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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 466 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑈) → 𝑊 ∈ LVec) |
7 | lspdisjb.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
8 | 7 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑈) → 𝑈 ∈ 𝑆) |
9 | lspdisjb.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
10 | 9 | eldifad 3735 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
11 | 10 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) |
12 | simpr 471 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑈) → ¬ 𝑋 ∈ 𝑈) | |
13 | 1, 2, 3, 4, 6, 8, 11, 12 | lspdisj 19338 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑈) → ((𝑁‘{𝑋}) ∩ 𝑈) = { 0 }) |
14 | eldifsni 4457 | . . . . 5 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 ) | |
15 | 9, 14 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
16 | 15 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ ((𝑁‘{𝑋}) ∩ 𝑈) = { 0 }) → 𝑋 ≠ 0 ) |
17 | lveclmod 19319 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
18 | 5, 17 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) |
19 | 1, 3 | lspsnid 19206 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) |
20 | 18, 10, 19 | syl2anc 573 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑋})) |
21 | elin 3947 | . . . . . . 7 ⊢ (𝑋 ∈ ((𝑁‘{𝑋}) ∩ 𝑈) ↔ (𝑋 ∈ (𝑁‘{𝑋}) ∧ 𝑋 ∈ 𝑈)) | |
22 | eleq2 2839 | . . . . . . . 8 ⊢ (((𝑁‘{𝑋}) ∩ 𝑈) = { 0 } → (𝑋 ∈ ((𝑁‘{𝑋}) ∩ 𝑈) ↔ 𝑋 ∈ { 0 })) | |
23 | elsni 4333 | . . . . . . . 8 ⊢ (𝑋 ∈ { 0 } → 𝑋 = 0 ) | |
24 | 22, 23 | syl6bi 243 | . . . . . . 7 ⊢ (((𝑁‘{𝑋}) ∩ 𝑈) = { 0 } → (𝑋 ∈ ((𝑁‘{𝑋}) ∩ 𝑈) → 𝑋 = 0 )) |
25 | 21, 24 | syl5bir 233 | . . . . . 6 ⊢ (((𝑁‘{𝑋}) ∩ 𝑈) = { 0 } → ((𝑋 ∈ (𝑁‘{𝑋}) ∧ 𝑋 ∈ 𝑈) → 𝑋 = 0 )) |
26 | 25 | expd 400 | . . . . 5 ⊢ (((𝑁‘{𝑋}) ∩ 𝑈) = { 0 } → (𝑋 ∈ (𝑁‘{𝑋}) → (𝑋 ∈ 𝑈 → 𝑋 = 0 ))) |
27 | 20, 26 | mpan9 496 | . . . 4 ⊢ ((𝜑 ∧ ((𝑁‘{𝑋}) ∩ 𝑈) = { 0 }) → (𝑋 ∈ 𝑈 → 𝑋 = 0 )) |
28 | 27 | necon3ad 2956 | . . 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 802 | 1 ⊢ (𝜑 → (¬ 𝑋 ∈ 𝑈 ↔ ((𝑁‘{𝑋}) ∩ 𝑈) = { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∖ cdif 3720 ∩ cin 3722 {csn 4316 ‘cfv 6031 Basecbs 16064 0gc0g 16308 LModclmod 19073 LSubSpclss 19142 LSpanclspn 19184 LVecclvec 19315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-tpos 7504 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-3 11282 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-0g 16310 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-grp 17633 df-minusg 17634 df-sbg 17635 df-mgp 18698 df-ur 18710 df-ring 18757 df-oppr 18831 df-dvdsr 18849 df-unit 18850 df-invr 18880 df-drng 18959 df-lmod 19075 df-lss 19143 df-lsp 19185 df-lvec 19316 |
This theorem is referenced by: mapdh6b0N 37546 hdmap1l6b0N 37620 |
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