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Mirrors > Home > HSE Home > Th. List > normge0 | Structured version Visualization version GIF version |
Description: The norm of a vector is nonnegative. (Contributed by NM, 29-May-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
normge0 | ⊢ (𝐴 ∈ ℋ → 0 ≤ (normℎ‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hiidrcl 30977 | . . 3 ⊢ (𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ) | |
2 | hiidge0 30980 | . . 3 ⊢ (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴)) | |
3 | 1, 2 | sqrtge0d 15403 | . 2 ⊢ (𝐴 ∈ ℋ → 0 ≤ (√‘(𝐴 ·ih 𝐴))) |
4 | normval 31006 | . 2 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) = (√‘(𝐴 ·ih 𝐴))) | |
5 | 3, 4 | breqtrrd 5177 | 1 ⊢ (𝐴 ∈ ℋ → 0 ≤ (normℎ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 0cc0 11140 ≤ cle 11281 √csqrt 15216 ℋchba 30801 ·ih csp 30804 normℎcno 30805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 ax-hv0cl 30885 ax-hvmul0 30892 ax-hfi 30961 ax-his1 30964 ax-his3 30966 ax-his4 30967 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9467 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-rp 13010 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-hnorm 30850 |
This theorem is referenced by: norm-i 31011 normpyc 31028 bcsiALT 31061 bcs2 31064 pjhthlem1 31273 chscllem2 31520 pjdifnormii 31565 pjneli 31605 nmopge0 31793 unopnorm 31799 lnconi 31915 cnlnadjlem2 31950 cnlnadjlem7 31955 nmopcoadji 31983 leopnmid 32020 pjnormssi 32050 pjssposi 32054 hstle1 32108 hstle 32112 strlem3a 32134 strlem5 32137 jplem1 32150 cdj1i 32315 cdj3lem1 32316 cdj3lem2b 32319 |
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