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Mirrors > Home > MPE Home > Th. List > cphsqrtcl | Structured version Visualization version GIF version |
Description: The scalar field of a subcomplex pre-Hilbert space is closed under square roots of positive reals (i.e. it is quadratically closed relative to ℝ). (Contributed by Mario Carneiro, 8-Oct-2015.) |
Ref | Expression |
---|---|
cphsca.f | ⊢ 𝐹 = (Scalar‘𝑊) |
cphsca.k | ⊢ 𝐾 = (Base‘𝐹) |
Ref | Expression |
---|---|
cphsqrtcl | ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) → (√‘𝐴) ∈ 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1061 | . . . 4 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ 𝐾) | |
2 | elrege0 12278 | . . . . . 6 ⊢ (𝐴 ∈ (0[,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) | |
3 | 2 | biimpri 218 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ (0[,)+∞)) |
4 | 3 | 3adant1 1079 | . . . 4 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ (0[,)+∞)) |
5 | 1, 4 | elind 3798 | . . 3 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ (𝐾 ∩ (0[,)+∞))) |
6 | sqrtf 14103 | . . . . 5 ⊢ √:ℂ⟶ℂ | |
7 | ffn 6045 | . . . . 5 ⊢ (√:ℂ⟶ℂ → √ Fn ℂ) | |
8 | 6, 7 | ax-mp 5 | . . . 4 ⊢ √ Fn ℂ |
9 | inss2 3834 | . . . . 5 ⊢ (𝐾 ∩ (0[,)+∞)) ⊆ (0[,)+∞) | |
10 | rge0ssre 12280 | . . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ | |
11 | ax-resscn 9993 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
12 | 10, 11 | sstri 3612 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℂ |
13 | 9, 12 | sstri 3612 | . . . 4 ⊢ (𝐾 ∩ (0[,)+∞)) ⊆ ℂ |
14 | fnfvima 6496 | . . . 4 ⊢ ((√ Fn ℂ ∧ (𝐾 ∩ (0[,)+∞)) ⊆ ℂ ∧ 𝐴 ∈ (𝐾 ∩ (0[,)+∞))) → (√‘𝐴) ∈ (√ “ (𝐾 ∩ (0[,)+∞)))) | |
15 | 8, 13, 14 | mp3an12 1414 | . . 3 ⊢ (𝐴 ∈ (𝐾 ∩ (0[,)+∞)) → (√‘𝐴) ∈ (√ “ (𝐾 ∩ (0[,)+∞)))) |
16 | 5, 15 | syl 17 | . 2 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘𝐴) ∈ (√ “ (𝐾 ∩ (0[,)+∞)))) |
17 | eqid 2622 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
18 | eqid 2622 | . . . . 5 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
19 | eqid 2622 | . . . . 5 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
20 | cphsca.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
21 | cphsca.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
22 | 17, 18, 19, 20, 21 | iscph 22970 | . . . 4 ⊢ (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂfld ↾s 𝐾)) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ (norm‘𝑊) = (𝑥 ∈ (Base‘𝑊) ↦ (√‘(𝑥(·𝑖‘𝑊)𝑥))))) |
23 | 22 | simp2bi 1077 | . . 3 ⊢ (𝑊 ∈ ℂPreHil → (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾) |
24 | 23 | sselda 3603 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ (√‘𝐴) ∈ (√ “ (𝐾 ∩ (0[,)+∞)))) → (√‘𝐴) ∈ 𝐾) |
25 | 16, 24 | sylan2 491 | 1 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) → (√‘𝐴) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∩ cin 3573 ⊆ wss 3574 class class class wbr 4653 ↦ cmpt 4729 “ cima 5117 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 +∞cpnf 10071 ≤ cle 10075 [,)cico 12177 √csqrt 13973 Basecbs 15857 ↾s cress 15858 Scalarcsca 15944 ·𝑖cip 15946 ℂfldccnfld 19746 PreHilcphl 19969 normcnm 22381 NrmModcnlm 22385 ℂPreHilccph 22966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ico 12181 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-cph 22968 |
This theorem is referenced by: cphabscl 22985 cphsqrtcl2 22986 cphsqrtcl3 22987 cphnmf 22995 ipcau 23037 |
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