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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 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑈) → 𝑊 ∈ LVec) |
7 | lspdisjb.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
8 | 7 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑈) → 𝑈 ∈ 𝑆) |
9 | lspdisjb.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
10 | 9 | eldifad 3950 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
11 | 10 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) |
12 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑈) → ¬ 𝑋 ∈ 𝑈) | |
13 | 1, 2, 3, 4, 6, 8, 11, 12 | lspdisj 19899 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑈) → ((𝑁‘{𝑋}) ∩ 𝑈) = { 0 }) |
14 | eldifsni 4724 | . . . . 5 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 ) | |
15 | 9, 14 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
16 | 15 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ((𝑁‘{𝑋}) ∩ 𝑈) = { 0 }) → 𝑋 ≠ 0 ) |
17 | lveclmod 19880 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
18 | 5, 17 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) |
19 | 1, 3 | lspsnid 19767 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) |
20 | 18, 10, 19 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑋})) |
21 | elin 4171 | . . . . . . 7 ⊢ (𝑋 ∈ ((𝑁‘{𝑋}) ∩ 𝑈) ↔ (𝑋 ∈ (𝑁‘{𝑋}) ∧ 𝑋 ∈ 𝑈)) | |
22 | eleq2 2903 | . . . . . . . 8 ⊢ (((𝑁‘{𝑋}) ∩ 𝑈) = { 0 } → (𝑋 ∈ ((𝑁‘{𝑋}) ∩ 𝑈) ↔ 𝑋 ∈ { 0 })) | |
23 | elsni 4586 | . . . . . . . 8 ⊢ (𝑋 ∈ { 0 } → 𝑋 = 0 ) | |
24 | 22, 23 | syl6bi 255 | . . . . . . 7 ⊢ (((𝑁‘{𝑋}) ∩ 𝑈) = { 0 } → (𝑋 ∈ ((𝑁‘{𝑋}) ∩ 𝑈) → 𝑋 = 0 )) |
25 | 21, 24 | syl5bir 245 | . . . . . 6 ⊢ (((𝑁‘{𝑋}) ∩ 𝑈) = { 0 } → ((𝑋 ∈ (𝑁‘{𝑋}) ∧ 𝑋 ∈ 𝑈) → 𝑋 = 0 )) |
26 | 25 | expd 418 | . . . . 5 ⊢ (((𝑁‘{𝑋}) ∩ 𝑈) = { 0 } → (𝑋 ∈ (𝑁‘{𝑋}) → (𝑋 ∈ 𝑈 → 𝑋 = 0 ))) |
27 | 20, 26 | mpan9 509 | . . . 4 ⊢ ((𝜑 ∧ ((𝑁‘{𝑋}) ∩ 𝑈) = { 0 }) → (𝑋 ∈ 𝑈 → 𝑋 = 0 )) |
28 | 27 | necon3ad 3031 | . . 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 799 | 1 ⊢ (𝜑 → (¬ 𝑋 ∈ 𝑈 ↔ ((𝑁‘{𝑋}) ∩ 𝑈) = { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∖ cdif 3935 ∩ cin 3937 {csn 4569 ‘cfv 6357 Basecbs 16485 0gc0g 16715 LModclmod 19636 LSubSpclss 19705 LSpanclspn 19745 LVecclvec 19876 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mgp 19242 df-ur 19254 df-ring 19301 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-drng 19506 df-lmod 19638 df-lss 19706 df-lsp 19746 df-lvec 19877 |
This theorem is referenced by: mapdh6b0N 38874 hdmap1l6b0N 38948 |
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