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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 12075 | . . 3 ⊢ 5 ∈ ℕ0 | |
2 | flsqrt 44661 | . . 3 ⊢ (((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ 5 ∈ ℕ0) → ((⌊‘(√‘𝑋)) = 5 ↔ ((5↑2) ≤ 𝑋 ∧ 𝑋 < ((5 + 1)↑2)))) | |
3 | 1, 2 | mpan2 691 | . 2 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → ((⌊‘(√‘𝑋)) = 5 ↔ ((5↑2) ≤ 𝑋 ∧ 𝑋 < ((5 + 1)↑2)))) |
4 | 5cn 11883 | . . . . . . 7 ⊢ 5 ∈ ℂ | |
5 | 4 | sqvali 13714 | . . . . . 6 ⊢ (5↑2) = (5 · 5) |
6 | 5t5e25 12361 | . . . . . 6 ⊢ (5 · 5) = ;25 | |
7 | 5, 6 | eqtri 2759 | . . . . 5 ⊢ (5↑2) = ;25 |
8 | 7 | breq1i 5046 | . . . 4 ⊢ ((5↑2) ≤ 𝑋 ↔ ;25 ≤ 𝑋) |
9 | 8 | a1i 11 | . . 3 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → ((5↑2) ≤ 𝑋 ↔ ;25 ≤ 𝑋)) |
10 | 5p1e6 11942 | . . . . . . 7 ⊢ (5 + 1) = 6 | |
11 | 10 | oveq1i 7201 | . . . . . 6 ⊢ ((5 + 1)↑2) = (6↑2) |
12 | 6cn 11886 | . . . . . . . 8 ⊢ 6 ∈ ℂ | |
13 | 12 | sqvali 13714 | . . . . . . 7 ⊢ (6↑2) = (6 · 6) |
14 | 6t6e36 12366 | . . . . . . 7 ⊢ (6 · 6) = ;36 | |
15 | 13, 14 | eqtri 2759 | . . . . . 6 ⊢ (6↑2) = ;36 |
16 | 11, 15 | eqtri 2759 | . . . . 5 ⊢ ((5 + 1)↑2) = ;36 |
17 | 16 | breq2i 5047 | . . . 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 634 | . 2 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → (((5↑2) ≤ 𝑋 ∧ 𝑋 < ((5 + 1)↑2)) ↔ (;25 ≤ 𝑋 ∧ 𝑋 < ;36))) |
20 | 3, 19 | bitr2d 283 | 1 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → ((;25 ≤ 𝑋 ∧ 𝑋 < ;36) ↔ (⌊‘(√‘𝑋)) = 5)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 ℝcr 10693 0cc0 10694 1c1 10695 + caddc 10697 · cmul 10699 < clt 10832 ≤ cle 10833 2c2 11850 3c3 11851 5c5 11853 6c6 11854 ℕ0cn0 12055 ;cdc 12258 ⌊cfl 13330 ↑cexp 13600 √csqrt 14761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-sup 9036 df-inf 9037 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-rp 12552 df-fl 13332 df-seq 13540 df-exp 13601 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 |
This theorem is referenced by: 31prm 44665 |
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