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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 20765 | . . . . . 6 ⊢ ℝfld ∈ Field | |
2 | fldidom 20080 | . . . . . . 7 ⊢ (ℝfld ∈ Field → ℝfld ∈ IDomn) | |
3 | isidom 20079 | . . . . . . . 8 ⊢ (ℝfld ∈ IDomn ↔ (ℝfld ∈ CRing ∧ ℝfld ∈ Domn)) | |
4 | crngring 19310 | . . . . . . . . 9 ⊢ (ℝfld ∈ CRing → ℝfld ∈ Ring) | |
5 | 4 | adantr 483 | . . . . . . . 8 ⊢ ((ℝfld ∈ CRing ∧ ℝfld ∈ Domn) → ℝfld ∈ Ring) |
6 | 3, 5 | sylbi 219 | . . . . . . 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 19979 | . . . . 5 ⊢ (ℝfld ∈ Ring → (ringLMod‘ℝfld) ∈ LMod) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ (ringLMod‘ℝfld) ∈ LMod |
11 | rlmsca 19974 | . . . . . 6 ⊢ (ℝfld ∈ Field → ℝfld = (Scalar‘(ringLMod‘ℝfld))) | |
12 | 1, 11 | ax-mp 5 | . . . . 5 ⊢ ℝfld = (Scalar‘(ringLMod‘ℝfld)) |
13 | df-refld 20751 | . . . . 5 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
14 | 12, 13 | eqtr3i 2848 | . . . 4 ⊢ (Scalar‘(ringLMod‘ℝfld)) = (ℂfld ↾s ℝ) |
15 | resubdrg 20754 | . . . . 5 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
16 | 15 | simpli 486 | . . . 4 ⊢ ℝ ∈ (SubRing‘ℂfld) |
17 | eqid 2823 | . . . . 5 ⊢ (Scalar‘(ringLMod‘ℝfld)) = (Scalar‘(ringLMod‘ℝfld)) | |
18 | 17 | isclmi 23683 | . . . 4 ⊢ (((ringLMod‘ℝfld) ∈ LMod ∧ (Scalar‘(ringLMod‘ℝfld)) = (ℂfld ↾s ℝ) ∧ ℝ ∈ (SubRing‘ℂfld)) → (ringLMod‘ℝfld) ∈ ℂMod) |
19 | 10, 14, 16, 18 | mp3an 1457 | . . 3 ⊢ (ringLMod‘ℝfld) ∈ ℂMod |
20 | 15 | simpri 488 | . . . 4 ⊢ ℝfld ∈ DivRing |
21 | rlmlvec 19980 | . . . 4 ⊢ (ℝfld ∈ DivRing → (ringLMod‘ℝfld) ∈ LVec) | |
22 | 20, 21 | ax-mp 5 | . . 3 ⊢ (ringLMod‘ℝfld) ∈ LVec |
23 | 19, 22 | elini 4172 | . 2 ⊢ (ringLMod‘ℝfld) ∈ (ℂMod ∩ LVec) |
24 | recvs.r | . 2 ⊢ 𝑅 = (ringLMod‘ℝfld) | |
25 | df-cvs 23730 | . 2 ⊢ ℂVec = (ℂMod ∩ LVec) | |
26 | 23, 24, 25 | 3eltr4i 2928 | 1 ⊢ 𝑅 ∈ ℂVec |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∩ cin 3937 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 ↾s cress 16486 Scalarcsca 16570 Ringcrg 19299 CRingccrg 19300 DivRingcdr 19504 Fieldcfield 19505 SubRingcsubrg 19533 LModclmod 19636 LVecclvec 19876 ringLModcrglmod 19943 Domncdomn 20055 IDomncidom 20056 ℂfldccnfld 20547 ℝfldcrefld 20750 ℂModcclm 23668 ℂVecccvs 23729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-subg 18278 df-cmn 18910 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-drng 19506 df-field 19507 df-subrg 19535 df-lmod 19638 df-lvec 19877 df-sra 19946 df-rgmod 19947 df-nzr 20033 df-rlreg 20058 df-domn 20059 df-idom 20060 df-cnfld 20548 df-refld 20751 df-clm 23669 df-cvs 23730 |
This theorem is referenced by: (None) |
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