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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.) |
Ref | Expression |
---|---|
recvs.r | ⊢ 𝑅 = (ringLMod‘ℝfld) |
Ref | Expression |
---|---|
recvs | ⊢ 𝑅 ∈ ℂVec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | refld 20308 | . . . . . 6 ⊢ ℝfld ∈ Field | |
2 | fldidom 20071 | . . . . . . 7 ⊢ (ℝfld ∈ Field → ℝfld ∈ IDomn) | |
3 | isidom 20070 | . . . . . . . 8 ⊢ (ℝfld ∈ IDomn ↔ (ℝfld ∈ CRing ∧ ℝfld ∈ Domn)) | |
4 | crngring 19302 | . . . . . . . . 9 ⊢ (ℝfld ∈ CRing → ℝfld ∈ Ring) | |
5 | 4 | adantr 484 | . . . . . . . 8 ⊢ ((ℝfld ∈ CRing ∧ ℝfld ∈ Domn) → ℝfld ∈ Ring) |
6 | 3, 5 | sylbi 220 | . . . . . . 7 ⊢ (ℝfld ∈ IDomn → ℝfld ∈ Ring) |
7 | 2, 6 | syl 17 | . . . . . 6 ⊢ (ℝfld ∈ Field → ℝfld ∈ Ring) |
8 | 1, 7 | ax-mp 5 | . . . . 5 ⊢ ℝfld ∈ Ring |
9 | rlmlmod 19970 | . . . . 5 ⊢ (ℝfld ∈ Ring → (ringLMod‘ℝfld) ∈ LMod) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ (ringLMod‘ℝfld) ∈ LMod |
11 | rlmsca 19965 | . . . . . 6 ⊢ (ℝfld ∈ Field → ℝfld = (Scalar‘(ringLMod‘ℝfld))) | |
12 | 1, 11 | ax-mp 5 | . . . . 5 ⊢ ℝfld = (Scalar‘(ringLMod‘ℝfld)) |
13 | df-refld 20294 | . . . . 5 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
14 | 12, 13 | eqtr3i 2823 | . . . 4 ⊢ (Scalar‘(ringLMod‘ℝfld)) = (ℂfld ↾s ℝ) |
15 | resubdrg 20297 | . . . . 5 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
16 | 15 | simpli 487 | . . . 4 ⊢ ℝ ∈ (SubRing‘ℂfld) |
17 | eqid 2798 | . . . . 5 ⊢ (Scalar‘(ringLMod‘ℝfld)) = (Scalar‘(ringLMod‘ℝfld)) | |
18 | 17 | isclmi 23682 | . . . 4 ⊢ (((ringLMod‘ℝfld) ∈ LMod ∧ (Scalar‘(ringLMod‘ℝfld)) = (ℂfld ↾s ℝ) ∧ ℝ ∈ (SubRing‘ℂfld)) → (ringLMod‘ℝfld) ∈ ℂMod) |
19 | 10, 14, 16, 18 | mp3an 1458 | . . 3 ⊢ (ringLMod‘ℝfld) ∈ ℂMod |
20 | 15 | simpri 489 | . . . 4 ⊢ ℝfld ∈ DivRing |
21 | rlmlvec 19971 | . . . 4 ⊢ (ℝfld ∈ DivRing → (ringLMod‘ℝfld) ∈ LVec) | |
22 | 20, 21 | ax-mp 5 | . . 3 ⊢ (ringLMod‘ℝfld) ∈ LVec |
23 | 19, 22 | elini 4120 | . 2 ⊢ (ringLMod‘ℝfld) ∈ (ℂMod ∩ LVec) |
24 | recvs.r | . 2 ⊢ 𝑅 = (ringLMod‘ℝfld) | |
25 | df-cvs 23729 | . 2 ⊢ ℂVec = (ℂMod ∩ LVec) | |
26 | 23, 24, 25 | 3eltr4i 2903 | 1 ⊢ 𝑅 ∈ ℂVec |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 ↾s cress 16476 Scalarcsca 16560 Ringcrg 19290 CRingccrg 19291 DivRingcdr 19495 Fieldcfield 19496 SubRingcsubrg 19524 LModclmod 19627 LVecclvec 19867 ringLModcrglmod 19934 Domncdomn 20046 IDomncidom 20047 ℂfldccnfld 20091 ℝfldcrefld 20293 ℂModcclm 23667 ℂVecccvs 23728 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-subg 18268 df-cmn 18900 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-drng 19497 df-field 19498 df-subrg 19526 df-lmod 19629 df-lvec 19868 df-sra 19937 df-rgmod 19938 df-nzr 20024 df-rlreg 20049 df-domn 20050 df-idom 20051 df-cnfld 20092 df-refld 20294 df-clm 23668 df-cvs 23729 |
This theorem is referenced by: (None) |
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