqrt 42290 | . . 3 ⊢ (((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ 5 ∈ ℕ0) → ((⌊‘(√‘𝑋)) = 5 ↔ ((5↑2) ≤ 𝑋 ∧ 𝑋 < ((5 + 1)↑2)))) | |
3 | 1, 2 | mpan2 683 | . 2 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → ((⌊‘(√‘𝑋)) = 5 ↔ ((5↑2) ≤ 𝑋 ∧ 𝑋 < ((5 + 1)↑2)))) |
4 | 5cn 11403 | . . . . . . 7 ⊢ 5 ∈ ℂ | |
5 | 4 | sqvali 13197 | . . . . . 6 ⊢ (5↑2) = (5 · 5) |
6 | 5t5e25 11888 | . . . . . 6 ⊢ (5 · 5) = ;25 | |
7 | 5, 6 | eqtri 2821 | . . . . 5 ⊢ (5↑2) = ;25 |
8 | 7 | breq1i 4850 | . . . 4 ⊢ ((5↑2) ≤ 𝑋 ↔ ;25 ≤ 𝑋) |
9 | 8 | a1i 11 | . . 3 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → ((5↑2) ≤ 𝑋 ↔ ;25 ≤ 𝑋)) |
10 | 5p1e6 11467 | . . . . . . 7 ⊢ (5 + 1) = 6 | |
11 | 10 | oveq1i 6888 | . . . . . 6 ⊢ ((5 + 1)↑2) = (6↑2) |
12 | 6cn 11407 | . . . . . . . 8 ⊢ 6 ∈ ℂ | |
13 | 12 | sqvali 13197 | . . . . . . 7 ⊢ (6↑2) = (6 · 6) |
14 | 6t6e36 11893 | . . . . . . 7 ⊢ (6 · 6) = ;36 | |
15 | 13, 14 | eqtri 2821 | . . . . . 6 ⊢ (6↑2) = ;36 |
16 | 11, 15 | eqtri 2821 | . . . . 5 ⊢ ((5 + 1)↑2) = ;36 |
17 | 16 | breq2i 4851 | . . . 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 625 | . 2 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → (((5↑2) ≤ 𝑋 ∧ 𝑋 < ((5 + 1)↑2)) ↔ (;25 ≤ 𝑋 ∧ 𝑋 < ;36))) |
20 | 3, 19 | bitr2d 272 | 1 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → ((;25 ≤ 𝑋 ∧ 𝑋 < ;36) ↔ (⌊‘(√‘𝑋)) = 5)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 class class class wbr 4843 ‘cfv 6101 (class class class)co 6878 ℝcr 10223 0cc0 10224 1c1 10225 + caddc 10227 · cmul 10229 < clt 10363 ≤ cle 10364 2c2 11368 3c3 11369 5c5 11371 6c6 11372 ℕ0cn0 11580 ;cdc 11783 ⌊cfl 12846 ↑cexp 13114 √csqrt 14314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-sup 8590 df-inf 8591 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-z 11667 df-dec 11784 df-uz 11931 df-rp 12075 df-fl 12848 df-seq 13056 df-exp 13115 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 |
This theorem is referenced by: 31prm 42294 |
Copyright terms: Public domain | W3C validator |