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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 23308 | . . . 4 ⊢ 𝑊 ∈ LMod |
3 | cnfldex 20108 | . . . . . 6 ⊢ ℂfld ∈ V | |
4 | cnfldbas 20109 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
5 | 4 | ressid 16297 | . . . . . 6 ⊢ (ℂfld ∈ V → (ℂfld ↾s ℂ) = ℂfld) |
6 | 3, 5 | ax-mp 5 | . . . . 5 ⊢ (ℂfld ↾s ℂ) = ℂfld |
7 | 6 | eqcomi 2833 | . . . 4 ⊢ ℂfld = (ℂfld ↾s ℂ) |
8 | id 22 | . . . . 5 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ) | |
9 | addcl 10333 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) | |
10 | negcl 10600 | . . . . 5 ⊢ (𝑥 ∈ ℂ → -𝑥 ∈ ℂ) | |
11 | ax-1cn 10309 | . . . . 5 ⊢ 1 ∈ ℂ | |
12 | mulcl 10335 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) | |
13 | 8, 9, 10, 11, 12 | cnsubrglem 20155 | . . . 4 ⊢ ℂ ∈ (SubRing‘ℂfld) |
14 | qdass 4505 | . . . . . . . 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 2848 | . . . . . . 7 ⊢ 𝑊 = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), ℂfld〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) |
16 | 15 | lmodsca 16378 | . . . . . 6 ⊢ (ℂfld ∈ V → ℂfld = (Scalar‘𝑊)) |
17 | 3, 16 | ax-mp 5 | . . . . 5 ⊢ ℂfld = (Scalar‘𝑊) |
18 | 17 | isclmi 23245 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ ℂfld = (ℂfld ↾s ℂ) ∧ ℂ ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod) |
19 | 2, 7, 13, 18 | mp3an 1591 | . . 3 ⊢ 𝑊 ∈ ℂMod |
20 | cndrng 20134 | . . . 4 ⊢ ℂfld ∈ DivRing | |
21 | 17 | islvec 19462 | . . . 4 ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ ℂfld ∈ DivRing)) |
22 | 2, 20, 21 | mpbir2an 704 | . . 3 ⊢ 𝑊 ∈ LVec |
23 | elin 4022 | . . 3 ⊢ (𝑊 ∈ (ℂMod ∩ LVec) ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec)) | |
24 | 19, 22, 23 | mpbir2an 704 | . 2 ⊢ 𝑊 ∈ (ℂMod ∩ LVec) |
25 | df-cvs 23292 | . 2 ⊢ ℂVec = (ℂMod ∩ LVec) | |
26 | 24, 25 | eleqtrri 2904 | 1 ⊢ 𝑊 ∈ ℂVec |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1658 ∈ wcel 2166 Vcvv 3413 ∪ cun 3795 ∩ cin 3796 {csn 4396 {cpr 4398 {ctp 4400 〈cop 4402 ‘cfv 6122 (class class class)co 6904 ℂcc 10249 + caddc 10254 · cmul 10256 ndxcnx 16218 Basecbs 16221 ↾s cress 16222 +gcplusg 16304 Scalarcsca 16307 ·𝑠 cvsca 16308 DivRingcdr 19102 SubRingcsubrg 19131 LModclmod 19218 LVecclvec 19460 ℂfldccnfld 20105 ℂModcclm 23230 ℂVecccvs 23291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 ax-addf 10330 ax-mulf 10331 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-1st 7427 df-2nd 7428 df-tpos 7616 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-oadd 7829 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 df-nn 11350 df-2 11413 df-3 11414 df-4 11415 df-5 11416 df-6 11417 df-7 11418 df-8 11419 df-9 11420 df-n0 11618 df-z 11704 df-dec 11821 df-uz 11968 df-fz 12619 df-struct 16223 df-ndx 16224 df-slot 16225 df-base 16227 df-sets 16228 df-ress 16229 df-plusg 16317 df-mulr 16318 df-starv 16319 df-sca 16320 df-vsca 16321 df-tset 16323 df-ple 16324 df-ds 16326 df-unif 16327 df-0g 16454 df-mgm 17594 df-sgrp 17636 df-mnd 17647 df-grp 17778 df-minusg 17779 df-subg 17941 df-cmn 18547 df-mgp 18843 df-ur 18855 df-ring 18902 df-cring 18903 df-oppr 18976 df-dvdsr 18994 df-unit 18995 df-invr 19025 df-dvr 19036 df-drng 19104 df-subrg 19133 df-lmod 19220 df-lvec 19461 df-cnfld 20106 df-clm 23231 df-cvs 23292 |
This theorem is referenced by: (None) |
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