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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 11189 | . . 3 ⊢ 5 ∈ ℕ0 | |
2 | flsqrt 40046 | . . 3 ⊢ (((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ 5 ∈ ℕ0) → ((⌊‘(√‘𝑋)) = 5 ↔ ((5↑2) ≤ 𝑋 ∧ 𝑋 < ((5 + 1)↑2)))) | |
3 | 1, 2 | mpan2 703 | . 2 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → ((⌊‘(√‘𝑋)) = 5 ↔ ((5↑2) ≤ 𝑋 ∧ 𝑋 < ((5 + 1)↑2)))) |
4 | 5cn 10977 | . . . . . . 7 ⊢ 5 ∈ ℂ | |
5 | 4 | sqvali 12805 | . . . . . 6 ⊢ (5↑2) = (5 · 5) |
6 | 5t5e25 11515 | . . . . . 6 ⊢ (5 · 5) = ;25 | |
7 | 5, 6 | eqtri 2632 | . . . . 5 ⊢ (5↑2) = ;25 |
8 | 7 | breq1i 4590 | . . . 4 ⊢ ((5↑2) ≤ 𝑋 ↔ ;25 ≤ 𝑋) |
9 | 8 | a1i 11 | . . 3 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → ((5↑2) ≤ 𝑋 ↔ ;25 ≤ 𝑋)) |
10 | 5p1e6 11032 | . . . . . . 7 ⊢ (5 + 1) = 6 | |
11 | 10 | oveq1i 6559 | . . . . . 6 ⊢ ((5 + 1)↑2) = (6↑2) |
12 | 6cn 10979 | . . . . . . . 8 ⊢ 6 ∈ ℂ | |
13 | 12 | sqvali 12805 | . . . . . . 7 ⊢ (6↑2) = (6 · 6) |
14 | 6t6e36 11522 | . . . . . . 7 ⊢ (6 · 6) = ;36 | |
15 | 13, 14 | eqtri 2632 | . . . . . 6 ⊢ (6↑2) = ;36 |
16 | 11, 15 | eqtri 2632 | . . . . 5 ⊢ ((5 + 1)↑2) = ;36 |
17 | 16 | breq2i 4591 | . . . 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 743 | . 2 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → (((5↑2) ≤ 𝑋 ∧ 𝑋 < ((5 + 1)↑2)) ↔ (;25 ≤ 𝑋 ∧ 𝑋 < ;36))) |
20 | 3, 19 | bitr2d 268 | 1 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → ((;25 ≤ 𝑋 ∧ 𝑋 < ;36) ↔ (⌊‘(√‘𝑋)) = 5)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 2c2 10947 3c3 10948 5c5 10950 6c6 10951 ℕ0cn0 11169 ;cdc 11369 ⌊cfl 12453 ↑cexp 12722 √csqrt 13821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-fl 12455 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 |
This theorem is referenced by: 31prm 40050 |
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