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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 21276 | . . . . . 6 ⊢ (ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing) | |
2 | drngring 20579 | . . . . . . 7 ⊢ ((ℂfld ↾s ℚ) ∈ DivRing → (ℂfld ↾s ℚ) ∈ Ring) | |
3 | 2 | adantl 481 | . . . . . 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 21044 | . . . . 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 485 | . . . . 5 ⊢ (ℂfld ↾s ℚ) ∈ DivRing |
8 | rlmsca 21039 | . . . . . 6 ⊢ ((ℂfld ↾s ℚ) ∈ DivRing → (ℂfld ↾s ℚ) = (Scalar‘(ringLMod‘(ℂfld ↾s ℚ)))) | |
9 | 8 | eqcomd 2730 | . . . . 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 483 | . . . 4 ⊢ ℚ ∈ (SubRing‘ℂfld) |
12 | eqid 2724 | . . . . 5 ⊢ (Scalar‘(ringLMod‘(ℂfld ↾s ℚ))) = (Scalar‘(ringLMod‘(ℂfld ↾s ℚ))) | |
13 | 12 | isclmi 24914 | . . . 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 21045 | . . . 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 4185 | . 2 ⊢ (ringLMod‘(ℂfld ↾s ℚ)) ∈ (ℂMod ∩ LVec) |
18 | qcvs.q | . 2 ⊢ 𝑄 = (ringLMod‘(ℂfld ↾s ℚ)) | |
19 | df-cvs 24961 | . 2 ⊢ ℂVec = (ℂMod ∩ LVec) | |
20 | 17, 18, 19 | 3eltr4i 2838 | 1 ⊢ 𝑄 ∈ ℂVec |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∩ cin 3939 ‘cfv 6533 (class class class)co 7401 ℚcq 12928 ↾s cress 17169 Scalarcsca 17196 Ringcrg 20123 SubRingcsubrg 20454 DivRingcdr 20572 LModclmod 20691 LVecclvec 20935 ringLModcrglmod 21005 ℂfldccnfld 21223 ℂModcclm 24899 ℂVecccvs 24960 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-addf 11184 ax-mulf 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-0g 17383 df-mgm 18560 df-sgrp 18639 df-mnd 18655 df-grp 18853 df-minusg 18854 df-subg 19035 df-cmn 19687 df-abl 19688 df-mgp 20025 df-rng 20043 df-ur 20072 df-ring 20125 df-cring 20126 df-oppr 20221 df-dvdsr 20244 df-unit 20245 df-invr 20275 df-dvr 20288 df-subrng 20431 df-subrg 20456 df-drng 20574 df-lmod 20693 df-lvec 20936 df-sra 21006 df-rgmod 21007 df-cnfld 21224 df-clm 24900 df-cvs 24961 |
This theorem is referenced by: (None) |
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