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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 20984) 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 47556. (Contributed by AV, 29-Apr-2019.) |
Ref | Expression |
---|---|
lvecpsslmod | ⊢ LVec ⊊ LMod |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lveclmod 20984 | . . 3 ⊢ (𝑣 ∈ LVec → 𝑣 ∈ LMod) | |
2 | 1 | ssriv 3982 | . 2 ⊢ LVec ⊆ LMod |
3 | vex 3474 | . . . 4 ⊢ 𝑖 ∈ V | |
4 | vex 3474 | . . . 4 ⊢ 𝑧 ∈ V | |
5 | 3, 4 | pm3.2i 470 | . . 3 ⊢ (𝑖 ∈ V ∧ 𝑧 ∈ V) |
6 | eqid 2728 | . . . . 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 2728 | . . . . 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 47555 | . . . 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 47556 | . . . . 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 3043 | . . . . 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 217 | . . . 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 511 | . . 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 3035 | . . . 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 2992 | . . 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 3964 | . 2 ⊢ (LVec ⊊ LMod ↔ (LVec ⊆ LMod ∧ LVec ≠ LMod)) | |
17 | 2, 15, 16 | mpbir2an 710 | 1 ⊢ LVec ⊊ LMod |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 ∈ wcel 2099 ≠ wne 2936 ∉ wnel 3042 Vcvv 3470 ∪ cun 3943 ⊆ wss 3945 ⊊ wpss 3946 {csn 4624 {ctp 4628 〈cop 4630 ‘cfv 6542 ndxcnx 17155 Basecbs 17173 +gcplusg 17226 .rcmulr 17227 Scalarcsca 17229 ·𝑠 cvsca 17230 LModclmod 20736 LVecclvec 20980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-dju 9918 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-n0 12497 df-xnn0 12569 df-z 12583 df-uz 12847 df-fz 13511 df-hash 14316 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-0g 17416 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-grp 18886 df-minusg 18887 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-oppr 20266 df-dvdsr 20289 df-unit 20290 df-nzr 20445 df-drng 20619 df-lmod 20738 df-lvec 20981 |
This theorem is referenced by: (None) |
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