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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 24655 | . . . 4 ⊢ 𝑊 ∈ LMod |
3 | cnfldex 20946 | . . . . . 6 ⊢ ℂfld ∈ V | |
4 | cnfldbas 20947 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
5 | 4 | ressid 17188 | . . . . . 6 ⊢ (ℂfld ∈ V → (ℂfld ↾s ℂ) = ℂfld) |
6 | 3, 5 | ax-mp 5 | . . . . 5 ⊢ (ℂfld ↾s ℂ) = ℂfld |
7 | 6 | eqcomi 2741 | . . . 4 ⊢ ℂfld = (ℂfld ↾s ℂ) |
8 | id 22 | . . . . 5 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ) | |
9 | addcl 11191 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) | |
10 | negcl 11459 | . . . . 5 ⊢ (𝑥 ∈ ℂ → -𝑥 ∈ ℂ) | |
11 | ax-1cn 11167 | . . . . 5 ⊢ 1 ∈ ℂ | |
12 | mulcl 11193 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) | |
13 | 8, 9, 10, 11, 12 | cnsubrglem 20994 | . . . 4 ⊢ ℂ ∈ (SubRing‘ℂfld) |
14 | qdass 4757 | . . . . . . . 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 2760 | . . . . . . 7 ⊢ 𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), ℂfld⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩}) |
16 | 15 | lmodsca 17272 | . . . . . 6 ⊢ (ℂfld ∈ V → ℂfld = (Scalar‘𝑊)) |
17 | 3, 16 | ax-mp 5 | . . . . 5 ⊢ ℂfld = (Scalar‘𝑊) |
18 | 17 | isclmi 24592 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ ℂfld = (ℂfld ↾s ℂ) ∧ ℂ ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod) |
19 | 2, 7, 13, 18 | mp3an 1461 | . . 3 ⊢ 𝑊 ∈ ℂMod |
20 | cndrng 20973 | . . . 4 ⊢ ℂfld ∈ DivRing | |
21 | 17 | islvec 20714 | . . . 4 ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ ℂfld ∈ DivRing)) |
22 | 2, 20, 21 | mpbir2an 709 | . . 3 ⊢ 𝑊 ∈ LVec |
23 | 19, 22 | elini 4193 | . 2 ⊢ 𝑊 ∈ (ℂMod ∩ LVec) |
24 | df-cvs 24639 | . 2 ⊢ ℂVec = (ℂMod ∩ LVec) | |
25 | 23, 24 | eleqtrri 2832 | 1 ⊢ 𝑊 ∈ ℂVec |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∪ cun 3946 ∩ cin 3947 {csn 4628 {cpr 4630 {ctp 4632 ⟨cop 4634 ‘cfv 6543 (class class class)co 7408 ℂcc 11107 + caddc 11112 · cmul 11114 ndxcnx 17125 Basecbs 17143 ↾s cress 17172 +gcplusg 17196 Scalarcsca 17199 ·𝑠 cvsca 17200 SubRingcsubrg 20314 DivRingcdr 20356 LModclmod 20470 LVecclvec 20712 ℂfldccnfld 20943 ℂModcclm 24577 ℂVecccvs 24638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-addf 11188 ax-mulf 11189 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-tpos 8210 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13484 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-0g 17386 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-grp 18821 df-minusg 18822 df-subg 19002 df-cmn 19649 df-mgp 19987 df-ur 20004 df-ring 20057 df-cring 20058 df-oppr 20149 df-dvdsr 20170 df-unit 20171 df-invr 20201 df-dvr 20214 df-subrg 20316 df-drng 20358 df-lmod 20472 df-lvec 20713 df-cnfld 20944 df-clm 24578 df-cvs 24639 |
This theorem is referenced by: (None) |
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