![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sqge0d | Structured version Visualization version GIF version |
Description: A square of a real is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
resqcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
sqge0d | ⊢ (𝜑 → 0 ≤ (𝐴↑2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resqcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | sqge0 13497 | . 2 ⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴↑2)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 0 ≤ (𝐴↑2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 0cc0 10526 ≤ cle 10665 2c2 11680 ↑cexp 13425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-seq 13365 df-exp 13426 |
This theorem is referenced by: cjmulge0 14497 sqrlem7 14600 absrele 14660 amgm2 14721 efgt0 15448 sinbnd 15525 cosbnd 15526 cphnmf 23800 ipge0 23803 csbren 24003 trirn 24004 rrxmet 24012 rrxdstprj1 24013 minveclem3b 24032 minveclem7 24039 pjthlem1 24041 dveflem 24582 loglesqrt 25347 2sq2 26017 2sqmod 26020 mulog2sumlem2 26119 log2sumbnd 26128 eqeelen 26698 brbtwn2 26699 colinearalglem4 26703 axcgrid 26710 axsegconlem3 26713 ax5seglem3 26725 minvecolem5 28664 minvecolem7 28666 normpyc 28929 pjhthlem1 29174 chscllem2 29421 pjige0i 29473 hstle1 30009 strlem3a 30035 sqsscirc1 31261 areacirclem1 35145 areacirclem4 35148 rrnmet 35267 rrndstprj1 35268 rrndstprj2 35269 3cubeslem1 39625 pellexlem2 39771 pellexlem6 39775 int-sqgeq0d 40892 sqrlearg 42190 rrndistlt 42932 hoiqssbllem2 43262 flsqrt 44110 2sphere 45163 itsclc0yqsollem2 45177 2itscp 45195 itscnhlinecirc02plem3 45198 itscnhlinecirc02p 45199 |
Copyright terms: Public domain | W3C validator |