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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 15402 | . . . 4 ⊢ √:ℂ⟶ℂ | |
| 2 | ffn 6736 | . . . 4 ⊢ (√:ℂ⟶ℂ → √ Fn ℂ) | |
| 3 | 1, 2 | ax-mp 5 | . . 3 ⊢ √ Fn ℂ |
| 4 | inss2 4238 | . . . 4 ⊢ (𝐾 ∩ (0[,)+∞)) ⊆ (0[,)+∞) | |
| 5 | rge0ssre 13496 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
| 6 | ax-resscn 11212 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
| 7 | 5, 6 | sstri 3993 | . . . 4 ⊢ (0[,)+∞) ⊆ ℂ |
| 8 | 4, 7 | sstri 3993 | . . 3 ⊢ (𝐾 ∩ (0[,)+∞)) ⊆ ℂ |
| 9 | simp1 1137 | . . . 4 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ 𝐾) | |
| 10 | elrege0 13494 | . . . . . 6 ⊢ (𝐴 ∈ (0[,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) | |
| 11 | 10 | biimpri 228 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ (0[,)+∞)) |
| 12 | 11 | 3adant1 1131 | . . . 4 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ (0[,)+∞)) |
| 13 | 9, 12 | elind 4200 | . . 3 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ (𝐾 ∩ (0[,)+∞))) |
| 14 | fnfvima 7253 | . . 3 ⊢ ((√ Fn ℂ ∧ (𝐾 ∩ (0[,)+∞)) ⊆ ℂ ∧ 𝐴 ∈ (𝐾 ∩ (0[,)+∞))) → (√‘𝐴) ∈ (√ “ (𝐾 ∩ (0[,)+∞)))) | |
| 15 | 3, 8, 13, 14 | mp3an12i 1467 | . 2 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘𝐴) ∈ (√ “ (𝐾 ∩ (0[,)+∞)))) |
| 16 | eqid 2737 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 17 | eqid 2737 | . . . . 5 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
| 18 | eqid 2737 | . . . . 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 25204 | . . . 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 1147 | . . 3 ⊢ (𝑊 ∈ ℂPreHil → (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾) |
| 23 | 22 | sselda 3983 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ (√‘𝐴) ∈ (√ “ (𝐾 ∩ (0[,)+∞)))) → (√‘𝐴) ∈ 𝐾) |
| 24 | 15, 23 | sylan2 593 | 1 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) → (√‘𝐴) ∈ 𝐾) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 “ cima 5688 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 +∞cpnf 11292 ≤ cle 11296 [,)cico 13389 √csqrt 15272 Basecbs 17247 ↾s cress 17274 Scalarcsca 17300 ·𝑖cip 17302 ℂfldccnfld 21364 PreHilcphl 21642 normcnm 24589 NrmModcnlm 24593 ℂPreHilccph 25200 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-ico 13393 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-cph 25202 |
| This theorem is referenced by: cphabscl 25219 cphsqrtcl2 25220 cphsqrtcl3 25221 cphnmf 25229 ipcau 25272 cphsscph 25285 |
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