Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cphsqrtcl | Structured version Visualization version GIF version |
Description: The scalar field of a complex 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 1054 | . . . 4 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ 𝐾) | |
2 | elrege0 12149 | . . . . . 6 ⊢ (𝐴 ∈ (0[,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) | |
3 | 2 | biimpri 217 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ (0[,)+∞)) |
4 | 3 | 3adant1 1072 | . . . 4 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ (0[,)+∞)) |
5 | 1, 4 | elind 3760 | . . 3 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ (𝐾 ∩ (0[,)+∞))) |
6 | sqrtf 13951 | . . . . 5 ⊢ √:ℂ⟶ℂ | |
7 | ffn 5958 | . . . . 5 ⊢ (√:ℂ⟶ℂ → √ Fn ℂ) | |
8 | 6, 7 | ax-mp 5 | . . . 4 ⊢ √ Fn ℂ |
9 | inss2 3796 | . . . . 5 ⊢ (𝐾 ∩ (0[,)+∞)) ⊆ (0[,)+∞) | |
10 | rge0ssre 12151 | . . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ | |
11 | ax-resscn 9872 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
12 | 10, 11 | sstri 3577 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℂ |
13 | 9, 12 | sstri 3577 | . . . 4 ⊢ (𝐾 ∩ (0[,)+∞)) ⊆ ℂ |
14 | fnfvima 6400 | . . . 4 ⊢ ((√ Fn ℂ ∧ (𝐾 ∩ (0[,)+∞)) ⊆ ℂ ∧ 𝐴 ∈ (𝐾 ∩ (0[,)+∞))) → (√‘𝐴) ∈ (√ “ (𝐾 ∩ (0[,)+∞)))) | |
15 | 8, 13, 14 | mp3an12 1406 | . . 3 ⊢ (𝐴 ∈ (𝐾 ∩ (0[,)+∞)) → (√‘𝐴) ∈ (√ “ (𝐾 ∩ (0[,)+∞)))) |
16 | 5, 15 | syl 17 | . 2 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘𝐴) ∈ (√ “ (𝐾 ∩ (0[,)+∞)))) |
17 | eqid 2610 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
18 | eqid 2610 | . . . . 5 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
19 | eqid 2610 | . . . . 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 22778 | . . . 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 1070 | . . 3 ⊢ (𝑊 ∈ ℂPreHil → (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾) |
24 | 23 | sselda 3568 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ (√‘𝐴) ∈ (√ “ (𝐾 ∩ (0[,)+∞)))) → (√‘𝐴) ∈ 𝐾) |
25 | 16, 24 | sylan2 490 | 1 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) → (√‘𝐴) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ⊆ wss 3540 class class class wbr 4583 ↦ cmpt 4643 “ cima 5041 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 +∞cpnf 9950 ≤ cle 9954 [,)cico 12048 √csqrt 13821 Basecbs 15695 ↾s cress 15696 Scalarcsca 15771 ·𝑖cip 15773 ℂfldccnfld 19567 PreHilcphl 19788 normcnm 22191 NrmModcnlm 22195 ℂPreHilccph 22774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-ico 12052 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-cph 22776 |
This theorem is referenced by: cphabscl 22793 cphsqrtcl2 22794 cphsqrtcl3 22795 cphnmf 22803 ipcau 22845 |
Copyright terms: Public domain | W3C validator |