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Mirrors > Home > MPE Home > Th. List > cphnmf | Structured version Visualization version GIF version |
Description: The norm of a vector is a member of the scalar field in a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 9-Oct-2015.) |
Ref | Expression |
---|---|
nmsq.v | ⊢ 𝑉 = (Base‘𝑊) |
nmsq.h | ⊢ , = (·𝑖‘𝑊) |
nmsq.n | ⊢ 𝑁 = (norm‘𝑊) |
cphnmcl.f | ⊢ 𝐹 = (Scalar‘𝑊) |
cphnmcl.k | ⊢ 𝐾 = (Base‘𝐹) |
Ref | Expression |
---|---|
cphnmf | ⊢ (𝑊 ∈ ℂPreHil → 𝑁:𝑉⟶𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmsq.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
2 | nmsq.h | . . 3 ⊢ , = (·𝑖‘𝑊) | |
3 | nmsq.n | . . 3 ⊢ 𝑁 = (norm‘𝑊) | |
4 | 1, 2, 3 | cphnmfval 23790 | . 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) |
5 | simpl 485 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑥 ∈ 𝑉) → 𝑊 ∈ ℂPreHil) | |
6 | cphphl 23769 | . . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil) | |
7 | 6 | adantr 483 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑥 ∈ 𝑉) → 𝑊 ∈ PreHil) |
8 | simpr 487 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉) | |
9 | cphnmcl.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
10 | cphnmcl.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
11 | 9, 2, 1, 10 | ipcl 20771 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → (𝑥 , 𝑥) ∈ 𝐾) |
12 | 7, 8, 8, 11 | syl3anc 1367 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑥 ∈ 𝑉) → (𝑥 , 𝑥) ∈ 𝐾) |
13 | 1, 2, 3 | nmsq 23792 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑥 ∈ 𝑉) → ((𝑁‘𝑥)↑2) = (𝑥 , 𝑥)) |
14 | cphngp 23771 | . . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp) | |
15 | 1, 3 | nmcl 23219 | . . . . . 6 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉) → (𝑁‘𝑥) ∈ ℝ) |
16 | 14, 15 | sylan 582 | . . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑥 ∈ 𝑉) → (𝑁‘𝑥) ∈ ℝ) |
17 | 16 | resqcld 13605 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑥 ∈ 𝑉) → ((𝑁‘𝑥)↑2) ∈ ℝ) |
18 | 13, 17 | eqeltrrd 2914 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑥 ∈ 𝑉) → (𝑥 , 𝑥) ∈ ℝ) |
19 | 16 | sqge0d 13606 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑥 ∈ 𝑉) → 0 ≤ ((𝑁‘𝑥)↑2)) |
20 | 19, 13 | breqtrd 5085 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥)) |
21 | 9, 10 | cphsqrtcl 23782 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ ((𝑥 , 𝑥) ∈ 𝐾 ∧ (𝑥 , 𝑥) ∈ ℝ ∧ 0 ≤ (𝑥 , 𝑥))) → (√‘(𝑥 , 𝑥)) ∈ 𝐾) |
22 | 5, 12, 18, 20, 21 | syl13anc 1368 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑥 ∈ 𝑉) → (√‘(𝑥 , 𝑥)) ∈ 𝐾) |
23 | 4, 22 | fmpt3d 6875 | 1 ⊢ (𝑊 ∈ ℂPreHil → 𝑁:𝑉⟶𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 class class class wbr 5059 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ℝcr 10530 0cc0 10531 ≤ cle 10670 2c2 11686 ↑cexp 13423 √csqrt 14586 Basecbs 16477 Scalarcsca 16562 ·𝑖cip 16564 PreHilcphl 20762 normcnm 23180 NrmGrpcngp 23181 ℂPreHilccph 23764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ico 12738 df-fz 12887 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-0g 16709 df-topgen 16711 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-subg 18270 df-ghm 18350 df-cmn 18902 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-drng 19498 df-subrg 19527 df-lmhm 19788 df-lvec 19869 df-sra 19938 df-rgmod 19939 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-cnfld 20540 df-phl 20764 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-xms 22924 df-ms 22925 df-nm 23186 df-ngp 23187 df-nlm 23190 df-cph 23766 |
This theorem is referenced by: cphnmcl 23794 |
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