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Mirrors > Home > MPE Home > Th. List > cnstrcvs | Structured version Visualization version GIF version |
Description: The set of complex numbers is a subcomplex vector space. The vector operation is +, and the scalar product is ·. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 20-Sep-2021.) |
Ref | Expression |
---|---|
cnlmod.w | ⊢ 𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩}) |
Ref | Expression |
---|---|
cnstrcvs | ⊢ 𝑊 ∈ ℂVec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnlmod.w | . . . . 5 ⊢ 𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩}) | |
2 | 1 | cnlmod 24887 | . . . 4 ⊢ 𝑊 ∈ LMod |
3 | cnfldex 21147 | . . . . . 6 ⊢ ℂfld ∈ V | |
4 | cnfldbas 21148 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
5 | 4 | ressid 17193 | . . . . . 6 ⊢ (ℂfld ∈ V → (ℂfld ↾s ℂ) = ℂfld) |
6 | 3, 5 | ax-mp 5 | . . . . 5 ⊢ (ℂfld ↾s ℂ) = ℂfld |
7 | 6 | eqcomi 2739 | . . . 4 ⊢ ℂfld = (ℂfld ↾s ℂ) |
8 | id 22 | . . . . 5 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ) | |
9 | addcl 11194 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) | |
10 | negcl 11464 | . . . . 5 ⊢ (𝑥 ∈ ℂ → -𝑥 ∈ ℂ) | |
11 | ax-1cn 11170 | . . . . 5 ⊢ 1 ∈ ℂ | |
12 | mulcl 11196 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) | |
13 | 8, 9, 10, 11, 12 | cnsubrglem 21195 | . . . 4 ⊢ ℂ ∈ (SubRing‘ℂfld) |
14 | qdass 4756 | . . . . . . . 8 ⊢ ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩}) = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), ℂfld⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩}) | |
15 | 1, 14 | eqtri 2758 | . . . . . . 7 ⊢ 𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), ℂfld⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩}) |
16 | 15 | lmodsca 17277 | . . . . . 6 ⊢ (ℂfld ∈ V → ℂfld = (Scalar‘𝑊)) |
17 | 3, 16 | ax-mp 5 | . . . . 5 ⊢ ℂfld = (Scalar‘𝑊) |
18 | 17 | isclmi 24824 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ ℂfld = (ℂfld ↾s ℂ) ∧ ℂ ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod) |
19 | 2, 7, 13, 18 | mp3an 1459 | . . 3 ⊢ 𝑊 ∈ ℂMod |
20 | cndrng 21174 | . . . 4 ⊢ ℂfld ∈ DivRing | |
21 | 17 | islvec 20859 | . . . 4 ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ ℂfld ∈ DivRing)) |
22 | 2, 20, 21 | mpbir2an 707 | . . 3 ⊢ 𝑊 ∈ LVec |
23 | 19, 22 | elini 4192 | . 2 ⊢ 𝑊 ∈ (ℂMod ∩ LVec) |
24 | df-cvs 24871 | . 2 ⊢ ℂVec = (ℂMod ∩ LVec) | |
25 | 23, 24 | eleqtrri 2830 | 1 ⊢ 𝑊 ∈ ℂVec |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∈ wcel 2104 Vcvv 3472 ∪ cun 3945 ∩ cin 3946 {csn 4627 {cpr 4629 {ctp 4631 ⟨cop 4633 ‘cfv 6542 (class class class)co 7411 ℂcc 11110 + caddc 11115 · cmul 11117 ndxcnx 17130 Basecbs 17148 ↾s cress 17177 +gcplusg 17201 Scalarcsca 17204 ·𝑠 cvsca 17205 SubRingcsubrg 20457 DivRingcdr 20500 LModclmod 20614 LVecclvec 20857 ℂfldccnfld 21144 ℂModcclm 24809 ℂVecccvs 24870 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-addf 11191 ax-mulf 11192 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13489 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-0g 17391 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 df-subg 19039 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-cring 20130 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-invr 20279 df-dvr 20292 df-subrng 20434 df-subrg 20459 df-drng 20502 df-lmod 20616 df-lvec 20858 df-cnfld 21145 df-clm 24810 df-cvs 24871 |
This theorem is referenced by: (None) |
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