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Mirrors > Home > MPE Home > Th. List > sqrt2gt1lt2 | Structured version Visualization version GIF version |
Description: The square root of 2 is bounded by 1 and 2. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 6-Sep-2013.) |
Ref | Expression |
---|---|
sqrt2gt1lt2 | ⊢ (1 < (√‘2) ∧ (√‘2) < 2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sqrt1 14056 | . . 3 ⊢ (√‘1) = 1 | |
2 | 1lt2 11232 | . . . 4 ⊢ 1 < 2 | |
3 | 1re 10077 | . . . . 5 ⊢ 1 ∈ ℝ | |
4 | 0le1 10589 | . . . . 5 ⊢ 0 ≤ 1 | |
5 | 2re 11128 | . . . . 5 ⊢ 2 ∈ ℝ | |
6 | 0le2 11149 | . . . . 5 ⊢ 0 ≤ 2 | |
7 | sqrtlt 14046 | . . . . 5 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ (2 ∈ ℝ ∧ 0 ≤ 2)) → (1 < 2 ↔ (√‘1) < (√‘2))) | |
8 | 3, 4, 5, 6, 7 | mp4an 709 | . . . 4 ⊢ (1 < 2 ↔ (√‘1) < (√‘2)) |
9 | 2, 8 | mpbi 220 | . . 3 ⊢ (√‘1) < (√‘2) |
10 | 1, 9 | eqbrtrri 4708 | . 2 ⊢ 1 < (√‘2) |
11 | 2lt4 11236 | . . . 4 ⊢ 2 < 4 | |
12 | 4re 11135 | . . . . 5 ⊢ 4 ∈ ℝ | |
13 | 0re 10078 | . . . . . 6 ⊢ 0 ∈ ℝ | |
14 | 4pos 11154 | . . . . . 6 ⊢ 0 < 4 | |
15 | 13, 12, 14 | ltleii 10198 | . . . . 5 ⊢ 0 ≤ 4 |
16 | sqrtlt 14046 | . . . . 5 ⊢ (((2 ∈ ℝ ∧ 0 ≤ 2) ∧ (4 ∈ ℝ ∧ 0 ≤ 4)) → (2 < 4 ↔ (√‘2) < (√‘4))) | |
17 | 5, 6, 12, 15, 16 | mp4an 709 | . . . 4 ⊢ (2 < 4 ↔ (√‘2) < (√‘4)) |
18 | 11, 17 | mpbi 220 | . . 3 ⊢ (√‘2) < (√‘4) |
19 | sqrt4 14057 | . . 3 ⊢ (√‘4) = 2 | |
20 | 18, 19 | breqtri 4710 | . 2 ⊢ (√‘2) < 2 |
21 | 10, 20 | pm3.2i 470 | 1 ⊢ (1 < (√‘2) ∧ (√‘2) < 2) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 383 ∈ wcel 2030 class class class wbr 4685 ‘cfv 5926 ℝcr 9973 0cc0 9974 1c1 9975 < clt 10112 ≤ cle 10113 2c2 11108 4c4 11110 √csqrt 14017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-sup 8389 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-n0 11331 df-z 11416 df-uz 11726 df-rp 11871 df-seq 12842 df-exp 12901 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 |
This theorem is referenced by: (None) |
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