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Mirrors > Home > MPE Home > Th. List > lspdisjb | Structured version Visualization version GIF version |
Description: A nonzero vector is not in a subspace iff its span is disjoint with the subspace. (Contributed by NM, 23-Apr-2015.) |
Ref | Expression |
---|---|
lspdisjb.v | ⊢ 𝑉 = (Base‘𝑊) |
lspdisjb.o | ⊢ 0 = (0g‘𝑊) |
lspdisjb.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lspdisjb.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lspdisjb.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lspdisjb.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lspdisjb.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
Ref | Expression |
---|---|
lspdisjb | ⊢ (𝜑 → (¬ 𝑋 ∈ 𝑈 ↔ ((𝑁‘{𝑋}) ∩ 𝑈) = { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspdisjb.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
2 | lspdisjb.o | . . 3 ⊢ 0 = (0g‘𝑊) | |
3 | lspdisjb.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
4 | lspdisjb.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
5 | lspdisjb.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
6 | 5 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑈) → 𝑊 ∈ LVec) |
7 | lspdisjb.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
8 | 7 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑈) → 𝑈 ∈ 𝑆) |
9 | lspdisjb.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
10 | 9 | eldifad 3945 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
11 | 10 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) |
12 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑈) → ¬ 𝑋 ∈ 𝑈) | |
13 | 1, 2, 3, 4, 6, 8, 11, 12 | lspdisj 19826 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑈) → ((𝑁‘{𝑋}) ∩ 𝑈) = { 0 }) |
14 | eldifsni 4714 | . . . . 5 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 ) | |
15 | 9, 14 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
16 | 15 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ((𝑁‘{𝑋}) ∩ 𝑈) = { 0 }) → 𝑋 ≠ 0 ) |
17 | lveclmod 19807 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
18 | 5, 17 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) |
19 | 1, 3 | lspsnid 19694 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) |
20 | 18, 10, 19 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑋})) |
21 | elin 4166 | . . . . . . 7 ⊢ (𝑋 ∈ ((𝑁‘{𝑋}) ∩ 𝑈) ↔ (𝑋 ∈ (𝑁‘{𝑋}) ∧ 𝑋 ∈ 𝑈)) | |
22 | eleq2 2898 | . . . . . . . 8 ⊢ (((𝑁‘{𝑋}) ∩ 𝑈) = { 0 } → (𝑋 ∈ ((𝑁‘{𝑋}) ∩ 𝑈) ↔ 𝑋 ∈ { 0 })) | |
23 | elsni 4574 | . . . . . . . 8 ⊢ (𝑋 ∈ { 0 } → 𝑋 = 0 ) | |
24 | 22, 23 | syl6bi 254 | . . . . . . 7 ⊢ (((𝑁‘{𝑋}) ∩ 𝑈) = { 0 } → (𝑋 ∈ ((𝑁‘{𝑋}) ∩ 𝑈) → 𝑋 = 0 )) |
25 | 21, 24 | syl5bir 244 | . . . . . 6 ⊢ (((𝑁‘{𝑋}) ∩ 𝑈) = { 0 } → ((𝑋 ∈ (𝑁‘{𝑋}) ∧ 𝑋 ∈ 𝑈) → 𝑋 = 0 )) |
26 | 25 | expd 416 | . . . . 5 ⊢ (((𝑁‘{𝑋}) ∩ 𝑈) = { 0 } → (𝑋 ∈ (𝑁‘{𝑋}) → (𝑋 ∈ 𝑈 → 𝑋 = 0 ))) |
27 | 20, 26 | mpan9 507 | . . . 4 ⊢ ((𝜑 ∧ ((𝑁‘{𝑋}) ∩ 𝑈) = { 0 }) → (𝑋 ∈ 𝑈 → 𝑋 = 0 )) |
28 | 27 | necon3ad 3026 | . . 3 ⊢ ((𝜑 ∧ ((𝑁‘{𝑋}) ∩ 𝑈) = { 0 }) → (𝑋 ≠ 0 → ¬ 𝑋 ∈ 𝑈)) |
29 | 16, 28 | mpd 15 | . 2 ⊢ ((𝜑 ∧ ((𝑁‘{𝑋}) ∩ 𝑈) = { 0 }) → ¬ 𝑋 ∈ 𝑈) |
30 | 13, 29 | impbida 797 | 1 ⊢ (𝜑 → (¬ 𝑋 ∈ 𝑈 ↔ ((𝑁‘{𝑋}) ∩ 𝑈) = { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∖ cdif 3930 ∩ cin 3932 {csn 4557 ‘cfv 6348 Basecbs 16471 0gc0g 16701 LModclmod 19563 LSubSpclss 19632 LSpanclspn 19672 LVecclvec 19803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-drng 19433 df-lmod 19565 df-lss 19633 df-lsp 19673 df-lvec 19804 |
This theorem is referenced by: mapdh6b0N 38752 hdmap1l6b0N 38826 |
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