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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 11312 | . . 3 ⊢ 5 ∈ ℕ0 | |
2 | flsqrt 41508 | . . 3 ⊢ (((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ 5 ∈ ℕ0) → ((⌊‘(√‘𝑋)) = 5 ↔ ((5↑2) ≤ 𝑋 ∧ 𝑋 < ((5 + 1)↑2)))) | |
3 | 1, 2 | mpan2 707 | . 2 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → ((⌊‘(√‘𝑋)) = 5 ↔ ((5↑2) ≤ 𝑋 ∧ 𝑋 < ((5 + 1)↑2)))) |
4 | 5cn 11100 | . . . . . . 7 ⊢ 5 ∈ ℂ | |
5 | 4 | sqvali 12943 | . . . . . 6 ⊢ (5↑2) = (5 · 5) |
6 | 5t5e25 11639 | . . . . . 6 ⊢ (5 · 5) = ;25 | |
7 | 5, 6 | eqtri 2644 | . . . . 5 ⊢ (5↑2) = ;25 |
8 | 7 | breq1i 4660 | . . . 4 ⊢ ((5↑2) ≤ 𝑋 ↔ ;25 ≤ 𝑋) |
9 | 8 | a1i 11 | . . 3 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → ((5↑2) ≤ 𝑋 ↔ ;25 ≤ 𝑋)) |
10 | 5p1e6 11155 | . . . . . . 7 ⊢ (5 + 1) = 6 | |
11 | 10 | oveq1i 6660 | . . . . . 6 ⊢ ((5 + 1)↑2) = (6↑2) |
12 | 6cn 11102 | . . . . . . . 8 ⊢ 6 ∈ ℂ | |
13 | 12 | sqvali 12943 | . . . . . . 7 ⊢ (6↑2) = (6 · 6) |
14 | 6t6e36 11646 | . . . . . . 7 ⊢ (6 · 6) = ;36 | |
15 | 13, 14 | eqtri 2644 | . . . . . 6 ⊢ (6↑2) = ;36 |
16 | 11, 15 | eqtri 2644 | . . . . 5 ⊢ ((5 + 1)↑2) = ;36 |
17 | 16 | breq2i 4661 | . . . 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 747 | . 2 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → (((5↑2) ≤ 𝑋 ∧ 𝑋 < ((5 + 1)↑2)) ↔ (;25 ≤ 𝑋 ∧ 𝑋 < ;36))) |
20 | 3, 19 | bitr2d 269 | 1 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → ((;25 ≤ 𝑋 ∧ 𝑋 < ;36) ↔ (⌊‘(√‘𝑋)) = 5)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 ≤ cle 10075 2c2 11070 3c3 11071 5c5 11073 6c6 11074 ℕ0cn0 11292 ;cdc 11493 ⌊cfl 12591 ↑cexp 12860 √csqrt 13973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-fl 12593 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 |
This theorem is referenced by: 31prm 41512 |
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