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Mirrors > Home > MPE Home > Th. List > rpsqrtcld | Structured version Visualization version GIF version |
Description: The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
sqrgt0d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
Ref | Expression |
---|---|
rpsqrtcld | ⊢ (𝜑 → (√‘𝐴) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sqrgt0d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
2 | rpsqrtcl 14343 | . 2 ⊢ (𝐴 ∈ ℝ+ → (√‘𝐴) ∈ ℝ+) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (√‘𝐴) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2157 ‘cfv 6099 ℝ+crp 12070 √csqrt 14311 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 ax-pre-sup 10300 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-sup 8588 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-div 10975 df-nn 11311 df-2 11372 df-3 11373 df-n0 11577 df-z 11663 df-uz 11927 df-rp 12071 df-seq 13052 df-exp 13111 df-cj 14177 df-re 14178 df-im 14179 df-sqrt 14313 |
This theorem is referenced by: sqrtgt0d 14489 prmreclem3 15952 prmreclem5 15954 cxpsqrt 24787 divsqrtsumlem 25055 bposlem7 25364 bposlem9 25366 chtppilim 25513 chpchtlim 25517 rplogsumlem1 25522 dchrisum0fno1 25549 dchrisum0lema 25552 dchrisum0lem1b 25553 dchrisum0lem1 25554 dchrisum0lem2a 25555 dchrisum0lem2 25556 dchrisum0lem3 25557 dchrisum0 25558 pntlemb 25635 pntlemh 25637 pntlemr 25640 pntlemj 25641 pntlemk 25644 minvecolem5 28254 logdivsqrle 31240 hgt750leme 31248 rrndstprj2 34109 rrncmslem 34110 rrnequiv 34113 pellexlem4 38170 pell1qrgaplem 38211 pell14qrgapw 38214 pellqrexplicit 38215 pellqrex 38217 pellfundge 38220 pellfundgt1 38221 rmspecfund 38247 rmxycomplete 38255 stirlinglem2 41023 stirlinglem4 41025 stirlinglem13 41034 stirlinglem15 41036 stirlingr 41038 qndenserrnbllem 41245 hoiqssbllem1 41570 hoiqssbllem2 41571 hoiqssbllem3 41572 |
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