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Mirrors > Home > MPE Home > Th. List > recvs | Structured version Visualization version GIF version |
Description: The field of the real numbers as left module over itself is a subcomplex vector space. The vector operation is +, and the scalar product is ·. (Contributed by AV, 22-Oct-2021.) (Proof shortened by SN, 23-Nov-2024.) |
Ref | Expression |
---|---|
recvs.r | ⊢ 𝑅 = (ringLMod‘ℝfld) |
Ref | Expression |
---|---|
recvs | ⊢ 𝑅 ∈ ℂVec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | refld 21046 | . . . . 5 ⊢ ℝfld ∈ Field | |
2 | isfld 20230 | . . . . . . 7 ⊢ (ℝfld ∈ Field ↔ (ℝfld ∈ DivRing ∧ ℝfld ∈ CRing)) | |
3 | 2 | simprbi 498 | . . . . . 6 ⊢ (ℝfld ∈ Field → ℝfld ∈ CRing) |
4 | 3 | crngringd 19985 | . . . . 5 ⊢ (ℝfld ∈ Field → ℝfld ∈ Ring) |
5 | rlmlmod 20719 | . . . . 5 ⊢ (ℝfld ∈ Ring → (ringLMod‘ℝfld) ∈ LMod) | |
6 | 1, 4, 5 | mp2b 10 | . . . 4 ⊢ (ringLMod‘ℝfld) ∈ LMod |
7 | rlmsca 20714 | . . . . . 6 ⊢ (ℝfld ∈ Field → ℝfld = (Scalar‘(ringLMod‘ℝfld))) | |
8 | 1, 7 | ax-mp 5 | . . . . 5 ⊢ ℝfld = (Scalar‘(ringLMod‘ℝfld)) |
9 | df-refld 21032 | . . . . 5 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
10 | 8, 9 | eqtr3i 2763 | . . . 4 ⊢ (Scalar‘(ringLMod‘ℝfld)) = (ℂfld ↾s ℝ) |
11 | resubdrg 21035 | . . . . 5 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
12 | 11 | simpli 485 | . . . 4 ⊢ ℝ ∈ (SubRing‘ℂfld) |
13 | eqid 2733 | . . . . 5 ⊢ (Scalar‘(ringLMod‘ℝfld)) = (Scalar‘(ringLMod‘ℝfld)) | |
14 | 13 | isclmi 24463 | . . . 4 ⊢ (((ringLMod‘ℝfld) ∈ LMod ∧ (Scalar‘(ringLMod‘ℝfld)) = (ℂfld ↾s ℝ) ∧ ℝ ∈ (SubRing‘ℂfld)) → (ringLMod‘ℝfld) ∈ ℂMod) |
15 | 6, 10, 12, 14 | mp3an 1462 | . . 3 ⊢ (ringLMod‘ℝfld) ∈ ℂMod |
16 | 11 | simpri 487 | . . . 4 ⊢ ℝfld ∈ DivRing |
17 | rlmlvec 20720 | . . . 4 ⊢ (ℝfld ∈ DivRing → (ringLMod‘ℝfld) ∈ LVec) | |
18 | 16, 17 | ax-mp 5 | . . 3 ⊢ (ringLMod‘ℝfld) ∈ LVec |
19 | 15, 18 | elini 4157 | . 2 ⊢ (ringLMod‘ℝfld) ∈ (ℂMod ∩ LVec) |
20 | recvs.r | . 2 ⊢ 𝑅 = (ringLMod‘ℝfld) | |
21 | df-cvs 24510 | . 2 ⊢ ℂVec = (ℂMod ∩ LVec) | |
22 | 19, 20, 21 | 3eltr4i 2847 | 1 ⊢ 𝑅 ∈ ℂVec |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ∩ cin 3913 ‘cfv 6500 (class class class)co 7361 ℝcr 11058 ↾s cress 17120 Scalarcsca 17144 Ringcrg 19972 CRingccrg 19973 DivRingcdr 20219 Fieldcfield 20220 SubRingcsubrg 20260 LModclmod 20365 LVecclvec 20607 ringLModcrglmod 20675 ℂfldccnfld 20819 ℝfldcrefld 21031 ℂModcclm 24448 ℂVecccvs 24509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-addf 11138 ax-mulf 11139 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-tpos 8161 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-fz 13434 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-minusg 18760 df-subg 18933 df-cmn 19572 df-mgp 19905 df-ur 19922 df-ring 19974 df-cring 19975 df-oppr 20057 df-dvdsr 20078 df-unit 20079 df-invr 20109 df-dvr 20120 df-drng 20221 df-field 20222 df-subrg 20262 df-lmod 20367 df-lvec 20608 df-sra 20678 df-rgmod 20679 df-cnfld 20820 df-refld 21032 df-clm 24449 df-cvs 24510 |
This theorem is referenced by: (None) |
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