Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > lmod1zrnlvec | Structured version Visualization version GIF version |
Description: There is a (left) module (a zero module) which is not a (left) vector space. (Contributed by AV, 29-Apr-2019.) |
Ref | Expression |
---|---|
lmod1zr.r | ⊢ 𝑅 = {〈(Base‘ndx), {𝑍}〉, 〈(+g‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉, 〈(.r‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉} |
lmod1zr.m | ⊢ 𝑀 = ({〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉, 〈(Scalar‘ndx), 𝑅〉} ∪ {〈( ·𝑠 ‘ndx), {〈〈𝑍, 𝐼〉, 𝐼〉}〉}) |
Ref | Expression |
---|---|
lmod1zrnlvec | ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑀 ∉ LVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmod1zr.r | . . . . . 6 ⊢ 𝑅 = {〈(Base‘ndx), {𝑍}〉, 〈(+g‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉, 〈(.r‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉} | |
2 | tpex 6855 | . . . . . 6 ⊢ {〈(Base‘ndx), {𝑍}〉, 〈(+g‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉, 〈(.r‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉} ∈ V | |
3 | 1, 2 | eqeltri 2684 | . . . . 5 ⊢ 𝑅 ∈ V |
4 | lmod1zr.m | . . . . . 6 ⊢ 𝑀 = ({〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉, 〈(Scalar‘ndx), 𝑅〉} ∪ {〈( ·𝑠 ‘ndx), {〈〈𝑍, 𝐼〉, 𝐼〉}〉}) | |
5 | 4 | lmodsca 15843 | . . . . 5 ⊢ (𝑅 ∈ V → 𝑅 = (Scalar‘𝑀)) |
6 | 3, 5 | mp1i 13 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑅 = (Scalar‘𝑀)) |
7 | 1 | rng1nnzr 19095 | . . . . . . 7 ⊢ (𝑍 ∈ 𝑊 → 𝑅 ∉ NzRing) |
8 | df-nel 2783 | . . . . . . 7 ⊢ (𝑅 ∉ NzRing ↔ ¬ 𝑅 ∈ NzRing) | |
9 | 7, 8 | sylib 207 | . . . . . 6 ⊢ (𝑍 ∈ 𝑊 → ¬ 𝑅 ∈ NzRing) |
10 | drngnzr 19083 | . . . . . 6 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ NzRing) | |
11 | 9, 10 | nsyl 134 | . . . . 5 ⊢ (𝑍 ∈ 𝑊 → ¬ 𝑅 ∈ DivRing) |
12 | 11 | adantl 481 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ¬ 𝑅 ∈ DivRing) |
13 | 6, 12 | eqneltrrd 2708 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ¬ (Scalar‘𝑀) ∈ DivRing) |
14 | 13 | intnand 953 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ¬ (𝑀 ∈ LMod ∧ (Scalar‘𝑀) ∈ DivRing)) |
15 | df-nel 2783 | . . 3 ⊢ (𝑀 ∉ LVec ↔ ¬ 𝑀 ∈ LVec) | |
16 | eqid 2610 | . . . 4 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀) | |
17 | 16 | islvec 18925 | . . 3 ⊢ (𝑀 ∈ LVec ↔ (𝑀 ∈ LMod ∧ (Scalar‘𝑀) ∈ DivRing)) |
18 | 15, 17 | xchbinx 323 | . 2 ⊢ (𝑀 ∉ LVec ↔ ¬ (𝑀 ∈ LMod ∧ (Scalar‘𝑀) ∈ DivRing)) |
19 | 14, 18 | sylibr 223 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑀 ∉ LVec) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∉ wnel 2781 Vcvv 3173 ∪ cun 3538 {csn 4125 {ctp 4129 〈cop 4131 ‘cfv 5804 ndxcnx 15692 Basecbs 15695 +gcplusg 15768 .rcmulr 15769 Scalarcsca 15771 ·𝑠 cvsca 15772 DivRingcdr 18570 LModclmod 18686 LVecclvec 18923 NzRingcnzr 19078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-drng 18572 df-lvec 18924 df-nzr 19079 |
This theorem is referenced by: lvecpsslmod 42090 |
Copyright terms: Public domain | W3C validator |