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| Mirrors > Home > MPE Home > Th. List > cphsqrtcl3 | Structured version Visualization version GIF version | ||
| Description: If the scalar field of a subcomplex pre-Hilbert space contains the imaginary unit i, then it is closed under square roots (i.e., it is quadratically closed). (Contributed by Mario Carneiro, 11-Oct-2015.) |
| Ref | Expression |
|---|---|
| cphsca.f | ⊢ 𝐹 = (Scalar‘𝑊) |
| cphsca.k | ⊢ 𝐾 = (Base‘𝐹) |
| Ref | Expression |
|---|---|
| cphsqrtcl3 | ⊢ ((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) → (√‘𝐴) ∈ 𝐾) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpl1 1210 | . . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) ∧ -𝐴 ∈ ℝ+) → 𝑊 ∈ ℂPreHil) | |
| 2 | cphsca.f | . . . . . . . . . . 11 ⊢ 𝐹 = (Scalar‘𝑊) | |
| 3 | cphsca.k | . . . . . . . . . . 11 ⊢ 𝐾 = (Base‘𝐹) | |
| 4 | 2, 3 | cphsubrg 25390 | . . . . . . . . . 10 ⊢ (𝑊 ∈ ℂPreHil → 𝐾 ∈ (SubRing‘ℂfld)) |
| 5 | 1, 4 | syl 18 | . . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) ∧ -𝐴 ∈ ℝ+) → 𝐾 ∈ (SubRing‘ℂfld)) |
| 6 | cnfldbas 21576 | . . . . . . . . . 10 ⊢ ℂ = (Base‘ℂfld) | |
| 7 | 6 | subrgss 20721 | . . . . . . . . 9 ⊢ (𝐾 ∈ (SubRing‘ℂfld) → 𝐾 ⊆ ℂ) |
| 8 | 5, 7 | syl 18 | . . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) ∧ -𝐴 ∈ ℝ+) → 𝐾 ⊆ ℂ) |
| 9 | simpl3 1212 | . . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) ∧ -𝐴 ∈ ℝ+) → 𝐴 ∈ 𝐾) | |
| 10 | 8, 9 | sseldd 3939 | . . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) ∧ -𝐴 ∈ ℝ+) → 𝐴 ∈ ℂ) |
| 11 | 10 | negnegd 11575 | . . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) ∧ -𝐴 ∈ ℝ+) → --𝐴 = 𝐴) |
| 12 | 11 | fveq2d 6889 | . . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) ∧ -𝐴 ∈ ℝ+) → (√‘--𝐴) = (√‘𝐴)) |
| 13 | rpre 13041 | . . . . . . 7 ⊢ (-𝐴 ∈ ℝ+ → -𝐴 ∈ ℝ) | |
| 14 | 13 | adantl 487 | . . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) ∧ -𝐴 ∈ ℝ+) → -𝐴 ∈ ℝ) |
| 15 | rpge0 13046 | . . . . . . 7 ⊢ (-𝐴 ∈ ℝ+ → 0 ≤ -𝐴) | |
| 16 | 15 | adantl 487 | . . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) ∧ -𝐴 ∈ ℝ+) → 0 ≤ -𝐴) |
| 17 | 14, 16 | sqrtnegd 15497 | . . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) ∧ -𝐴 ∈ ℝ+) → (√‘--𝐴) = (i · (√‘-𝐴))) |
| 18 | 12, 17 | eqtr3d 2802 | . . . 4 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) ∧ -𝐴 ∈ ℝ+) → (√‘𝐴) = (i · (√‘-𝐴))) |
| 19 | simpl2 1211 | . . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) ∧ -𝐴 ∈ ℝ+) → i ∈ 𝐾) | |
| 20 | cnfldneg 21598 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((invg‘ℂfld)‘𝐴) = -𝐴) | |
| 21 | 10, 20 | syl 18 | . . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) ∧ -𝐴 ∈ ℝ+) → ((invg‘ℂfld)‘𝐴) = -𝐴) |
| 22 | subrgsubg 20726 | . . . . . . . . 9 ⊢ (𝐾 ∈ (SubRing‘ℂfld) → 𝐾 ∈ (SubGrp‘ℂfld)) | |
| 23 | 5, 22 | syl 18 | . . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) ∧ -𝐴 ∈ ℝ+) → 𝐾 ∈ (SubGrp‘ℂfld)) |
| 24 | eqid 2765 | . . . . . . . . 9 ⊢ (invg‘ℂfld) = (invg‘ℂfld) | |
| 25 | 24 | subginvcl 19245 | . . . . . . . 8 ⊢ ((𝐾 ∈ (SubGrp‘ℂfld) ∧ 𝐴 ∈ 𝐾) → ((invg‘ℂfld)‘𝐴) ∈ 𝐾) |
| 26 | 23, 9, 25 | syl2anc 596 | . . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) ∧ -𝐴 ∈ ℝ+) → ((invg‘ℂfld)‘𝐴) ∈ 𝐾) |
| 27 | 21, 26 | eqeltrrd 2866 | . . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) ∧ -𝐴 ∈ ℝ+) → -𝐴 ∈ 𝐾) |
| 28 | 2, 3 | cphsqrtcl 25394 | . . . . . 6 ⊢ ((𝑊 ∈ ℂPreHil ∧ (-𝐴 ∈ 𝐾 ∧ -𝐴 ∈ ℝ ∧ 0 ≤ -𝐴)) → (√‘-𝐴) ∈ 𝐾) |
| 29 | 1, 27, 14, 16, 28 | syl13anc 1399 | . . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) ∧ -𝐴 ∈ ℝ+) → (√‘-𝐴) ∈ 𝐾) |
| 30 | cnfldmul 21580 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
| 31 | 30 | subrgmcl 20733 | . . . . 5 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ i ∈ 𝐾 ∧ (√‘-𝐴) ∈ 𝐾) → (i · (√‘-𝐴)) ∈ 𝐾) |
| 32 | 5, 19, 29, 31 | syl3anc 1398 | . . . 4 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) ∧ -𝐴 ∈ ℝ+) → (i · (√‘-𝐴)) ∈ 𝐾) |
| 33 | 18, 32 | eqeltrd 2865 | . . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) ∧ -𝐴 ∈ ℝ+) → (√‘𝐴) ∈ 𝐾) |
| 34 | 33 | ex 418 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) → (-𝐴 ∈ ℝ+ → (√‘𝐴) ∈ 𝐾)) |
| 35 | 2, 3 | cphsqrtcl2 25396 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝐾 ∧ ¬ -𝐴 ∈ ℝ+) → (√‘𝐴) ∈ 𝐾) |
| 36 | 35 | 3expia 1139 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝐾) → (¬ -𝐴 ∈ ℝ+ → (√‘𝐴) ∈ 𝐾)) |
| 37 | 36 | 3adant2 1149 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) → (¬ -𝐴 ∈ ℝ+ → (√‘𝐴) ∈ 𝐾)) |
| 38 | 34, 37 | pm2.61d 181 | 1 ⊢ ((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) → (√‘𝐴) ∈ 𝐾) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ⊆ wss 3906 class class class wbr 5111 ‘cfv 6540 (class class class)co 7419 ℂcc 11113 ℝcr 11114 0cc0 11115 ici 11117 · cmul 11120 ≤ cle 11259 -cneg 11457 ℝ+crp 13032 √csqrt 15308 Basecbs 17291 Scalarcsca 17335 invgcminusg 19045 SubGrpcsubg 19230 SubRingcsubrg 20718 ℂfldccnfld 21572 ℂPreHilccph 25376 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193 ax-addf 11194 ax-mulf 11195 |
| This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-tpos 8228 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-sup 9409 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12520 df-z 12607 df-dec 12728 df-uz 12879 df-rp 13033 df-ico 13394 df-fz 13552 df-seq 14056 df-exp 14116 df-cj 15174 df-re 15175 df-im 15176 df-sqrt 15310 df-abs 15311 df-struct 17229 df-sets 17246 df-slot 17264 df-ndx 17276 df-base 17292 df-ress 17313 df-plusg 17345 df-mulr 17346 df-starv 17347 df-tset 17351 df-ple 17352 df-ds 17354 df-unif 17355 df-0g 17516 df-mgm 18720 df-sgrp 18809 df-mnd 18825 df-mhm 18878 df-grp 19047 df-minusg 19048 df-subg 19233 df-ghm 19328 df-cmn 19896 df-abl 19897 df-mgp 20261 df-rng 20275 df-ur 20308 df-ring 20361 df-cring 20362 df-oppr 20465 df-dvdsr 20485 df-unit 20486 df-invr 20516 df-dvr 20529 df-rhm 20600 df-subrng 20695 df-subrg 20719 df-drng 20879 df-staf 20992 df-srng 20993 df-lvec 21274 df-cnfld 21573 df-phl 21826 df-cph 25378 |
| This theorem is used by: (None) |
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