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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 4323 | . . 3 ⊢ (𝑈 ⊆ {(0g‘𝑊)} ↔ (𝑈 ∖ {(0g‘𝑊)}) = ∅) | |
2 | lspprat.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
3 | lveclmod 19878 | . . . . . . . 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 19663 | . . . . . . 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 19720 | . . . . . . . . 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 3984 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑈 ⊆ {(0g‘𝑊)}) → 𝑈 = {(0g‘𝑊)}) |
17 | lspprat.n | . . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑊) | |
18 | 6, 17 | lspsn0 19780 | . . . . . . . 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 4577 | . . . . . . 7 ⊢ (𝑧 = (0g‘𝑊) → {𝑧} = {(0g‘𝑊)}) | |
23 | 22 | fveq2d 6674 | . . . . . 6 ⊢ (𝑧 = (0g‘𝑊) → (𝑁‘{𝑧}) = (𝑁‘{(0g‘𝑊)})) |
24 | 23 | rspceeqv 3638 | . . . . 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 19708 | . . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉) |
29 | 11, 28 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ 𝑉) |
30 | 29 | ssdifssd 4119 | . . . . . 6 ⊢ (𝜑 → (𝑈 ∖ {(0g‘𝑊)}) ⊆ 𝑉) |
31 | 30 | sseld 3966 | . . . . 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 19924 | . . . . 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 1918 | . . 3 ⊢ (𝜑 → (∃𝑧 𝑧 ∈ (𝑈 ∖ {(0g‘𝑊)}) → ∃𝑧(𝑧 ∈ 𝑉 ∧ 𝑈 = (𝑁‘{𝑧})))) |
38 | n0 4310 | . . 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 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 ∖ cdif 3933 ⊆ wss 3936 ⊊ wpss 3937 ∅c0 4291 {csn 4567 {cpr 4569 ‘cfv 6355 Basecbs 16483 0gc0g 16713 LModclmod 19634 LSubSpclss 19703 LSpanclspn 19743 LVecclvec 19874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-drng 19504 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lvec 19875 |
This theorem is referenced by: dvh3dim3N 38600 |
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