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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 25141 | . . . 4 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp) | |
2 | reipcl.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
3 | eqid 2725 | . . . . 5 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
4 | 2, 3 | nmcl 24565 | . . . 4 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉) → ((norm‘𝑊)‘𝐴) ∈ ℝ) |
5 | 1, 4 | sylan 578 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → ((norm‘𝑊)‘𝐴) ∈ ℝ) |
6 | 5 | sqge0d 14135 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → 0 ≤ (((norm‘𝑊)‘𝐴)↑2)) |
7 | reipcl.h | . . 3 ⊢ , = (·𝑖‘𝑊) | |
8 | 2, 7, 3 | nmsq 25162 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → (((norm‘𝑊)‘𝐴)↑2) = (𝐴 , 𝐴)) |
9 | 6, 8 | breqtrd 5175 | 1 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → 0 ≤ (𝐴 , 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 class class class wbr 5149 ‘cfv 6548 (class class class)co 7418 ℝcr 11138 0cc0 11139 ≤ cle 11280 2c2 12298 ↑cexp 14060 Basecbs 17181 ·𝑖cip 17239 normcnm 24525 NrmGrpcngp 24526 ℂPreHilccph 25134 |
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-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 ax-mulf 11219 |
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-tp 4635 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 6306 df-ord 6373 df-on 6374 df-lim 6375 df-suc 6376 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7374 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7871 df-1st 7993 df-2nd 7994 df-tpos 8231 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-sup 9466 df-inf 9467 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-fz 13518 df-seq 14001 df-exp 14061 df-cj 15080 df-re 15081 df-im 15082 df-sqrt 15216 df-abs 15217 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17182 df-ress 17211 df-plusg 17247 df-mulr 17248 df-starv 17249 df-sca 17250 df-vsca 17251 df-ip 17252 df-tset 17253 df-ple 17254 df-ds 17256 df-unif 17257 df-0g 17424 df-topgen 17426 df-mgm 18601 df-sgrp 18680 df-mnd 18696 df-grp 18899 df-minusg 18900 df-subg 19084 df-ghm 19174 df-cmn 19746 df-abl 19747 df-mgp 20084 df-rng 20102 df-ur 20131 df-ring 20184 df-cring 20185 df-oppr 20282 df-dvdsr 20305 df-unit 20306 df-subrg 20517 df-drng 20635 df-lmhm 20916 df-lvec 20997 df-sra 21067 df-rgmod 21068 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-cnfld 21294 df-phl 21572 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22889 df-xms 24266 df-ms 24267 df-nm 24531 df-ngp 24532 df-nlm 24535 df-cph 25136 |
This theorem is referenced by: ipcau 25206 pjthlem1 25405 |
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