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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 21654 | . . . . 5 ⊢ ℝfld ∈ Field | |
2 | isfld 20756 | . . . . . . 7 ⊢ (ℝfld ∈ Field ↔ (ℝfld ∈ DivRing ∧ ℝfld ∈ CRing)) | |
3 | 2 | simprbi 496 | . . . . . 6 ⊢ (ℝfld ∈ Field → ℝfld ∈ CRing) |
4 | 3 | crngringd 20263 | . . . . 5 ⊢ (ℝfld ∈ Field → ℝfld ∈ Ring) |
5 | rlmlmod 21227 | . . . . 5 ⊢ (ℝfld ∈ Ring → (ringLMod‘ℝfld) ∈ LMod) | |
6 | 1, 4, 5 | mp2b 10 | . . . 4 ⊢ (ringLMod‘ℝfld) ∈ LMod |
7 | rlmsca 21222 | . . . . . 6 ⊢ (ℝfld ∈ Field → ℝfld = (Scalar‘(ringLMod‘ℝfld))) | |
8 | 1, 7 | ax-mp 5 | . . . . 5 ⊢ ℝfld = (Scalar‘(ringLMod‘ℝfld)) |
9 | df-refld 21640 | . . . . 5 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
10 | 8, 9 | eqtr3i 2764 | . . . 4 ⊢ (Scalar‘(ringLMod‘ℝfld)) = (ℂfld ↾s ℝ) |
11 | resubdrg 21643 | . . . . 5 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
12 | 11 | simpli 483 | . . . 4 ⊢ ℝ ∈ (SubRing‘ℂfld) |
13 | eqid 2734 | . . . . 5 ⊢ (Scalar‘(ringLMod‘ℝfld)) = (Scalar‘(ringLMod‘ℝfld)) | |
14 | 13 | isclmi 25123 | . . . 4 ⊢ (((ringLMod‘ℝfld) ∈ LMod ∧ (Scalar‘(ringLMod‘ℝfld)) = (ℂfld ↾s ℝ) ∧ ℝ ∈ (SubRing‘ℂfld)) → (ringLMod‘ℝfld) ∈ ℂMod) |
15 | 6, 10, 12, 14 | mp3an 1460 | . . 3 ⊢ (ringLMod‘ℝfld) ∈ ℂMod |
16 | 11 | simpri 485 | . . . 4 ⊢ ℝfld ∈ DivRing |
17 | rlmlvec 21228 | . . . 4 ⊢ (ℝfld ∈ DivRing → (ringLMod‘ℝfld) ∈ LVec) | |
18 | 16, 17 | ax-mp 5 | . . 3 ⊢ (ringLMod‘ℝfld) ∈ LVec |
19 | 15, 18 | elini 4208 | . 2 ⊢ (ringLMod‘ℝfld) ∈ (ℂMod ∩ LVec) |
20 | recvs.r | . 2 ⊢ 𝑅 = (ringLMod‘ℝfld) | |
21 | df-cvs 25170 | . 2 ⊢ ℂVec = (ℂMod ∩ LVec) | |
22 | 19, 20, 21 | 3eltr4i 2851 | 1 ⊢ 𝑅 ∈ ℂVec |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2105 ∩ cin 3961 ‘cfv 6562 (class class class)co 7430 ℝcr 11151 ↾s cress 17273 Scalarcsca 17300 Ringcrg 20250 CRingccrg 20251 SubRingcsubrg 20585 DivRingcdr 20745 Fieldcfield 20746 LModclmod 20874 LVecclvec 21118 ringLModcrglmod 21188 ℂfldccnfld 21381 ℝfldcrefld 21639 ℂModcclm 25108 ℂVecccvs 25169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-addf 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-subg 19153 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-dvr 20417 df-subrng 20562 df-subrg 20586 df-drng 20747 df-field 20748 df-lmod 20876 df-lvec 21119 df-sra 21189 df-rgmod 21190 df-cnfld 21382 df-refld 21640 df-clm 25109 df-cvs 25170 |
This theorem is referenced by: (None) |
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