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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 nonnegative reals. (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 | sqrtf 14717 | . . . 4 ⊢ √:ℂ⟶ℂ | |
2 | ffn 6508 | . . . 4 ⊢ (√:ℂ⟶ℂ → √ Fn ℂ) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ √ Fn ℂ |
4 | inss2 4205 | . . . 4 ⊢ (𝐾 ∩ (0[,)+∞)) ⊆ (0[,)+∞) | |
5 | rge0ssre 12838 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
6 | ax-resscn 10588 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
7 | 5, 6 | sstri 3975 | . . . 4 ⊢ (0[,)+∞) ⊆ ℂ |
8 | 4, 7 | sstri 3975 | . . 3 ⊢ (𝐾 ∩ (0[,)+∞)) ⊆ ℂ |
9 | simp1 1132 | . . . 4 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ 𝐾) | |
10 | elrege0 12836 | . . . . . 6 ⊢ (𝐴 ∈ (0[,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) | |
11 | 10 | biimpri 230 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ (0[,)+∞)) |
12 | 11 | 3adant1 1126 | . . . 4 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ (0[,)+∞)) |
13 | 9, 12 | elind 4170 | . . 3 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ (𝐾 ∩ (0[,)+∞))) |
14 | fnfvima 6989 | . . 3 ⊢ ((√ Fn ℂ ∧ (𝐾 ∩ (0[,)+∞)) ⊆ ℂ ∧ 𝐴 ∈ (𝐾 ∩ (0[,)+∞))) → (√‘𝐴) ∈ (√ “ (𝐾 ∩ (0[,)+∞)))) | |
15 | 3, 8, 13, 14 | mp3an12i 1461 | . 2 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘𝐴) ∈ (√ “ (𝐾 ∩ (0[,)+∞)))) |
16 | eqid 2821 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
17 | eqid 2821 | . . . . 5 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
18 | eqid 2821 | . . . . 5 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
19 | cphsca.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
20 | cphsca.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
21 | 16, 17, 18, 19, 20 | iscph 23768 | . . . 4 ⊢ (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂfld ↾s 𝐾)) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ (norm‘𝑊) = (𝑥 ∈ (Base‘𝑊) ↦ (√‘(𝑥(·𝑖‘𝑊)𝑥))))) |
22 | 21 | simp2bi 1142 | . . 3 ⊢ (𝑊 ∈ ℂPreHil → (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾) |
23 | 22 | sselda 3966 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ (√‘𝐴) ∈ (√ “ (𝐾 ∩ (0[,)+∞)))) → (√‘𝐴) ∈ 𝐾) |
24 | 15, 23 | sylan2 594 | 1 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) → (√‘𝐴) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∩ cin 3934 ⊆ wss 3935 class class class wbr 5058 ↦ cmpt 5138 “ cima 5552 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 ℝcr 10530 0cc0 10531 +∞cpnf 10666 ≤ cle 10670 [,)cico 12734 √csqrt 14586 Basecbs 16477 ↾s cress 16478 Scalarcsca 16562 ·𝑖cip 16564 ℂfldccnfld 20539 PreHilcphl 20762 normcnm 23180 NrmModcnlm 23184 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 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 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 |
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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 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-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-ico 12738 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-cph 23766 |
This theorem is referenced by: cphabscl 23783 cphsqrtcl2 23784 cphsqrtcl3 23785 cphnmf 23793 ipcau 23835 cphsscph 23848 |
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