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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 14725 | . . . 4 ⊢ √:ℂ⟶ℂ | |
2 | ffn 6516 | . . . 4 ⊢ (√:ℂ⟶ℂ → √ Fn ℂ) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ √ Fn ℂ |
4 | inss2 4208 | . . . 4 ⊢ (𝐾 ∩ (0[,)+∞)) ⊆ (0[,)+∞) | |
5 | rge0ssre 12847 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
6 | ax-resscn 10596 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
7 | 5, 6 | sstri 3978 | . . . 4 ⊢ (0[,)+∞) ⊆ ℂ |
8 | 4, 7 | sstri 3978 | . . 3 ⊢ (𝐾 ∩ (0[,)+∞)) ⊆ ℂ |
9 | simp1 1132 | . . . 4 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ 𝐾) | |
10 | elrege0 12845 | . . . . . 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 4173 | . . 3 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ (𝐾 ∩ (0[,)+∞))) |
14 | fnfvima 6997 | . . 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 2823 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
17 | eqid 2823 | . . . . 5 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
18 | eqid 2823 | . . . . 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 23776 | . . . 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 3969 | . 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 1537 ∈ wcel 2114 ∩ cin 3937 ⊆ wss 3938 class class class wbr 5068 ↦ cmpt 5148 “ cima 5560 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 0cc0 10539 +∞cpnf 10674 ≤ cle 10678 [,)cico 12743 √csqrt 14594 Basecbs 16485 ↾s cress 16486 Scalarcsca 16570 ·𝑖cip 16572 ℂfldccnfld 20547 PreHilcphl 20770 normcnm 23188 NrmModcnlm 23192 ℂPreHilccph 23772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-ico 12747 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-cph 23774 |
This theorem is referenced by: cphabscl 23791 cphsqrtcl2 23792 cphsqrtcl3 23793 cphnmf 23801 ipcau 23843 cphsscph 23856 |
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