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Mirrors > Home > MPE Home > Th. List > cphsubrg | Structured version Visualization version GIF version |
Description: The scalar field of a subcomplex pre-Hilbert space is a subring of ℂfld. (Contributed by Mario Carneiro, 8-Oct-2015.) |
Ref | Expression |
---|---|
cphsca.f | ⊢ 𝐹 = (Scalar‘𝑊) |
cphsca.k | ⊢ 𝐾 = (Base‘𝐹) |
Ref | Expression |
---|---|
cphsubrg | ⊢ (𝑊 ∈ ℂPreHil → 𝐾 ∈ (SubRing‘ℂfld)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cphsca.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
2 | cphsca.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | 2, 1 | cphsca 23710 | . . 3 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 = (ℂfld ↾s 𝐾)) |
4 | cphlvec 23706 | . . . 4 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec) | |
5 | 2 | lvecdrng 19806 | . . . 4 ⊢ (𝑊 ∈ LVec → 𝐹 ∈ DivRing) |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 ∈ DivRing) |
7 | 1, 3, 6 | cphsubrglem 23708 | . 2 ⊢ (𝑊 ∈ ℂPreHil → (𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 = (𝐾 ∩ ℂ) ∧ 𝐾 ∈ (SubRing‘ℂfld))) |
8 | 7 | simp3d 1136 | 1 ⊢ (𝑊 ∈ ℂPreHil → 𝐾 ∈ (SubRing‘ℂfld)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∩ cin 3932 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 Basecbs 16471 ↾s cress 16472 Scalarcsca 16556 DivRingcdr 19431 SubRingcsubrg 19460 LVecclvec 19803 ℂfldccnfld 20473 ℂPreHilccph 23697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-seq 13358 df-exp 13418 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-subg 18214 df-cmn 18837 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-drng 19433 df-subrg 19462 df-lvec 19804 df-cnfld 20474 df-phl 20698 df-cph 23699 |
This theorem is referenced by: cphdivcl 23713 cphabscl 23716 cphsqrtcl2 23717 cphsqrtcl3 23718 cphqss 23719 cphclm 23720 cphipcl 23722 4cphipval2 23772 hlprlem 23897 ishl2 23900 |
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