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Mirrors > Home > ILE Home > Th. List > sqrtgt0 | 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 | simpr 110 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 < 𝐴) | |
2 | sq0 10628 | . . . 4 ⊢ (0↑2) = 0 | |
3 | 2 | a1i 9 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (0↑2) = 0) |
4 | 0red 7975 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 ∈ ℝ) | |
5 | simpl 109 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ∈ ℝ) | |
6 | 4, 5, 1 | ltled 8093 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 ≤ 𝐴) |
7 | resqrtth 11057 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√‘𝐴)↑2) = 𝐴) | |
8 | 6, 7 | syldan 282 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ((√‘𝐴)↑2) = 𝐴) |
9 | 1, 3, 8 | 3brtr4d 4049 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (0↑2) < ((√‘𝐴)↑2)) |
10 | resqrtcl 11055 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘𝐴) ∈ ℝ) | |
11 | 6, 10 | syldan 282 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (√‘𝐴) ∈ ℝ) |
12 | 4 | leidd 8488 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 ≤ 0) |
13 | sqrtge0 11059 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ≤ (√‘𝐴)) | |
14 | 6, 13 | syldan 282 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 ≤ (√‘𝐴)) |
15 | 4, 11, 12, 14 | lt2sqd 10702 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (0 < (√‘𝐴) ↔ (0↑2) < ((√‘𝐴)↑2))) |
16 | 9, 15 | mpbird 167 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 < (√‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1363 ∈ wcel 2159 class class class wbr 4017 ‘cfv 5230 (class class class)co 5890 ℝcr 7827 0cc0 7828 < clt 8009 ≤ cle 8010 2c2 8987 ↑cexp 10536 √csqrt 11022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2161 ax-14 2162 ax-ext 2170 ax-coll 4132 ax-sep 4135 ax-nul 4143 ax-pow 4188 ax-pr 4223 ax-un 4447 ax-setind 4550 ax-iinf 4601 ax-cnex 7919 ax-resscn 7920 ax-1cn 7921 ax-1re 7922 ax-icn 7923 ax-addcl 7924 ax-addrcl 7925 ax-mulcl 7926 ax-mulrcl 7927 ax-addcom 7928 ax-mulcom 7929 ax-addass 7930 ax-mulass 7931 ax-distr 7932 ax-i2m1 7933 ax-0lt1 7934 ax-1rid 7935 ax-0id 7936 ax-rnegex 7937 ax-precex 7938 ax-cnre 7939 ax-pre-ltirr 7940 ax-pre-ltwlin 7941 ax-pre-lttrn 7942 ax-pre-apti 7943 ax-pre-ltadd 7944 ax-pre-mulgt0 7945 ax-pre-mulext 7946 ax-arch 7947 ax-caucvg 7948 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-nel 2455 df-ral 2472 df-rex 2473 df-reu 2474 df-rmo 2475 df-rab 2476 df-v 2753 df-sbc 2977 df-csb 3072 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-nul 3437 df-if 3549 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-int 3859 df-iun 3902 df-br 4018 df-opab 4079 df-mpt 4080 df-tr 4116 df-id 4307 df-po 4310 df-iso 4311 df-iord 4380 df-on 4382 df-ilim 4383 df-suc 4385 df-iom 4604 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-rn 4651 df-res 4652 df-ima 4653 df-iota 5192 df-fun 5232 df-fn 5233 df-f 5234 df-f1 5235 df-fo 5236 df-f1o 5237 df-fv 5238 df-riota 5846 df-ov 5893 df-oprab 5894 df-mpo 5895 df-1st 6158 df-2nd 6159 df-recs 6323 df-frec 6409 df-pnf 8011 df-mnf 8012 df-xr 8013 df-ltxr 8014 df-le 8015 df-sub 8147 df-neg 8148 df-reap 8549 df-ap 8556 df-div 8647 df-inn 8937 df-2 8995 df-3 8996 df-4 8997 df-n0 9194 df-z 9271 df-uz 9546 df-rp 9671 df-seqfrec 10463 df-exp 10537 df-rsqrt 11024 |
This theorem is referenced by: rpsqrtcl 11067 sqrtgt0i 11147 |
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