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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 20166 | . . . . 5 ⊢ ℤring ∈ Ring | |
2 | rlmlmod 19970 | . . . . 5 ⊢ (ℤring ∈ Ring → (ringLMod‘ℤring) ∈ LMod) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (ringLMod‘ℤring) ∈ LMod |
4 | rlmsca 19965 | . . . . . 6 ⊢ (ℤring ∈ Ring → ℤring = (Scalar‘(ringLMod‘ℤring))) | |
5 | 1, 4 | ax-mp 5 | . . . . 5 ⊢ ℤring = (Scalar‘(ringLMod‘ℤring)) |
6 | df-zring 20164 | . . . . 5 ⊢ ℤring = (ℂfld ↾s ℤ) | |
7 | 5, 6 | eqtr3i 2823 | . . . 4 ⊢ (Scalar‘(ringLMod‘ℤring)) = (ℂfld ↾s ℤ) |
8 | zsubrg 20144 | . . . 4 ⊢ ℤ ∈ (SubRing‘ℂfld) | |
9 | eqid 2798 | . . . . 5 ⊢ (Scalar‘(ringLMod‘ℤring)) = (Scalar‘(ringLMod‘ℤring)) | |
10 | 9 | isclmi 23682 | . . . 4 ⊢ (((ringLMod‘ℤring) ∈ LMod ∧ (Scalar‘(ringLMod‘ℤring)) = (ℂfld ↾s ℤ) ∧ ℤ ∈ (SubRing‘ℂfld)) → (ringLMod‘ℤring) ∈ ℂMod) |
11 | 3, 7, 8, 10 | mp3an 1458 | . . 3 ⊢ (ringLMod‘ℤring) ∈ ℂMod |
12 | zclmncvs.z | . . . 4 ⊢ 𝑍 = (ringLMod‘ℤring) | |
13 | 12 | eleq1i 2880 | . . 3 ⊢ (𝑍 ∈ ℂMod ↔ (ringLMod‘ℤring) ∈ ℂMod) |
14 | 11, 13 | mpbir 234 | . 2 ⊢ 𝑍 ∈ ℂMod |
15 | zringndrg 20183 | . . . . . . . 8 ⊢ ℤring ∉ DivRing | |
16 | 15 | neli 3093 | . . . . . . 7 ⊢ ¬ ℤring ∈ DivRing |
17 | 4 | eqcomd 2804 | . . . . . . . . 9 ⊢ (ℤring ∈ Ring → (Scalar‘(ringLMod‘ℤring)) = ℤring) |
18 | 1, 17 | ax-mp 5 | . . . . . . . 8 ⊢ (Scalar‘(ringLMod‘ℤring)) = ℤring |
19 | 18 | eleq1i 2880 | . . . . . . 7 ⊢ ((Scalar‘(ringLMod‘ℤring)) ∈ DivRing ↔ ℤring ∈ DivRing) |
20 | 16, 19 | mtbir 326 | . . . . . 6 ⊢ ¬ (Scalar‘(ringLMod‘ℤring)) ∈ DivRing |
21 | 20 | intnan 490 | . . . . 5 ⊢ ¬ ((ringLMod‘ℤring) ∈ LMod ∧ (Scalar‘(ringLMod‘ℤring)) ∈ DivRing) |
22 | 9 | islvec 19869 | . . . . 5 ⊢ ((ringLMod‘ℤring) ∈ LVec ↔ ((ringLMod‘ℤring) ∈ LMod ∧ (Scalar‘(ringLMod‘ℤring)) ∈ DivRing)) |
23 | 21, 22 | mtbir 326 | . . . 4 ⊢ ¬ (ringLMod‘ℤring) ∈ LVec |
24 | 23 | olci 863 | . . 3 ⊢ (¬ (ringLMod‘ℤring) ∈ ℂMod ∨ ¬ (ringLMod‘ℤring) ∈ LVec) |
25 | df-nel 3092 | . . . 4 ⊢ (𝑍 ∉ ℂVec ↔ ¬ 𝑍 ∈ ℂVec) | |
26 | ianor 979 | . . . . . 6 ⊢ (¬ ((ringLMod‘ℤring) ∈ ℂMod ∧ (ringLMod‘ℤring) ∈ LVec) ↔ (¬ (ringLMod‘ℤring) ∈ ℂMod ∨ ¬ (ringLMod‘ℤring) ∈ LVec)) | |
27 | elin 3897 | . . . . . 6 ⊢ ((ringLMod‘ℤring) ∈ (ℂMod ∩ LVec) ↔ ((ringLMod‘ℤring) ∈ ℂMod ∧ (ringLMod‘ℤring) ∈ LVec)) | |
28 | 26, 27 | xchnxbir 336 | . . . . 5 ⊢ (¬ (ringLMod‘ℤring) ∈ (ℂMod ∩ LVec) ↔ (¬ (ringLMod‘ℤring) ∈ ℂMod ∨ ¬ (ringLMod‘ℤring) ∈ LVec)) |
29 | df-cvs 23729 | . . . . . 6 ⊢ ℂVec = (ℂMod ∩ LVec) | |
30 | 12, 29 | eleq12i 2882 | . . . . 5 ⊢ (𝑍 ∈ ℂVec ↔ (ringLMod‘ℤring) ∈ (ℂMod ∩ LVec)) |
31 | 28, 30 | xchnxbir 336 | . . . 4 ⊢ (¬ 𝑍 ∈ ℂVec ↔ (¬ (ringLMod‘ℤring) ∈ ℂMod ∨ ¬ (ringLMod‘ℤring) ∈ LVec)) |
32 | 25, 31 | bitri 278 | . . 3 ⊢ (𝑍 ∉ ℂVec ↔ (¬ (ringLMod‘ℤring) ∈ ℂMod ∨ ¬ (ringLMod‘ℤring) ∈ LVec)) |
33 | 24, 32 | mpbir 234 | . 2 ⊢ 𝑍 ∉ ℂVec |
34 | 14, 33 | pm3.2i 474 | 1 ⊢ (𝑍 ∈ ℂMod ∧ 𝑍 ∉ ℂVec) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ∉ wnel 3091 ∩ cin 3880 ‘cfv 6324 (class class class)co 7135 ℤcz 11969 ↾s cress 16476 Scalarcsca 16560 Ringcrg 19290 DivRingcdr 19495 SubRingcsubrg 19524 LModclmod 19627 LVecclvec 19867 ringLModcrglmod 19934 ℂfldccnfld 20091 ℤringzring 20163 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-pre-sup 10604 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-sup 8890 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-rp 12378 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-gz 16256 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-subrg 19526 df-lmod 19629 df-lvec 19868 df-sra 19937 df-rgmod 19938 df-cnfld 20092 df-zring 20164 df-clm 23668 df-cvs 23729 |
This theorem is referenced by: (None) |
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