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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 4365 | . . 3 ⊢ (𝑈 ⊆ {(0g‘𝑊)} ↔ (𝑈 ∖ {(0g‘𝑊)}) = ∅) | |
| 2 | lspprat.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
| 3 | lveclmod 21106 | . . . . . . . 8 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
| 4 | 2, 3 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod) | 
| 5 | lspprat.v | . . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊) | |
| 6 | eqid 2736 | . . . . . . . 8 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
| 7 | 5, 6 | lmod0vcl 20890 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → (0g‘𝑊) ∈ 𝑉) | 
| 8 | 4, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → (0g‘𝑊) ∈ 𝑉) | 
| 9 | 8 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑈 ⊆ {(0g‘𝑊)}) → (0g‘𝑊) ∈ 𝑉) | 
| 10 | simpr 484 | . . . . . . 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 20948 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → {(0g‘𝑊)} ⊆ 𝑈) | 
| 14 | 4, 11, 13 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → {(0g‘𝑊)} ⊆ 𝑈) | 
| 15 | 14 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑈 ⊆ {(0g‘𝑊)}) → {(0g‘𝑊)} ⊆ 𝑈) | 
| 16 | 10, 15 | eqssd 4000 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑈 ⊆ {(0g‘𝑊)}) → 𝑈 = {(0g‘𝑊)}) | 
| 17 | lspprat.n | . . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑊) | |
| 18 | 6, 17 | lspsn0 21007 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → (𝑁‘{(0g‘𝑊)}) = {(0g‘𝑊)}) | 
| 19 | 4, 18 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{(0g‘𝑊)}) = {(0g‘𝑊)}) | 
| 20 | 19 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑈 ⊆ {(0g‘𝑊)}) → (𝑁‘{(0g‘𝑊)}) = {(0g‘𝑊)}) | 
| 21 | 16, 20 | eqtr4d 2779 | . . . . 5 ⊢ ((𝜑 ∧ 𝑈 ⊆ {(0g‘𝑊)}) → 𝑈 = (𝑁‘{(0g‘𝑊)})) | 
| 22 | sneq 4635 | . . . . . . 7 ⊢ (𝑧 = (0g‘𝑊) → {𝑧} = {(0g‘𝑊)}) | |
| 23 | 22 | fveq2d 6909 | . . . . . 6 ⊢ (𝑧 = (0g‘𝑊) → (𝑁‘{𝑧}) = (𝑁‘{(0g‘𝑊)})) | 
| 24 | 23 | rspceeqv 3644 | . . . . 5 ⊢ (((0g‘𝑊) ∈ 𝑉 ∧ 𝑈 = (𝑁‘{(0g‘𝑊)})) → ∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧})) | 
| 25 | 9, 21, 24 | syl2anc 584 | . . . 4 ⊢ ((𝜑 ∧ 𝑈 ⊆ {(0g‘𝑊)}) → ∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧})) | 
| 26 | 25 | ex 412 | . . 3 ⊢ (𝜑 → (𝑈 ⊆ {(0g‘𝑊)} → ∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧}))) | 
| 27 | 1, 26 | biimtrrid 243 | . 2 ⊢ (𝜑 → ((𝑈 ∖ {(0g‘𝑊)}) = ∅ → ∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧}))) | 
| 28 | 5, 12 | lssss 20935 | . . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉) | 
| 29 | 11, 28 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ 𝑉) | 
| 30 | 29 | ssdifssd 4146 | . . . . . 6 ⊢ (𝜑 → (𝑈 ∖ {(0g‘𝑊)}) ⊆ 𝑉) | 
| 31 | 30 | sseld 3981 | . . . . 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 21155 | . . . . 5 ⊢ (𝜑 → (𝑧 ∈ (𝑈 ∖ {(0g‘𝑊)}) → 𝑈 = (𝑁‘{𝑧}))) | 
| 36 | 31, 35 | jcad 512 | . . . 4 ⊢ (𝜑 → (𝑧 ∈ (𝑈 ∖ {(0g‘𝑊)}) → (𝑧 ∈ 𝑉 ∧ 𝑈 = (𝑁‘{𝑧})))) | 
| 37 | 36 | eximdv 1916 | . . 3 ⊢ (𝜑 → (∃𝑧 𝑧 ∈ (𝑈 ∖ {(0g‘𝑊)}) → ∃𝑧(𝑧 ∈ 𝑉 ∧ 𝑈 = (𝑁‘{𝑧})))) | 
| 38 | n0 4352 | . . 3 ⊢ ((𝑈 ∖ {(0g‘𝑊)}) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑈 ∖ {(0g‘𝑊)})) | |
| 39 | df-rex 3070 | . . 3 ⊢ (∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧}) ↔ ∃𝑧(𝑧 ∈ 𝑉 ∧ 𝑈 = (𝑁‘{𝑧}))) | |
| 40 | 37, 38, 39 | 3imtr4g 296 | . 2 ⊢ (𝜑 → ((𝑈 ∖ {(0g‘𝑊)}) ≠ ∅ → ∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧}))) | 
| 41 | 27, 40 | pm2.61dne 3027 | 1 ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧})) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ≠ wne 2939 ∃wrex 3069 ∖ cdif 3947 ⊆ wss 3950 ⊊ wpss 3951 ∅c0 4332 {csn 4625 {cpr 4627 ‘cfv 6560 Basecbs 17248 0gc0g 17485 LModclmod 20859 LSubSpclss 20930 LSpanclspn 20970 LVecclvec 21102 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-minusg 18956 df-sbg 18957 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-invr 20389 df-drng 20732 df-lmod 20861 df-lss 20931 df-lsp 20971 df-lvec 21103 | 
| This theorem is referenced by: dvh3dim3N 41452 | 
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