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 24061 | . . . 4 ⊢ 𝑊 ∈ LMod |
3 | cnfldex 20391 | . . . . . 6 ⊢ ℂfld ∈ V | |
4 | cnfldbas 20392 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
5 | 4 | ressid 16821 | . . . . . 6 ⊢ (ℂfld ∈ V → (ℂfld ↾s ℂ) = ℂfld) |
6 | 3, 5 | ax-mp 5 | . . . . 5 ⊢ (ℂfld ↾s ℂ) = ℂfld |
7 | 6 | eqcomi 2747 | . . . 4 ⊢ ℂfld = (ℂfld ↾s ℂ) |
8 | id 22 | . . . . 5 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ) | |
9 | addcl 10836 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) | |
10 | negcl 11103 | . . . . 5 ⊢ (𝑥 ∈ ℂ → -𝑥 ∈ ℂ) | |
11 | ax-1cn 10812 | . . . . 5 ⊢ 1 ∈ ℂ | |
12 | mulcl 10838 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) | |
13 | 8, 9, 10, 11, 12 | cnsubrglem 20438 | . . . 4 ⊢ ℂ ∈ (SubRing‘ℂfld) |
14 | qdass 4684 | . . . . . . . 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 2766 | . . . . . . 7 ⊢ 𝑊 = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), ℂfld〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) |
16 | 15 | lmodsca 16889 | . . . . . 6 ⊢ (ℂfld ∈ V → ℂfld = (Scalar‘𝑊)) |
17 | 3, 16 | ax-mp 5 | . . . . 5 ⊢ ℂfld = (Scalar‘𝑊) |
18 | 17 | isclmi 23998 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ ℂfld = (ℂfld ↾s ℂ) ∧ ℂ ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod) |
19 | 2, 7, 13, 18 | mp3an 1463 | . . 3 ⊢ 𝑊 ∈ ℂMod |
20 | cndrng 20417 | . . . 4 ⊢ ℂfld ∈ DivRing | |
21 | 17 | islvec 20166 | . . . 4 ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ ℂfld ∈ DivRing)) |
22 | 2, 20, 21 | mpbir2an 711 | . . 3 ⊢ 𝑊 ∈ LVec |
23 | 19, 22 | elini 4122 | . 2 ⊢ 𝑊 ∈ (ℂMod ∩ LVec) |
24 | df-cvs 24045 | . 2 ⊢ ℂVec = (ℂMod ∩ LVec) | |
25 | 23, 24 | eleqtrri 2838 | 1 ⊢ 𝑊 ∈ ℂVec |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2111 Vcvv 3421 ∪ cun 3879 ∩ cin 3880 {csn 4556 {cpr 4558 {ctp 4560 〈cop 4562 ‘cfv 6398 (class class class)co 7232 ℂcc 10752 + caddc 10757 · cmul 10759 ndxcnx 16769 Basecbs 16785 ↾s cress 16809 +gcplusg 16827 Scalarcsca 16830 ·𝑠 cvsca 16831 DivRingcdr 19792 SubRingcsubrg 19821 LModclmod 19924 LVecclvec 20164 ℂfldccnfld 20388 ℂModcclm 23983 ℂVecccvs 24044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5194 ax-sep 5207 ax-nul 5214 ax-pow 5273 ax-pr 5337 ax-un 7542 ax-cnex 10810 ax-resscn 10811 ax-1cn 10812 ax-icn 10813 ax-addcl 10814 ax-addrcl 10815 ax-mulcl 10816 ax-mulrcl 10817 ax-mulcom 10818 ax-addass 10819 ax-mulass 10820 ax-distr 10821 ax-i2m1 10822 ax-1ne0 10823 ax-1rid 10824 ax-rnegex 10825 ax-rrecex 10826 ax-cnre 10827 ax-pre-lttri 10828 ax-pre-lttrn 10829 ax-pre-ltadd 10830 ax-pre-mulgt0 10831 ax-addf 10833 ax-mulf 10834 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4835 df-iun 4921 df-br 5069 df-opab 5131 df-mpt 5151 df-tr 5177 df-id 5470 df-eprel 5475 df-po 5483 df-so 5484 df-fr 5524 df-we 5526 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-pred 6176 df-ord 6234 df-on 6235 df-lim 6236 df-suc 6237 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fo 6404 df-f1o 6405 df-fv 6406 df-riota 7189 df-ov 7235 df-oprab 7236 df-mpo 7237 df-om 7664 df-1st 7780 df-2nd 7781 df-tpos 7989 df-wrecs 8068 df-recs 8129 df-rdg 8167 df-1o 8223 df-er 8412 df-en 8648 df-dom 8649 df-sdom 8650 df-fin 8651 df-pnf 10894 df-mnf 10895 df-xr 10896 df-ltxr 10897 df-le 10898 df-sub 11089 df-neg 11090 df-div 11515 df-nn 11856 df-2 11918 df-3 11919 df-4 11920 df-5 11921 df-6 11922 df-7 11923 df-8 11924 df-9 11925 df-n0 12116 df-z 12202 df-dec 12319 df-uz 12464 df-fz 13121 df-struct 16725 df-sets 16742 df-slot 16760 df-ndx 16770 df-base 16786 df-ress 16810 df-plusg 16840 df-mulr 16841 df-starv 16842 df-sca 16843 df-vsca 16844 df-tset 16846 df-ple 16847 df-ds 16849 df-unif 16850 df-0g 16971 df-mgm 18139 df-sgrp 18188 df-mnd 18199 df-grp 18393 df-minusg 18394 df-subg 18565 df-cmn 19197 df-mgp 19530 df-ur 19542 df-ring 19589 df-cring 19590 df-oppr 19666 df-dvdsr 19684 df-unit 19685 df-invr 19715 df-dvr 19726 df-drng 19794 df-subrg 19823 df-lmod 19926 df-lvec 20165 df-cnfld 20389 df-clm 23984 df-cvs 24045 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |