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Mirrors > Home > MPE Home > Th. List > lspprat | Structured version Visualization version GIF version |
Description: A proper subspace of the span of a pair of vectors is the span of a singleton (an atom) or the zero subspace (if 𝑧 is zero). Proof suggested by Mario Carneiro, 28-Aug-2014. (Contributed by NM, 29-Aug-2014.) |
Ref | Expression |
---|---|
lspprat.v | ⊢ 𝑉 = (Base‘𝑊) |
lspprat.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lspprat.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lspprat.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lspprat.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lspprat.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lspprat.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
lspprat.p | ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) |
Ref | Expression |
---|---|
lspprat | ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssdif0 4050 | . . 3 ⊢ (𝑈 ⊆ {(0g‘𝑊)} ↔ (𝑈 ∖ {(0g‘𝑊)}) = ∅) | |
2 | lspprat.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
3 | lveclmod 19229 | . . . . . . . 8 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
4 | 2, 3 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod) |
5 | lspprat.v | . . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊) | |
6 | eqid 2724 | . . . . . . . 8 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
7 | 5, 6 | lmod0vcl 19015 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → (0g‘𝑊) ∈ 𝑉) |
8 | 4, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → (0g‘𝑊) ∈ 𝑉) |
9 | 8 | adantr 472 | . . . . 5 ⊢ ((𝜑 ∧ 𝑈 ⊆ {(0g‘𝑊)}) → (0g‘𝑊) ∈ 𝑉) |
10 | simpr 479 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑈 ⊆ {(0g‘𝑊)}) → 𝑈 ⊆ {(0g‘𝑊)}) | |
11 | lspprat.u | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
12 | lspprat.s | . . . . . . . . . 10 ⊢ 𝑆 = (LSubSp‘𝑊) | |
13 | 6, 12 | lss0ss 19072 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → {(0g‘𝑊)} ⊆ 𝑈) |
14 | 4, 11, 13 | syl2anc 696 | . . . . . . . 8 ⊢ (𝜑 → {(0g‘𝑊)} ⊆ 𝑈) |
15 | 14 | adantr 472 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑈 ⊆ {(0g‘𝑊)}) → {(0g‘𝑊)} ⊆ 𝑈) |
16 | 10, 15 | eqssd 3726 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑈 ⊆ {(0g‘𝑊)}) → 𝑈 = {(0g‘𝑊)}) |
17 | lspprat.n | . . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑊) | |
18 | 6, 17 | lspsn0 19131 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → (𝑁‘{(0g‘𝑊)}) = {(0g‘𝑊)}) |
19 | 4, 18 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{(0g‘𝑊)}) = {(0g‘𝑊)}) |
20 | 19 | adantr 472 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑈 ⊆ {(0g‘𝑊)}) → (𝑁‘{(0g‘𝑊)}) = {(0g‘𝑊)}) |
21 | 16, 20 | eqtr4d 2761 | . . . . 5 ⊢ ((𝜑 ∧ 𝑈 ⊆ {(0g‘𝑊)}) → 𝑈 = (𝑁‘{(0g‘𝑊)})) |
22 | sneq 4295 | . . . . . . . 8 ⊢ (𝑧 = (0g‘𝑊) → {𝑧} = {(0g‘𝑊)}) | |
23 | 22 | fveq2d 6308 | . . . . . . 7 ⊢ (𝑧 = (0g‘𝑊) → (𝑁‘{𝑧}) = (𝑁‘{(0g‘𝑊)})) |
24 | 23 | eqeq2d 2734 | . . . . . 6 ⊢ (𝑧 = (0g‘𝑊) → (𝑈 = (𝑁‘{𝑧}) ↔ 𝑈 = (𝑁‘{(0g‘𝑊)}))) |
25 | 24 | rspcev 3413 | . . . . 5 ⊢ (((0g‘𝑊) ∈ 𝑉 ∧ 𝑈 = (𝑁‘{(0g‘𝑊)})) → ∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧})) |
26 | 9, 21, 25 | syl2anc 696 | . . . 4 ⊢ ((𝜑 ∧ 𝑈 ⊆ {(0g‘𝑊)}) → ∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧})) |
27 | 26 | ex 449 | . . 3 ⊢ (𝜑 → (𝑈 ⊆ {(0g‘𝑊)} → ∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧}))) |
28 | 1, 27 | syl5bir 233 | . 2 ⊢ (𝜑 → ((𝑈 ∖ {(0g‘𝑊)}) = ∅ → ∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧}))) |
29 | 5, 12 | lssss 19060 | . . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉) |
30 | 11, 29 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ 𝑉) |
31 | 30 | ssdifssd 3856 | . . . . . 6 ⊢ (𝜑 → (𝑈 ∖ {(0g‘𝑊)}) ⊆ 𝑉) |
32 | 31 | sseld 3708 | . . . . 5 ⊢ (𝜑 → (𝑧 ∈ (𝑈 ∖ {(0g‘𝑊)}) → 𝑧 ∈ 𝑉)) |
33 | lspprat.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
34 | lspprat.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
35 | lspprat.p | . . . . . 6 ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) | |
36 | 5, 12, 17, 2, 11, 33, 34, 35, 6 | lsppratlem6 19275 | . . . . 5 ⊢ (𝜑 → (𝑧 ∈ (𝑈 ∖ {(0g‘𝑊)}) → 𝑈 = (𝑁‘{𝑧}))) |
37 | 32, 36 | jcad 556 | . . . 4 ⊢ (𝜑 → (𝑧 ∈ (𝑈 ∖ {(0g‘𝑊)}) → (𝑧 ∈ 𝑉 ∧ 𝑈 = (𝑁‘{𝑧})))) |
38 | 37 | eximdv 1959 | . . 3 ⊢ (𝜑 → (∃𝑧 𝑧 ∈ (𝑈 ∖ {(0g‘𝑊)}) → ∃𝑧(𝑧 ∈ 𝑉 ∧ 𝑈 = (𝑁‘{𝑧})))) |
39 | n0 4039 | . . 3 ⊢ ((𝑈 ∖ {(0g‘𝑊)}) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑈 ∖ {(0g‘𝑊)})) | |
40 | df-rex 3020 | . . 3 ⊢ (∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧}) ↔ ∃𝑧(𝑧 ∈ 𝑉 ∧ 𝑈 = (𝑁‘{𝑧}))) | |
41 | 38, 39, 40 | 3imtr4g 285 | . 2 ⊢ (𝜑 → ((𝑈 ∖ {(0g‘𝑊)}) ≠ ∅ → ∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧}))) |
42 | 28, 41 | pm2.61dne 2982 | 1 ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1596 ∃wex 1817 ∈ wcel 2103 ≠ wne 2896 ∃wrex 3015 ∖ cdif 3677 ⊆ wss 3680 ⊊ wpss 3681 ∅c0 4023 {csn 4285 {cpr 4287 ‘cfv 6001 Basecbs 15980 0gc0g 16223 LModclmod 18986 LSubSpclss 19055 LSpanclspn 19094 LVecclvec 19225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1835 ax-4 1850 ax-5 1952 ax-6 2018 ax-7 2054 ax-8 2105 ax-9 2112 ax-10 2132 ax-11 2147 ax-12 2160 ax-13 2355 ax-ext 2704 ax-rep 4879 ax-sep 4889 ax-nul 4897 ax-pow 4948 ax-pr 5011 ax-un 7066 ax-cnex 10105 ax-resscn 10106 ax-1cn 10107 ax-icn 10108 ax-addcl 10109 ax-addrcl 10110 ax-mulcl 10111 ax-mulrcl 10112 ax-mulcom 10113 ax-addass 10114 ax-mulass 10115 ax-distr 10116 ax-i2m1 10117 ax-1ne0 10118 ax-1rid 10119 ax-rnegex 10120 ax-rrecex 10121 ax-cnre 10122 ax-pre-lttri 10123 ax-pre-lttrn 10124 ax-pre-ltadd 10125 ax-pre-mulgt0 10126 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1599 df-ex 1818 df-nf 1823 df-sb 2011 df-eu 2575 df-mo 2576 df-clab 2711 df-cleq 2717 df-clel 2720 df-nfc 2855 df-ne 2897 df-nel 3000 df-ral 3019 df-rex 3020 df-reu 3021 df-rmo 3022 df-rab 3023 df-v 3306 df-sbc 3542 df-csb 3640 df-dif 3683 df-un 3685 df-in 3687 df-ss 3694 df-pss 3696 df-nul 4024 df-if 4195 df-pw 4268 df-sn 4286 df-pr 4288 df-tp 4290 df-op 4292 df-uni 4545 df-int 4584 df-iun 4630 df-br 4761 df-opab 4821 df-mpt 4838 df-tr 4861 df-id 5128 df-eprel 5133 df-po 5139 df-so 5140 df-fr 5177 df-we 5179 df-xp 5224 df-rel 5225 df-cnv 5226 df-co 5227 df-dm 5228 df-rn 5229 df-res 5230 df-ima 5231 df-pred 5793 df-ord 5839 df-on 5840 df-lim 5841 df-suc 5842 df-iota 5964 df-fun 6003 df-fn 6004 df-f 6005 df-f1 6006 df-fo 6007 df-f1o 6008 df-fv 6009 df-riota 6726 df-ov 6768 df-oprab 6769 df-mpt2 6770 df-om 7183 df-1st 7285 df-2nd 7286 df-tpos 7472 df-wrecs 7527 df-recs 7588 df-rdg 7626 df-er 7862 df-en 8073 df-dom 8074 df-sdom 8075 df-pnf 10189 df-mnf 10190 df-xr 10191 df-ltxr 10192 df-le 10193 df-sub 10381 df-neg 10382 df-nn 11134 df-2 11192 df-3 11193 df-ndx 15983 df-slot 15984 df-base 15986 df-sets 15987 df-ress 15988 df-plusg 16077 df-mulr 16078 df-0g 16225 df-mgm 17364 df-sgrp 17406 df-mnd 17417 df-grp 17547 df-minusg 17548 df-sbg 17549 df-cmn 18316 df-abl 18317 df-mgp 18611 df-ur 18623 df-ring 18670 df-oppr 18744 df-dvdsr 18762 df-unit 18763 df-invr 18793 df-drng 18872 df-lmod 18988 df-lss 19056 df-lsp 19095 df-lvec 19226 |
This theorem is referenced by: dvh3dim3N 37157 |
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