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Mirrors > Home > MPE Home > Th. List > lspsnel5a | Structured version Visualization version GIF version |
Description: Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 20-Feb-2015.) |
Ref | Expression |
---|---|
lspsnel5a.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lspsnel5a.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lspsnel5a.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lspsnel5a.a | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lspsnel5a.x | ⊢ (𝜑 → 𝑋 ∈ 𝑈) |
Ref | Expression |
---|---|
lspsnel5a | ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspsnel5a.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑈) | |
2 | eqid 2823 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
3 | lspsnel5a.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
4 | lspsnel5a.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
5 | lspsnel5a.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
6 | lspsnel5a.a | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
7 | 2, 3 | lssel 19711 | . . . 4 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑊)) |
8 | 6, 1, 7 | syl2anc 586 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊)) |
9 | 2, 3, 4, 5, 6, 8 | lspsnel5 19769 | . 2 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑁‘{𝑋}) ⊆ 𝑈)) |
10 | 1, 9 | mpbid 234 | 1 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 {csn 4569 ‘cfv 6357 Basecbs 16485 LModclmod 19636 LSubSpclss 19705 LSpanclspn 19745 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-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-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 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-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-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-lmod 19638 df-lss 19706 df-lsp 19746 |
This theorem is referenced by: lssats2 19774 lspsn 19776 lspsnvsi 19778 lsmelval2 19859 lspprabs 19869 lspvadd 19870 lspabs3 19895 lsmcv 19915 lspsnat 19919 lsppratlem6 19926 issubassa2 20123 lshpnel 36121 lsatel 36143 lsmsat 36146 lssatomic 36149 lssats 36150 lsat0cv 36171 dia2dimlem10 38211 dochsatshpb 38590 lclkrlem2f 38650 lcfrlem25 38705 lcfrlem35 38715 mapdval2N 38768 mapdrvallem2 38783 mapdpglem8 38817 mapdpglem13 38822 mapdindp0 38857 mapdh6aN 38873 mapdh8e 38922 mapdh9a 38927 hdmap1l6a 38947 hdmapval0 38971 hdmapval3lemN 38975 hdmap10lem 38977 hdmap11lem1 38979 hdmap11lem2 38980 hdmaprnlem4N 38991 hdmaprnlem3eN 38996 |
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