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Mirrors > Home > MPE Home > Th. List > sqrtgt0 | Structured version Visualization version GIF version |
Description: The square root function is positive for positive input. (Contributed by Mario Carneiro, 10-Jul-2013.) (Revised by Mario Carneiro, 6-Sep-2013.) |
Ref | Expression |
---|---|
sqrtgt0 | ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 < (√‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 10357 | . . . . 5 ⊢ 0 ∈ ℝ | |
2 | ltle 10444 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 → 0 ≤ 𝐴)) | |
3 | 1, 2 | mpan 683 | . . . 4 ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 → 0 ≤ 𝐴)) |
4 | 3 | imp 397 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 ≤ 𝐴) |
5 | resqrtcl 14370 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘𝐴) ∈ ℝ) | |
6 | 4, 5 | syldan 587 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (√‘𝐴) ∈ ℝ) |
7 | sqrtge0 14374 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ≤ (√‘𝐴)) | |
8 | 4, 7 | syldan 587 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 ≤ (√‘𝐴)) |
9 | gt0ne0 10816 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ≠ 0) | |
10 | sq0i 13249 | . . . . 5 ⊢ ((√‘𝐴) = 0 → ((√‘𝐴)↑2) = 0) | |
11 | resqrtth 14372 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√‘𝐴)↑2) = 𝐴) | |
12 | 4, 11 | syldan 587 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ((√‘𝐴)↑2) = 𝐴) |
13 | 12 | eqeq1d 2826 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (((√‘𝐴)↑2) = 0 ↔ 𝐴 = 0)) |
14 | 10, 13 | syl5ib 236 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ((√‘𝐴) = 0 → 𝐴 = 0)) |
15 | 14 | necon3d 3019 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 ≠ 0 → (√‘𝐴) ≠ 0)) |
16 | 9, 15 | mpd 15 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (√‘𝐴) ≠ 0) |
17 | 6, 8, 16 | ne0gt0d 10492 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 < (√‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ≠ wne 2998 class class class wbr 4872 ‘cfv 6122 (class class class)co 6904 ℝcr 10250 0cc0 10251 < clt 10390 ≤ cle 10391 2c2 11405 ↑cexp 13153 √csqrt 14349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 ax-pre-sup 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-2nd 7428 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-sup 8616 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 df-nn 11350 df-2 11413 df-3 11414 df-n0 11618 df-z 11704 df-uz 11968 df-rp 12112 df-seq 13095 df-exp 13154 df-cj 14215 df-re 14216 df-im 14217 df-sqrt 14351 |
This theorem is referenced by: rpsqrtcl 14381 sqrtgt0i 14488 normgt0 28538 pellexlem2 38237 qndenserrnbllem 41304 |
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