Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > lspsnel5 | Structured version Visualization version GIF version |
Description: Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.) |
Ref | Expression |
---|---|
lspsnel5.v | ⊢ 𝑉 = (Base‘𝑊) |
lspsnel5.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lspsnel5.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lspsnel5.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lspsnel5.a | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lspsnel5.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
Ref | Expression |
---|---|
lspsnel5 | ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑁‘{𝑋}) ⊆ 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspsnel5.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
2 | lspsnel5.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | lspsnel5.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
4 | lspsnel5.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
5 | lspsnel5.a | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
6 | 1, 2, 3, 4, 5 | lspsnel6 18815 | . 2 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈))) |
7 | lspsnel5.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
8 | 7 | biantrurd 528 | . 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊆ 𝑈 ↔ (𝑋 ∈ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈))) |
9 | 6, 8 | bitr4d 270 | 1 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑁‘{𝑋}) ⊆ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 {csn 4125 ‘cfv 5804 Basecbs 15695 LModclmod 18686 LSubSpclss 18753 LSpanclspn 18792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-lmod 18688 df-lss 18754 df-lsp 18793 |
This theorem is referenced by: lspsnel5a 18817 lspprid1 18818 lspsnss2 18826 lsmelpr 18912 lspsncmp 18937 lspsnne1 18938 lspsnne2 18939 lspsneq 18943 lspindpi 18953 islbs2 18975 lindsenlbs 32574 lsatelbN 33311 lsmsat 33313 lsatfixedN 33314 l1cvpat 33359 dia2dimlem5 35375 dochsncom 35689 dihjat1lem 35735 dvh4dimlem 35750 lclkrlem2a 35814 lcfrlem6 35854 lcfrlem20 35869 lcfrlem26 35875 lcfrlem36 35885 mapdval2N 35937 mapdrvallem2 35952 mapdindp 35978 mapdh6aN 36042 lspindp5 36077 mapdh8ab 36084 mapdh8e 36091 hdmap1l6a 36117 hdmaprnlem3eN 36168 hdmapoc 36241 |
Copyright terms: Public domain | W3C validator |