![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 25054 | . . . 4 ⊢ 𝑊 ∈ LMod |
3 | cnfldex 21269 | . . . . . 6 ⊢ ℂfld ∈ V | |
4 | cnfldbas 21270 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
5 | 4 | ressid 17216 | . . . . . 6 ⊢ (ℂfld ∈ V → (ℂfld ↾s ℂ) = ℂfld) |
6 | 3, 5 | ax-mp 5 | . . . . 5 ⊢ (ℂfld ↾s ℂ) = ℂfld |
7 | 6 | eqcomi 2736 | . . . 4 ⊢ ℂfld = (ℂfld ↾s ℂ) |
8 | id 22 | . . . . 5 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ) | |
9 | addcl 11212 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) | |
10 | negcl 11482 | . . . . 5 ⊢ (𝑥 ∈ ℂ → -𝑥 ∈ ℂ) | |
11 | ax-1cn 11188 | . . . . 5 ⊢ 1 ∈ ℂ | |
12 | mulcl 11214 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) | |
13 | 8, 9, 10, 11, 12 | cnsubrglem 21336 | . . . 4 ⊢ ℂ ∈ (SubRing‘ℂfld) |
14 | qdass 4753 | . . . . . . . 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 2755 | . . . . . . 7 ⊢ 𝑊 = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), ℂfld〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) |
16 | 15 | lmodsca 17300 | . . . . . 6 ⊢ (ℂfld ∈ V → ℂfld = (Scalar‘𝑊)) |
17 | 3, 16 | ax-mp 5 | . . . . 5 ⊢ ℂfld = (Scalar‘𝑊) |
18 | 17 | isclmi 24991 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ ℂfld = (ℂfld ↾s ℂ) ∧ ℂ ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod) |
19 | 2, 7, 13, 18 | mp3an 1458 | . . 3 ⊢ 𝑊 ∈ ℂMod |
20 | cndrng 21313 | . . . 4 ⊢ ℂfld ∈ DivRing | |
21 | 17 | islvec 20978 | . . . 4 ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ ℂfld ∈ DivRing)) |
22 | 2, 20, 21 | mpbir2an 710 | . . 3 ⊢ 𝑊 ∈ LVec |
23 | 19, 22 | elini 4189 | . 2 ⊢ 𝑊 ∈ (ℂMod ∩ LVec) |
24 | df-cvs 25038 | . 2 ⊢ ℂVec = (ℂMod ∩ LVec) | |
25 | 23, 24 | eleqtrri 2827 | 1 ⊢ 𝑊 ∈ ℂVec |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 Vcvv 3469 ∪ cun 3942 ∩ cin 3943 {csn 4624 {cpr 4626 {ctp 4628 〈cop 4630 ‘cfv 6542 (class class class)co 7414 ℂcc 11128 + caddc 11133 · cmul 11135 ndxcnx 17153 Basecbs 17171 ↾s cress 17200 +gcplusg 17224 Scalarcsca 17227 ·𝑠 cvsca 17228 SubRingcsubrg 20495 DivRingcdr 20613 LModclmod 20732 LVecclvec 20976 ℂfldccnfld 21266 ℂModcclm 24976 ℂVecccvs 25037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-addf 11209 ax-mulf 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-fz 13509 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-starv 17239 df-sca 17240 df-vsca 17241 df-tset 17243 df-ple 17244 df-ds 17246 df-unif 17247 df-0g 17414 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-grp 18884 df-minusg 18885 df-subg 19069 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-cring 20167 df-oppr 20262 df-dvdsr 20285 df-unit 20286 df-invr 20316 df-dvr 20329 df-subrng 20472 df-subrg 20497 df-drng 20615 df-lmod 20734 df-lvec 20977 df-cnfld 21267 df-clm 24977 df-cvs 25038 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |