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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 24337 | . 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) |
5 | simpl 482 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑥 ∈ 𝑉) → 𝑊 ∈ ℂPreHil) | |
6 | cphphl 24316 | . . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil) | |
7 | 6 | adantr 480 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑥 ∈ 𝑉) → 𝑊 ∈ PreHil) |
8 | simpr 484 | . . . 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 20819 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → (𝑥 , 𝑥) ∈ 𝐾) |
12 | 7, 8, 8, 11 | syl3anc 1369 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑥 ∈ 𝑉) → (𝑥 , 𝑥) ∈ 𝐾) |
13 | 1, 2, 3 | nmsq 24339 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑥 ∈ 𝑉) → ((𝑁‘𝑥)↑2) = (𝑥 , 𝑥)) |
14 | cphngp 24318 | . . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp) | |
15 | 1, 3 | nmcl 23753 | . . . . . 6 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉) → (𝑁‘𝑥) ∈ ℝ) |
16 | 14, 15 | sylan 579 | . . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑥 ∈ 𝑉) → (𝑁‘𝑥) ∈ ℝ) |
17 | 16 | resqcld 13946 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑥 ∈ 𝑉) → ((𝑁‘𝑥)↑2) ∈ ℝ) |
18 | 13, 17 | eqeltrrd 2841 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑥 ∈ 𝑉) → (𝑥 , 𝑥) ∈ ℝ) |
19 | 16 | sqge0d 13947 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑥 ∈ 𝑉) → 0 ≤ ((𝑁‘𝑥)↑2)) |
20 | 19, 13 | breqtrd 5104 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥)) |
21 | 9, 10 | cphsqrtcl 24329 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ ((𝑥 , 𝑥) ∈ 𝐾 ∧ (𝑥 , 𝑥) ∈ ℝ ∧ 0 ≤ (𝑥 , 𝑥))) → (√‘(𝑥 , 𝑥)) ∈ 𝐾) |
22 | 5, 12, 18, 20, 21 | syl13anc 1370 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑥 ∈ 𝑉) → (√‘(𝑥 , 𝑥)) ∈ 𝐾) |
23 | 4, 22 | fmpt3d 6984 | 1 ⊢ (𝑊 ∈ ℂPreHil → 𝑁:𝑉⟶𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 class class class wbr 5078 ⟶wf 6426 ‘cfv 6430 (class class class)co 7268 ℝcr 10854 0cc0 10855 ≤ cle 10994 2c2 12011 ↑cexp 13763 √csqrt 14925 Basecbs 16893 Scalarcsca 16946 ·𝑖cip 16948 PreHilcphl 20810 normcnm 23713 NrmGrpcngp 23714 ℂPreHilccph 24311 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933 ax-addf 10934 ax-mulf 10935 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-tpos 8026 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-map 8591 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-sup 9162 df-inf 9163 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-z 12303 df-dec 12420 df-uz 12565 df-q 12671 df-rp 12713 df-xneg 12830 df-xadd 12831 df-xmul 12832 df-ico 13067 df-fz 13222 df-seq 13703 df-exp 13764 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-starv 16958 df-sca 16959 df-vsca 16960 df-ip 16961 df-tset 16962 df-ple 16963 df-ds 16965 df-unif 16966 df-0g 17133 df-topgen 17135 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-grp 18561 df-subg 18733 df-ghm 18813 df-cmn 19369 df-mgp 19702 df-ur 19719 df-ring 19766 df-cring 19767 df-oppr 19843 df-dvdsr 19864 df-unit 19865 df-drng 19974 df-subrg 20003 df-lmhm 20265 df-lvec 20346 df-sra 20415 df-rgmod 20416 df-psmet 20570 df-xmet 20571 df-met 20572 df-bl 20573 df-mopn 20574 df-cnfld 20579 df-phl 20812 df-top 22024 df-topon 22041 df-topsp 22063 df-bases 22077 df-xms 23454 df-ms 23455 df-nm 23719 df-ngp 23720 df-nlm 23723 df-cph 24313 |
This theorem is referenced by: cphnmcl 24341 |
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