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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 14458 | . 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 2081 ‘cfv 6225 ℝ+crp 12239 √csqrt 14426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-sup 8752 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-n0 11746 df-z 11830 df-uz 12094 df-rp 12240 df-seq 13220 df-exp 13280 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 |
This theorem is referenced by: sqrtgt0d 14606 prmreclem3 16083 prmreclem5 16085 cxpsqrt 24967 divsqrtsumlem 25239 bposlem7 25548 bposlem9 25550 chtppilim 25733 chpchtlim 25737 rplogsumlem1 25742 dchrisum0fno1 25769 dchrisum0lema 25772 dchrisum0lem1b 25773 dchrisum0lem1 25774 dchrisum0lem2a 25775 dchrisum0lem2 25776 dchrisum0lem3 25777 dchrisum0 25778 pntlemb 25855 pntlemh 25857 pntlemr 25860 pntlemj 25861 pntlemk 25864 minvecolem5 28349 logdivsqrle 31538 hgt750leme 31546 rrndstprj2 34641 rrncmslem 34642 rrnequiv 34645 pellexlem4 38914 pell1qrgaplem 38955 pell14qrgapw 38958 pellqrexplicit 38959 pellqrex 38961 pellfundge 38964 pellfundgt1 38965 rmspecfund 38991 rmxycomplete 38999 stirlinglem2 41902 stirlinglem4 41904 stirlinglem13 41913 stirlinglem15 41915 stirlingr 41917 qndenserrnbllem 42121 hoiqssbllem1 42446 hoiqssbllem2 42447 hoiqssbllem3 42448 |
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