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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 4322 | . . 3 ⊢ (𝑈 ⊆ {(0g‘𝑊)} ↔ (𝑈 ∖ {(0g‘𝑊)}) = ∅) | |
2 | lspprat.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
3 | lveclmod 19872 | . . . . . . . 8 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
4 | 2, 3 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod) |
5 | lspprat.v | . . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊) | |
6 | eqid 2821 | . . . . . . . 8 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
7 | 5, 6 | lmod0vcl 19657 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → (0g‘𝑊) ∈ 𝑉) |
8 | 4, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → (0g‘𝑊) ∈ 𝑉) |
9 | 8 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑈 ⊆ {(0g‘𝑊)}) → (0g‘𝑊) ∈ 𝑉) |
10 | simpr 487 | . . . . . . 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 19714 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → {(0g‘𝑊)} ⊆ 𝑈) |
14 | 4, 11, 13 | syl2anc 586 | . . . . . . . 8 ⊢ (𝜑 → {(0g‘𝑊)} ⊆ 𝑈) |
15 | 14 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑈 ⊆ {(0g‘𝑊)}) → {(0g‘𝑊)} ⊆ 𝑈) |
16 | 10, 15 | eqssd 3983 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑈 ⊆ {(0g‘𝑊)}) → 𝑈 = {(0g‘𝑊)}) |
17 | lspprat.n | . . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑊) | |
18 | 6, 17 | lspsn0 19774 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → (𝑁‘{(0g‘𝑊)}) = {(0g‘𝑊)}) |
19 | 4, 18 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{(0g‘𝑊)}) = {(0g‘𝑊)}) |
20 | 19 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑈 ⊆ {(0g‘𝑊)}) → (𝑁‘{(0g‘𝑊)}) = {(0g‘𝑊)}) |
21 | 16, 20 | eqtr4d 2859 | . . . . 5 ⊢ ((𝜑 ∧ 𝑈 ⊆ {(0g‘𝑊)}) → 𝑈 = (𝑁‘{(0g‘𝑊)})) |
22 | sneq 4570 | . . . . . . 7 ⊢ (𝑧 = (0g‘𝑊) → {𝑧} = {(0g‘𝑊)}) | |
23 | 22 | fveq2d 6668 | . . . . . 6 ⊢ (𝑧 = (0g‘𝑊) → (𝑁‘{𝑧}) = (𝑁‘{(0g‘𝑊)})) |
24 | 23 | rspceeqv 3637 | . . . . 5 ⊢ (((0g‘𝑊) ∈ 𝑉 ∧ 𝑈 = (𝑁‘{(0g‘𝑊)})) → ∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧})) |
25 | 9, 21, 24 | syl2anc 586 | . . . 4 ⊢ ((𝜑 ∧ 𝑈 ⊆ {(0g‘𝑊)}) → ∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧})) |
26 | 25 | ex 415 | . . 3 ⊢ (𝜑 → (𝑈 ⊆ {(0g‘𝑊)} → ∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧}))) |
27 | 1, 26 | syl5bir 245 | . 2 ⊢ (𝜑 → ((𝑈 ∖ {(0g‘𝑊)}) = ∅ → ∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧}))) |
28 | 5, 12 | lssss 19702 | . . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉) |
29 | 11, 28 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ 𝑉) |
30 | 29 | ssdifssd 4118 | . . . . . 6 ⊢ (𝜑 → (𝑈 ∖ {(0g‘𝑊)}) ⊆ 𝑉) |
31 | 30 | sseld 3965 | . . . . 5 ⊢ (𝜑 → (𝑧 ∈ (𝑈 ∖ {(0g‘𝑊)}) → 𝑧 ∈ 𝑉)) |
32 | lspprat.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
33 | lspprat.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
34 | lspprat.p | . . . . . 6 ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) | |
35 | 5, 12, 17, 2, 11, 32, 33, 34, 6 | lsppratlem6 19918 | . . . . 5 ⊢ (𝜑 → (𝑧 ∈ (𝑈 ∖ {(0g‘𝑊)}) → 𝑈 = (𝑁‘{𝑧}))) |
36 | 31, 35 | jcad 515 | . . . 4 ⊢ (𝜑 → (𝑧 ∈ (𝑈 ∖ {(0g‘𝑊)}) → (𝑧 ∈ 𝑉 ∧ 𝑈 = (𝑁‘{𝑧})))) |
37 | 36 | eximdv 1914 | . . 3 ⊢ (𝜑 → (∃𝑧 𝑧 ∈ (𝑈 ∖ {(0g‘𝑊)}) → ∃𝑧(𝑧 ∈ 𝑉 ∧ 𝑈 = (𝑁‘{𝑧})))) |
38 | n0 4309 | . . 3 ⊢ ((𝑈 ∖ {(0g‘𝑊)}) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑈 ∖ {(0g‘𝑊)})) | |
39 | df-rex 3144 | . . 3 ⊢ (∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧}) ↔ ∃𝑧(𝑧 ∈ 𝑉 ∧ 𝑈 = (𝑁‘{𝑧}))) | |
40 | 37, 38, 39 | 3imtr4g 298 | . 2 ⊢ (𝜑 → ((𝑈 ∖ {(0g‘𝑊)}) ≠ ∅ → ∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧}))) |
41 | 27, 40 | pm2.61dne 3103 | 1 ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ≠ wne 3016 ∃wrex 3139 ∖ cdif 3932 ⊆ wss 3935 ⊊ wpss 3936 ∅c0 4290 {csn 4560 {cpr 4562 ‘cfv 6349 Basecbs 16477 0gc0g 16707 LModclmod 19628 LSubSpclss 19697 LSpanclspn 19737 LVecclvec 19868 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-sbg 18102 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-drng 19498 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lvec 19869 |
This theorem is referenced by: dvh3dim3N 38579 |
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