![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 15236 | . . 3 ⊢ (√‘1) = 1 | |
2 | 1lt2 12399 | . . . 4 ⊢ 1 < 2 | |
3 | 1re 11230 | . . . . 5 ⊢ 1 ∈ ℝ | |
4 | 0le1 11753 | . . . . 5 ⊢ 0 ≤ 1 | |
5 | 2re 12302 | . . . . 5 ⊢ 2 ∈ ℝ | |
6 | 0le2 12330 | . . . . 5 ⊢ 0 ≤ 2 | |
7 | sqrtlt 15226 | . . . . 5 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ (2 ∈ ℝ ∧ 0 ≤ 2)) → (1 < 2 ↔ (√‘1) < (√‘2))) | |
8 | 3, 4, 5, 6, 7 | mp4an 692 | . . . 4 ⊢ (1 < 2 ↔ (√‘1) < (√‘2)) |
9 | 2, 8 | mpbi 229 | . . 3 ⊢ (√‘1) < (√‘2) |
10 | 1, 9 | eqbrtrri 5165 | . 2 ⊢ 1 < (√‘2) |
11 | 2lt4 12403 | . . . 4 ⊢ 2 < 4 | |
12 | 4re 12312 | . . . . 5 ⊢ 4 ∈ ℝ | |
13 | 0re 11232 | . . . . . 6 ⊢ 0 ∈ ℝ | |
14 | 4pos 12335 | . . . . . 6 ⊢ 0 < 4 | |
15 | 13, 12, 14 | ltleii 11353 | . . . . 5 ⊢ 0 ≤ 4 |
16 | sqrtlt 15226 | . . . . 5 ⊢ (((2 ∈ ℝ ∧ 0 ≤ 2) ∧ (4 ∈ ℝ ∧ 0 ≤ 4)) → (2 < 4 ↔ (√‘2) < (√‘4))) | |
17 | 5, 6, 12, 15, 16 | mp4an 692 | . . . 4 ⊢ (2 < 4 ↔ (√‘2) < (√‘4)) |
18 | 11, 17 | mpbi 229 | . . 3 ⊢ (√‘2) < (√‘4) |
19 | sqrt4 15237 | . . 3 ⊢ (√‘4) = 2 | |
20 | 18, 19 | breqtri 5167 | . 2 ⊢ (√‘2) < 2 |
21 | 10, 20 | pm3.2i 470 | 1 ⊢ (1 < (√‘2) ∧ (√‘2) < 2) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∈ wcel 2099 class class class wbr 5142 ‘cfv 6542 ℝcr 11123 0cc0 11124 1c1 11125 < clt 11264 ≤ cle 11265 2c2 12283 4c4 12285 √csqrt 15198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-sup 9451 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-n0 12489 df-z 12575 df-uz 12839 df-rp 12993 df-seq 13985 df-exp 14045 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |