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Mirrors > Home > MPE Home > Th. List > qcvs | Structured version Visualization version GIF version |
Description: The field of rational 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 |
---|---|
qcvs.q | ⊢ 𝑄 = (ringLMod‘(ℂfld ↾s ℚ)) |
Ref | Expression |
---|---|
qcvs | ⊢ 𝑄 ∈ ℂVec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qsubdrg 20591 | . . . . . 6 ⊢ (ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing) | |
2 | drngring 19503 | . . . . . . 7 ⊢ ((ℂfld ↾s ℚ) ∈ DivRing → (ℂfld ↾s ℚ) ∈ Ring) | |
3 | 2 | adantl 484 | . . . . . 6 ⊢ ((ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing) → (ℂfld ↾s ℚ) ∈ Ring) |
4 | 1, 3 | ax-mp 5 | . . . . 5 ⊢ (ℂfld ↾s ℚ) ∈ Ring |
5 | rlmlmod 19971 | . . . . 5 ⊢ ((ℂfld ↾s ℚ) ∈ Ring → (ringLMod‘(ℂfld ↾s ℚ)) ∈ LMod) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ (ringLMod‘(ℂfld ↾s ℚ)) ∈ LMod |
7 | 1 | simpri 488 | . . . . 5 ⊢ (ℂfld ↾s ℚ) ∈ DivRing |
8 | rlmsca 19966 | . . . . . 6 ⊢ ((ℂfld ↾s ℚ) ∈ DivRing → (ℂfld ↾s ℚ) = (Scalar‘(ringLMod‘(ℂfld ↾s ℚ)))) | |
9 | 8 | eqcomd 2827 | . . . . 5 ⊢ ((ℂfld ↾s ℚ) ∈ DivRing → (Scalar‘(ringLMod‘(ℂfld ↾s ℚ))) = (ℂfld ↾s ℚ)) |
10 | 7, 9 | ax-mp 5 | . . . 4 ⊢ (Scalar‘(ringLMod‘(ℂfld ↾s ℚ))) = (ℂfld ↾s ℚ) |
11 | 1 | simpli 486 | . . . 4 ⊢ ℚ ∈ (SubRing‘ℂfld) |
12 | eqid 2821 | . . . . 5 ⊢ (Scalar‘(ringLMod‘(ℂfld ↾s ℚ))) = (Scalar‘(ringLMod‘(ℂfld ↾s ℚ))) | |
13 | 12 | isclmi 23675 | . . . 4 ⊢ (((ringLMod‘(ℂfld ↾s ℚ)) ∈ LMod ∧ (Scalar‘(ringLMod‘(ℂfld ↾s ℚ))) = (ℂfld ↾s ℚ) ∧ ℚ ∈ (SubRing‘ℂfld)) → (ringLMod‘(ℂfld ↾s ℚ)) ∈ ℂMod) |
14 | 6, 10, 11, 13 | mp3an 1457 | . . 3 ⊢ (ringLMod‘(ℂfld ↾s ℚ)) ∈ ℂMod |
15 | rlmlvec 19972 | . . . 4 ⊢ ((ℂfld ↾s ℚ) ∈ DivRing → (ringLMod‘(ℂfld ↾s ℚ)) ∈ LVec) | |
16 | 7, 15 | ax-mp 5 | . . 3 ⊢ (ringLMod‘(ℂfld ↾s ℚ)) ∈ LVec |
17 | 14, 16 | elini 4169 | . 2 ⊢ (ringLMod‘(ℂfld ↾s ℚ)) ∈ (ℂMod ∩ LVec) |
18 | qcvs.q | . 2 ⊢ 𝑄 = (ringLMod‘(ℂfld ↾s ℚ)) | |
19 | df-cvs 23722 | . 2 ⊢ ℂVec = (ℂMod ∩ LVec) | |
20 | 17, 18, 19 | 3eltr4i 2926 | 1 ⊢ 𝑄 ∈ ℂVec |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∩ cin 3934 ‘cfv 6349 (class class class)co 7150 ℚcq 12342 ↾s cress 16478 Scalarcsca 16562 Ringcrg 19291 DivRingcdr 19496 SubRingcsubrg 19525 LModclmod 19628 LVecclvec 19868 ringLModcrglmod 19935 ℂfldccnfld 20539 ℂModcclm 23660 ℂVecccvs 23721 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-subg 18270 df-cmn 18902 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-dvr 19427 df-drng 19498 df-subrg 19527 df-lmod 19630 df-lvec 19869 df-sra 19938 df-rgmod 19939 df-cnfld 20540 df-clm 23661 df-cvs 23722 |
This theorem is referenced by: (None) |
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