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Mirrors > Home > MPE Home > Th. List > zclmncvs | Structured version Visualization version GIF version |
Description: The ring of integers as left module over itself is a subcomplex module, but not a subcomplex vector space. The vector operation is +, and the scalar product is ·. (Contributed by AV, 22-Oct-2021.) |
Ref | Expression |
---|---|
zclmncvs.z | ⊢ 𝑍 = (ringLMod‘ℤring) |
Ref | Expression |
---|---|
zclmncvs | ⊢ (𝑍 ∈ ℂMod ∧ 𝑍 ∉ ℂVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zringring 20548 | . . . . 5 ⊢ ℤring ∈ Ring | |
2 | rlmlmod 19906 | . . . . 5 ⊢ (ℤring ∈ Ring → (ringLMod‘ℤring) ∈ LMod) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (ringLMod‘ℤring) ∈ LMod |
4 | rlmsca 19901 | . . . . . 6 ⊢ (ℤring ∈ Ring → ℤring = (Scalar‘(ringLMod‘ℤring))) | |
5 | 1, 4 | ax-mp 5 | . . . . 5 ⊢ ℤring = (Scalar‘(ringLMod‘ℤring)) |
6 | df-zring 20546 | . . . . 5 ⊢ ℤring = (ℂfld ↾s ℤ) | |
7 | 5, 6 | eqtr3i 2843 | . . . 4 ⊢ (Scalar‘(ringLMod‘ℤring)) = (ℂfld ↾s ℤ) |
8 | zsubrg 20526 | . . . 4 ⊢ ℤ ∈ (SubRing‘ℂfld) | |
9 | eqid 2818 | . . . . 5 ⊢ (Scalar‘(ringLMod‘ℤring)) = (Scalar‘(ringLMod‘ℤring)) | |
10 | 9 | isclmi 23608 | . . . 4 ⊢ (((ringLMod‘ℤring) ∈ LMod ∧ (Scalar‘(ringLMod‘ℤring)) = (ℂfld ↾s ℤ) ∧ ℤ ∈ (SubRing‘ℂfld)) → (ringLMod‘ℤring) ∈ ℂMod) |
11 | 3, 7, 8, 10 | mp3an 1452 | . . 3 ⊢ (ringLMod‘ℤring) ∈ ℂMod |
12 | zclmncvs.z | . . . 4 ⊢ 𝑍 = (ringLMod‘ℤring) | |
13 | 12 | eleq1i 2900 | . . 3 ⊢ (𝑍 ∈ ℂMod ↔ (ringLMod‘ℤring) ∈ ℂMod) |
14 | 11, 13 | mpbir 232 | . 2 ⊢ 𝑍 ∈ ℂMod |
15 | zringndrg 20565 | . . . . . . . 8 ⊢ ℤring ∉ DivRing | |
16 | 15 | neli 3122 | . . . . . . 7 ⊢ ¬ ℤring ∈ DivRing |
17 | 4 | eqcomd 2824 | . . . . . . . . 9 ⊢ (ℤring ∈ Ring → (Scalar‘(ringLMod‘ℤring)) = ℤring) |
18 | 1, 17 | ax-mp 5 | . . . . . . . 8 ⊢ (Scalar‘(ringLMod‘ℤring)) = ℤring |
19 | 18 | eleq1i 2900 | . . . . . . 7 ⊢ ((Scalar‘(ringLMod‘ℤring)) ∈ DivRing ↔ ℤring ∈ DivRing) |
20 | 16, 19 | mtbir 324 | . . . . . 6 ⊢ ¬ (Scalar‘(ringLMod‘ℤring)) ∈ DivRing |
21 | 20 | intnan 487 | . . . . 5 ⊢ ¬ ((ringLMod‘ℤring) ∈ LMod ∧ (Scalar‘(ringLMod‘ℤring)) ∈ DivRing) |
22 | 9 | islvec 19805 | . . . . 5 ⊢ ((ringLMod‘ℤring) ∈ LVec ↔ ((ringLMod‘ℤring) ∈ LMod ∧ (Scalar‘(ringLMod‘ℤring)) ∈ DivRing)) |
23 | 21, 22 | mtbir 324 | . . . 4 ⊢ ¬ (ringLMod‘ℤring) ∈ LVec |
24 | 23 | olci 860 | . . 3 ⊢ (¬ (ringLMod‘ℤring) ∈ ℂMod ∨ ¬ (ringLMod‘ℤring) ∈ LVec) |
25 | df-nel 3121 | . . . 4 ⊢ (𝑍 ∉ ℂVec ↔ ¬ 𝑍 ∈ ℂVec) | |
26 | ianor 975 | . . . . . 6 ⊢ (¬ ((ringLMod‘ℤring) ∈ ℂMod ∧ (ringLMod‘ℤring) ∈ LVec) ↔ (¬ (ringLMod‘ℤring) ∈ ℂMod ∨ ¬ (ringLMod‘ℤring) ∈ LVec)) | |
27 | elin 4166 | . . . . . 6 ⊢ ((ringLMod‘ℤring) ∈ (ℂMod ∩ LVec) ↔ ((ringLMod‘ℤring) ∈ ℂMod ∧ (ringLMod‘ℤring) ∈ LVec)) | |
28 | 26, 27 | xchnxbir 334 | . . . . 5 ⊢ (¬ (ringLMod‘ℤring) ∈ (ℂMod ∩ LVec) ↔ (¬ (ringLMod‘ℤring) ∈ ℂMod ∨ ¬ (ringLMod‘ℤring) ∈ LVec)) |
29 | df-cvs 23655 | . . . . . 6 ⊢ ℂVec = (ℂMod ∩ LVec) | |
30 | 12, 29 | eleq12i 2902 | . . . . 5 ⊢ (𝑍 ∈ ℂVec ↔ (ringLMod‘ℤring) ∈ (ℂMod ∩ LVec)) |
31 | 28, 30 | xchnxbir 334 | . . . 4 ⊢ (¬ 𝑍 ∈ ℂVec ↔ (¬ (ringLMod‘ℤring) ∈ ℂMod ∨ ¬ (ringLMod‘ℤring) ∈ LVec)) |
32 | 25, 31 | bitri 276 | . . 3 ⊢ (𝑍 ∉ ℂVec ↔ (¬ (ringLMod‘ℤring) ∈ ℂMod ∨ ¬ (ringLMod‘ℤring) ∈ LVec)) |
33 | 24, 32 | mpbir 232 | . 2 ⊢ 𝑍 ∉ ℂVec |
34 | 14, 33 | pm3.2i 471 | 1 ⊢ (𝑍 ∈ ℂMod ∧ 𝑍 ∉ ℂVec) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 ∉ wnel 3120 ∩ cin 3932 ‘cfv 6348 (class class class)co 7145 ℤcz 11969 ↾s cress 16472 Scalarcsca 16556 Ringcrg 19226 DivRingcdr 19431 SubRingcsubrg 19460 LModclmod 19563 LVecclvec 19803 ringLModcrglmod 19870 ℂfldccnfld 20473 ℤringzring 20545 ℂModcclm 23593 ℂVecccvs 23654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 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-rp 12378 df-fz 12881 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-gz 16254 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-subg 18214 df-cmn 18837 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-subrg 19462 df-lmod 19565 df-lvec 19804 df-sra 19873 df-rgmod 19874 df-cnfld 20474 df-zring 20546 df-clm 23594 df-cvs 23655 |
This theorem is referenced by: (None) |
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