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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 13233 | . 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 2166 class class class wbr 4872 (class class class)co 6904 ℝcr 10250 0cc0 10251 ≤ cle 10391 2c2 11405 ↑cexp 13153 |
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 |
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-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-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-nn 11350 df-2 11413 df-n0 11618 df-z 11704 df-uz 11968 df-seq 13095 df-exp 13154 |
This theorem is referenced by: cjmulge0 14262 sqrlem7 14365 absrele 14424 amgm2 14485 efgt0 15204 sinbnd 15281 cosbnd 15282 cphnmf 23363 ipge0 23366 csbren 23566 trirn 23567 rrxmet 23575 rrxdstprj1 23576 minveclem3b 23595 minveclem7 23602 pjthlem1 23604 dveflem 24140 loglesqrt 24900 mulog2sumlem2 25636 log2sumbnd 25645 eqeelen 26202 brbtwn2 26203 colinearalglem4 26207 axcgrid 26214 axsegconlem3 26217 ax5seglem3 26229 minvecolem5 28291 minvecolem7 28293 normpyc 28557 pjhthlem1 28804 chscllem2 29051 pjige0i 29103 hstle1 29639 strlem3a 29665 2sqmod 30192 sqsscirc1 30498 areacirclem1 34042 areacirclem4 34045 rrnmet 34169 rrndstprj1 34170 rrndstprj2 34171 pellexlem2 38237 pellexlem6 38241 int-sqgeq0d 39328 sqrlearg 40574 rrndistlt 41300 hoiqssbllem2 41630 flsqrt 42337 2sphere 43300 itsclc0lem3 43309 |
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