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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 21478 | . . . . 5 ⊢ ℤring ∈ Ring | |
2 | rlmlmod 21228 | . . . . 5 ⊢ (ℤring ∈ Ring → (ringLMod‘ℤring) ∈ LMod) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (ringLMod‘ℤring) ∈ LMod |
4 | rlmsca 21223 | . . . . . 6 ⊢ (ℤring ∈ Ring → ℤring = (Scalar‘(ringLMod‘ℤring))) | |
5 | 1, 4 | ax-mp 5 | . . . . 5 ⊢ ℤring = (Scalar‘(ringLMod‘ℤring)) |
6 | df-zring 21476 | . . . . 5 ⊢ ℤring = (ℂfld ↾s ℤ) | |
7 | 5, 6 | eqtr3i 2765 | . . . 4 ⊢ (Scalar‘(ringLMod‘ℤring)) = (ℂfld ↾s ℤ) |
8 | zsubrg 21456 | . . . 4 ⊢ ℤ ∈ (SubRing‘ℂfld) | |
9 | eqid 2735 | . . . . 5 ⊢ (Scalar‘(ringLMod‘ℤring)) = (Scalar‘(ringLMod‘ℤring)) | |
10 | 9 | isclmi 25124 | . . . 4 ⊢ (((ringLMod‘ℤring) ∈ LMod ∧ (Scalar‘(ringLMod‘ℤring)) = (ℂfld ↾s ℤ) ∧ ℤ ∈ (SubRing‘ℂfld)) → (ringLMod‘ℤring) ∈ ℂMod) |
11 | 3, 7, 8, 10 | mp3an 1460 | . . 3 ⊢ (ringLMod‘ℤring) ∈ ℂMod |
12 | zclmncvs.z | . . . 4 ⊢ 𝑍 = (ringLMod‘ℤring) | |
13 | 12 | eleq1i 2830 | . . 3 ⊢ (𝑍 ∈ ℂMod ↔ (ringLMod‘ℤring) ∈ ℂMod) |
14 | 11, 13 | mpbir 231 | . 2 ⊢ 𝑍 ∈ ℂMod |
15 | zringndrg 21497 | . . . . . . . 8 ⊢ ℤring ∉ DivRing | |
16 | 15 | neli 3046 | . . . . . . 7 ⊢ ¬ ℤring ∈ DivRing |
17 | 4 | eqcomd 2741 | . . . . . . . . 9 ⊢ (ℤring ∈ Ring → (Scalar‘(ringLMod‘ℤring)) = ℤring) |
18 | 1, 17 | ax-mp 5 | . . . . . . . 8 ⊢ (Scalar‘(ringLMod‘ℤring)) = ℤring |
19 | 18 | eleq1i 2830 | . . . . . . 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 21121 | . . . . 5 ⊢ ((ringLMod‘ℤring) ∈ LVec ↔ ((ringLMod‘ℤring) ∈ LMod ∧ (Scalar‘(ringLMod‘ℤring)) ∈ DivRing)) |
23 | 21, 22 | mtbir 323 | . . . 4 ⊢ ¬ (ringLMod‘ℤring) ∈ LVec |
24 | 23 | olci 866 | . . 3 ⊢ (¬ (ringLMod‘ℤring) ∈ ℂMod ∨ ¬ (ringLMod‘ℤring) ∈ LVec) |
25 | df-nel 3045 | . . . 4 ⊢ (𝑍 ∉ ℂVec ↔ ¬ 𝑍 ∈ ℂVec) | |
26 | ianor 983 | . . . . . 6 ⊢ (¬ ((ringLMod‘ℤring) ∈ ℂMod ∧ (ringLMod‘ℤring) ∈ LVec) ↔ (¬ (ringLMod‘ℤring) ∈ ℂMod ∨ ¬ (ringLMod‘ℤring) ∈ LVec)) | |
27 | elin 3979 | . . . . . 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 25171 | . . . . . 6 ⊢ ℂVec = (ℂMod ∩ LVec) | |
30 | 12, 29 | eleq12i 2832 | . . . . 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 847 = wceq 1537 ∈ wcel 2106 ∉ wnel 3044 ∩ cin 3962 ‘cfv 6563 (class class class)co 7431 ℤcz 12611 ↾s cress 17274 Scalarcsca 17301 Ringcrg 20251 SubRingcsubrg 20586 DivRingcdr 20746 LModclmod 20875 LVecclvec 21119 ringLModcrglmod 21189 ℂfldccnfld 21382 ℤringczring 21475 ℂModcclm 25109 ℂVecccvs 25170 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-fz 13545 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-gz 16964 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-subg 19154 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-subrng 20563 df-subrg 20587 df-drng 20748 df-lmod 20877 df-lvec 21120 df-sra 21190 df-rgmod 21191 df-cnfld 21383 df-zring 21476 df-clm 25110 df-cvs 25171 |
This theorem is referenced by: (None) |
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