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Mirrors > Home > MPE Home > Th. List > ipge0 | Structured version Visualization version GIF version |
Description: The inner product in a subcomplex pre-Hilbert space is positive definite. (Contributed by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
reipcl.v | ⊢ 𝑉 = (Base‘𝑊) |
reipcl.h | ⊢ , = (·𝑖‘𝑊) |
Ref | Expression |
---|---|
ipge0 | ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → 0 ≤ (𝐴 , 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cphngp 23193 | . . . 4 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp) | |
2 | reipcl.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
3 | eqid 2760 | . . . . 5 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
4 | 2, 3 | nmcl 22641 | . . . 4 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉) → ((norm‘𝑊)‘𝐴) ∈ ℝ) |
5 | 1, 4 | sylan 489 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → ((norm‘𝑊)‘𝐴) ∈ ℝ) |
6 | 5 | sqge0d 13250 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → 0 ≤ (((norm‘𝑊)‘𝐴)↑2)) |
7 | reipcl.h | . . 3 ⊢ , = (·𝑖‘𝑊) | |
8 | 2, 7, 3 | nmsq 23214 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → (((norm‘𝑊)‘𝐴)↑2) = (𝐴 , 𝐴)) |
9 | 6, 8 | breqtrd 4830 | 1 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → 0 ≤ (𝐴 , 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 class class class wbr 4804 ‘cfv 6049 (class class class)co 6814 ℝcr 10147 0cc0 10148 ≤ cle 10287 2c2 11282 ↑cexp 13074 Basecbs 16079 ·𝑖cip 16168 normcnm 22602 NrmGrpcngp 22603 ℂPreHilccph 23186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 ax-pre-sup 10226 ax-addf 10227 ax-mulf 10228 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-1st 7334 df-2nd 7335 df-tpos 7522 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-oadd 7734 df-er 7913 df-map 8027 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-sup 8515 df-inf 8516 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-div 10897 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-5 11294 df-6 11295 df-7 11296 df-8 11297 df-9 11298 df-n0 11505 df-z 11590 df-dec 11706 df-uz 11900 df-q 12002 df-rp 12046 df-xneg 12159 df-xadd 12160 df-xmul 12161 df-fz 12540 df-seq 13016 df-exp 13075 df-cj 14058 df-re 14059 df-im 14060 df-sqrt 14194 df-abs 14195 df-struct 16081 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-ress 16087 df-plusg 16176 df-mulr 16177 df-starv 16178 df-sca 16179 df-vsca 16180 df-ip 16181 df-tset 16182 df-ple 16183 df-ds 16186 df-unif 16187 df-0g 16324 df-topgen 16326 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-grp 17646 df-subg 17812 df-ghm 17879 df-cmn 18415 df-mgp 18710 df-ur 18722 df-ring 18769 df-cring 18770 df-oppr 18843 df-dvdsr 18861 df-unit 18862 df-drng 18971 df-subrg 19000 df-lmhm 19244 df-lvec 19325 df-sra 19394 df-rgmod 19395 df-psmet 19960 df-xmet 19961 df-met 19962 df-bl 19963 df-mopn 19964 df-cnfld 19969 df-phl 20193 df-top 20921 df-topon 20938 df-topsp 20959 df-bases 20972 df-xms 22346 df-ms 22347 df-nm 22608 df-ngp 22609 df-nlm 22612 df-cph 23188 |
This theorem is referenced by: ipcau 23257 pjthlem1 23428 |
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