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Mirrors > Home > MPE Home > Th. List > Mathboxes > lvecpsslmod | Structured version Visualization version GIF version |
Description: The class of all (left) vector spaces is a proper subclass of the class of all (left) modules. Although it is obvious (and proven by lveclmod 19807) that every left vector space is a left module, there is (at least) one left module which is no left vector space, for example the zero module over the zero ring, see lmod1zrnlvec 44477. (Contributed by AV, 29-Apr-2019.) |
Ref | Expression |
---|---|
lvecpsslmod | ⊢ LVec ⊊ LMod |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lveclmod 19807 | . . 3 ⊢ (𝑣 ∈ LVec → 𝑣 ∈ LMod) | |
2 | 1 | ssriv 3968 | . 2 ⊢ LVec ⊆ LMod |
3 | vex 3495 | . . . 4 ⊢ 𝑖 ∈ V | |
4 | vex 3495 | . . . 4 ⊢ 𝑧 ∈ V | |
5 | 3, 4 | pm3.2i 471 | . . 3 ⊢ (𝑖 ∈ V ∧ 𝑧 ∈ V) |
6 | eqid 2818 | . . . . 5 ⊢ {〈(Base‘ndx), {𝑧}〉, 〈(+g‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉, 〈(.r‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉} = {〈(Base‘ndx), {𝑧}〉, 〈(+g‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉, 〈(.r‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉} | |
7 | eqid 2818 | . . . . 5 ⊢ ({〈(Base‘ndx), {𝑖}〉, 〈(+g‘ndx), {〈〈𝑖, 𝑖〉, 𝑖〉}〉, 〈(Scalar‘ndx), {〈(Base‘ndx), {𝑧}〉, 〈(+g‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉, 〈(.r‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉}〉} ∪ {〈( ·𝑠 ‘ndx), {〈〈𝑧, 𝑖〉, 𝑖〉}〉}) = ({〈(Base‘ndx), {𝑖}〉, 〈(+g‘ndx), {〈〈𝑖, 𝑖〉, 𝑖〉}〉, 〈(Scalar‘ndx), {〈(Base‘ndx), {𝑧}〉, 〈(+g‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉, 〈(.r‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉}〉} ∪ {〈( ·𝑠 ‘ndx), {〈〈𝑧, 𝑖〉, 𝑖〉}〉}) | |
8 | 6, 7 | lmod1zr 44476 | . . . 4 ⊢ ((𝑖 ∈ V ∧ 𝑧 ∈ V) → ({〈(Base‘ndx), {𝑖}〉, 〈(+g‘ndx), {〈〈𝑖, 𝑖〉, 𝑖〉}〉, 〈(Scalar‘ndx), {〈(Base‘ndx), {𝑧}〉, 〈(+g‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉, 〈(.r‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉}〉} ∪ {〈( ·𝑠 ‘ndx), {〈〈𝑧, 𝑖〉, 𝑖〉}〉}) ∈ LMod) |
9 | 6, 7 | lmod1zrnlvec 44477 | . . . . 5 ⊢ ((𝑖 ∈ V ∧ 𝑧 ∈ V) → ({〈(Base‘ndx), {𝑖}〉, 〈(+g‘ndx), {〈〈𝑖, 𝑖〉, 𝑖〉}〉, 〈(Scalar‘ndx), {〈(Base‘ndx), {𝑧}〉, 〈(+g‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉, 〈(.r‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉}〉} ∪ {〈( ·𝑠 ‘ndx), {〈〈𝑧, 𝑖〉, 𝑖〉}〉}) ∉ LVec) |
10 | df-nel 3121 | . . . . 5 ⊢ (({〈(Base‘ndx), {𝑖}〉, 〈(+g‘ndx), {〈〈𝑖, 𝑖〉, 𝑖〉}〉, 〈(Scalar‘ndx), {〈(Base‘ndx), {𝑧}〉, 〈(+g‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉, 〈(.r‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉}〉} ∪ {〈( ·𝑠 ‘ndx), {〈〈𝑧, 𝑖〉, 𝑖〉}〉}) ∉ LVec ↔ ¬ ({〈(Base‘ndx), {𝑖}〉, 〈(+g‘ndx), {〈〈𝑖, 𝑖〉, 𝑖〉}〉, 〈(Scalar‘ndx), {〈(Base‘ndx), {𝑧}〉, 〈(+g‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉, 〈(.r‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉}〉} ∪ {〈( ·𝑠 ‘ndx), {〈〈𝑧, 𝑖〉, 𝑖〉}〉}) ∈ LVec) | |
11 | 9, 10 | sylib 219 | . . . 4 ⊢ ((𝑖 ∈ V ∧ 𝑧 ∈ V) → ¬ ({〈(Base‘ndx), {𝑖}〉, 〈(+g‘ndx), {〈〈𝑖, 𝑖〉, 𝑖〉}〉, 〈(Scalar‘ndx), {〈(Base‘ndx), {𝑧}〉, 〈(+g‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉, 〈(.r‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉}〉} ∪ {〈( ·𝑠 ‘ndx), {〈〈𝑧, 𝑖〉, 𝑖〉}〉}) ∈ LVec) |
12 | 8, 11 | jca 512 | . . 3 ⊢ ((𝑖 ∈ V ∧ 𝑧 ∈ V) → (({〈(Base‘ndx), {𝑖}〉, 〈(+g‘ndx), {〈〈𝑖, 𝑖〉, 𝑖〉}〉, 〈(Scalar‘ndx), {〈(Base‘ndx), {𝑧}〉, 〈(+g‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉, 〈(.r‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉}〉} ∪ {〈( ·𝑠 ‘ndx), {〈〈𝑧, 𝑖〉, 𝑖〉}〉}) ∈ LMod ∧ ¬ ({〈(Base‘ndx), {𝑖}〉, 〈(+g‘ndx), {〈〈𝑖, 𝑖〉, 𝑖〉}〉, 〈(Scalar‘ndx), {〈(Base‘ndx), {𝑧}〉, 〈(+g‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉, 〈(.r‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉}〉} ∪ {〈( ·𝑠 ‘ndx), {〈〈𝑧, 𝑖〉, 𝑖〉}〉}) ∈ LVec)) |
13 | nelne1 3110 | . . . 4 ⊢ ((({〈(Base‘ndx), {𝑖}〉, 〈(+g‘ndx), {〈〈𝑖, 𝑖〉, 𝑖〉}〉, 〈(Scalar‘ndx), {〈(Base‘ndx), {𝑧}〉, 〈(+g‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉, 〈(.r‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉}〉} ∪ {〈( ·𝑠 ‘ndx), {〈〈𝑧, 𝑖〉, 𝑖〉}〉}) ∈ LMod ∧ ¬ ({〈(Base‘ndx), {𝑖}〉, 〈(+g‘ndx), {〈〈𝑖, 𝑖〉, 𝑖〉}〉, 〈(Scalar‘ndx), {〈(Base‘ndx), {𝑧}〉, 〈(+g‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉, 〈(.r‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉}〉} ∪ {〈( ·𝑠 ‘ndx), {〈〈𝑧, 𝑖〉, 𝑖〉}〉}) ∈ LVec) → LMod ≠ LVec) | |
14 | 13 | necomd 3068 | . . 3 ⊢ ((({〈(Base‘ndx), {𝑖}〉, 〈(+g‘ndx), {〈〈𝑖, 𝑖〉, 𝑖〉}〉, 〈(Scalar‘ndx), {〈(Base‘ndx), {𝑧}〉, 〈(+g‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉, 〈(.r‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉}〉} ∪ {〈( ·𝑠 ‘ndx), {〈〈𝑧, 𝑖〉, 𝑖〉}〉}) ∈ LMod ∧ ¬ ({〈(Base‘ndx), {𝑖}〉, 〈(+g‘ndx), {〈〈𝑖, 𝑖〉, 𝑖〉}〉, 〈(Scalar‘ndx), {〈(Base‘ndx), {𝑧}〉, 〈(+g‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉, 〈(.r‘ndx), {〈〈𝑧, 𝑧〉, 𝑧〉}〉}〉} ∪ {〈( ·𝑠 ‘ndx), {〈〈𝑧, 𝑖〉, 𝑖〉}〉}) ∈ LVec) → LVec ≠ LMod) |
15 | 5, 12, 14 | mp2b 10 | . 2 ⊢ LVec ≠ LMod |
16 | df-pss 3951 | . 2 ⊢ (LVec ⊊ LMod ↔ (LVec ⊆ LMod ∧ LVec ≠ LMod)) | |
17 | 2, 15, 16 | mpbir2an 707 | 1 ⊢ LVec ⊊ LMod |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 ∈ wcel 2105 ≠ wne 3013 ∉ wnel 3120 Vcvv 3492 ∪ cun 3931 ⊆ wss 3933 ⊊ wpss 3934 {csn 4557 {ctp 4561 〈cop 4563 ‘cfv 6348 ndxcnx 16468 Basecbs 16471 +gcplusg 16553 .rcmulr 16554 Scalarcsca 16556 ·𝑠 cvsca 16557 LModclmod 19563 LVecclvec 19803 |
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 |
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-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12881 df-hash 13679 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-drng 19433 df-lmod 19565 df-lvec 19804 df-nzr 19959 |
This theorem is referenced by: (None) |
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