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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 21460 | . . . . 5 ⊢ ℤring ∈ Ring | |
| 2 | rlmlmod 21210 | . . . . 5 ⊢ (ℤring ∈ Ring → (ringLMod‘ℤring) ∈ LMod) | |
| 3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (ringLMod‘ℤring) ∈ LMod |
| 4 | rlmsca 21205 | . . . . . 6 ⊢ (ℤring ∈ Ring → ℤring = (Scalar‘(ringLMod‘ℤring))) | |
| 5 | 1, 4 | ax-mp 5 | . . . . 5 ⊢ ℤring = (Scalar‘(ringLMod‘ℤring)) |
| 6 | df-zring 21458 | . . . . 5 ⊢ ℤring = (ℂfld ↾s ℤ) | |
| 7 | 5, 6 | eqtr3i 2767 | . . . 4 ⊢ (Scalar‘(ringLMod‘ℤring)) = (ℂfld ↾s ℤ) |
| 8 | zsubrg 21438 | . . . 4 ⊢ ℤ ∈ (SubRing‘ℂfld) | |
| 9 | eqid 2737 | . . . . 5 ⊢ (Scalar‘(ringLMod‘ℤring)) = (Scalar‘(ringLMod‘ℤring)) | |
| 10 | 9 | isclmi 25110 | . . . 4 ⊢ (((ringLMod‘ℤring) ∈ LMod ∧ (Scalar‘(ringLMod‘ℤring)) = (ℂfld ↾s ℤ) ∧ ℤ ∈ (SubRing‘ℂfld)) → (ringLMod‘ℤring) ∈ ℂMod) |
| 11 | 3, 7, 8, 10 | mp3an 1463 | . . 3 ⊢ (ringLMod‘ℤring) ∈ ℂMod |
| 12 | zclmncvs.z | . . . 4 ⊢ 𝑍 = (ringLMod‘ℤring) | |
| 13 | 12 | eleq1i 2832 | . . 3 ⊢ (𝑍 ∈ ℂMod ↔ (ringLMod‘ℤring) ∈ ℂMod) |
| 14 | 11, 13 | mpbir 231 | . 2 ⊢ 𝑍 ∈ ℂMod |
| 15 | zringndrg 21479 | . . . . . . . 8 ⊢ ℤring ∉ DivRing | |
| 16 | 15 | neli 3048 | . . . . . . 7 ⊢ ¬ ℤring ∈ DivRing |
| 17 | 4 | eqcomd 2743 | . . . . . . . . 9 ⊢ (ℤring ∈ Ring → (Scalar‘(ringLMod‘ℤring)) = ℤring) |
| 18 | 1, 17 | ax-mp 5 | . . . . . . . 8 ⊢ (Scalar‘(ringLMod‘ℤring)) = ℤring |
| 19 | 18 | eleq1i 2832 | . . . . . . 7 ⊢ ((Scalar‘(ringLMod‘ℤring)) ∈ DivRing ↔ ℤring ∈ DivRing) |
| 20 | 16, 19 | mtbir 323 | . . . . . 6 ⊢ ¬ (Scalar‘(ringLMod‘ℤring)) ∈ DivRing |
| 21 | 20 | intnan 486 | . . . . 5 ⊢ ¬ ((ringLMod‘ℤring) ∈ LMod ∧ (Scalar‘(ringLMod‘ℤring)) ∈ DivRing) |
| 22 | 9 | islvec 21103 | . . . . 5 ⊢ ((ringLMod‘ℤring) ∈ LVec ↔ ((ringLMod‘ℤring) ∈ LMod ∧ (Scalar‘(ringLMod‘ℤring)) ∈ DivRing)) |
| 23 | 21, 22 | mtbir 323 | . . . 4 ⊢ ¬ (ringLMod‘ℤring) ∈ LVec |
| 24 | 23 | olci 867 | . . 3 ⊢ (¬ (ringLMod‘ℤring) ∈ ℂMod ∨ ¬ (ringLMod‘ℤring) ∈ LVec) |
| 25 | df-nel 3047 | . . . 4 ⊢ (𝑍 ∉ ℂVec ↔ ¬ 𝑍 ∈ ℂVec) | |
| 26 | ianor 984 | . . . . . 6 ⊢ (¬ ((ringLMod‘ℤring) ∈ ℂMod ∧ (ringLMod‘ℤring) ∈ LVec) ↔ (¬ (ringLMod‘ℤring) ∈ ℂMod ∨ ¬ (ringLMod‘ℤring) ∈ LVec)) | |
| 27 | elin 3967 | . . . . . 6 ⊢ ((ringLMod‘ℤring) ∈ (ℂMod ∩ LVec) ↔ ((ringLMod‘ℤring) ∈ ℂMod ∧ (ringLMod‘ℤring) ∈ LVec)) | |
| 28 | 26, 27 | xchnxbir 333 | . . . . 5 ⊢ (¬ (ringLMod‘ℤring) ∈ (ℂMod ∩ LVec) ↔ (¬ (ringLMod‘ℤring) ∈ ℂMod ∨ ¬ (ringLMod‘ℤring) ∈ LVec)) |
| 29 | df-cvs 25157 | . . . . . 6 ⊢ ℂVec = (ℂMod ∩ LVec) | |
| 30 | 12, 29 | eleq12i 2834 | . . . . 5 ⊢ (𝑍 ∈ ℂVec ↔ (ringLMod‘ℤring) ∈ (ℂMod ∩ LVec)) |
| 31 | 28, 30 | xchnxbir 333 | . . . 4 ⊢ (¬ 𝑍 ∈ ℂVec ↔ (¬ (ringLMod‘ℤring) ∈ ℂMod ∨ ¬ (ringLMod‘ℤring) ∈ LVec)) |
| 32 | 25, 31 | bitri 275 | . . 3 ⊢ (𝑍 ∉ ℂVec ↔ (¬ (ringLMod‘ℤring) ∈ ℂMod ∨ ¬ (ringLMod‘ℤring) ∈ LVec)) |
| 33 | 24, 32 | mpbir 231 | . 2 ⊢ 𝑍 ∉ ℂVec |
| 34 | 14, 33 | pm3.2i 470 | 1 ⊢ (𝑍 ∈ ℂMod ∧ 𝑍 ∉ ℂVec) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∉ wnel 3046 ∩ cin 3950 ‘cfv 6561 (class class class)co 7431 ℤcz 12613 ↾s cress 17274 Scalarcsca 17300 Ringcrg 20230 SubRingcsubrg 20569 DivRingcdr 20729 LModclmod 20858 LVecclvec 21101 ringLModcrglmod 21171 ℂfldccnfld 21364 ℤringczring 21457 ℂModcclm 25095 ℂVecccvs 25156 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-gz 16968 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-subrng 20546 df-subrg 20570 df-drng 20731 df-lmod 20860 df-lvec 21102 df-sra 21172 df-rgmod 21173 df-cnfld 21365 df-zring 21458 df-clm 25096 df-cvs 25157 |
| This theorem is referenced by: (None) |
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