![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > lspsnel6 | 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.) (Revised by Mario Carneiro, 8-Jan-2015.) |
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) |
Ref | Expression |
---|---|
lspsnel6 | ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspsnel5.a | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
2 | lspsnel5.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
3 | lspsnel5.s | . . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊) | |
4 | 2, 3 | lssel 20535 | . . . 4 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) |
5 | 1, 4 | sylan 581 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) |
6 | lspsnel5.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
7 | 6 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑊 ∈ LMod) |
8 | 1 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑈 ∈ 𝑆) |
9 | simpr 486 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑈) | |
10 | lspsnel5.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑊) | |
11 | 3, 10 | lspsnss 20588 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈) |
12 | 7, 8, 9, 11 | syl3anc 1372 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈) |
13 | 5, 12 | jca 513 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑋 ∈ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈)) |
14 | 2, 10 | lspsnid 20591 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) |
15 | 6, 14 | sylan 581 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) |
16 | ssel 3973 | . . . 4 ⊢ ((𝑁‘{𝑋}) ⊆ 𝑈 → (𝑋 ∈ (𝑁‘{𝑋}) → 𝑋 ∈ 𝑈)) | |
17 | 15, 16 | syl5com 31 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉) → ((𝑁‘{𝑋}) ⊆ 𝑈 → 𝑋 ∈ 𝑈)) |
18 | 17 | impr 456 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈)) → 𝑋 ∈ 𝑈) |
19 | 13, 18 | impbida 800 | 1 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3946 {csn 4626 ‘cfv 6539 Basecbs 17139 LModclmod 20458 LSubSpclss 20529 LSpanclspn 20569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-int 4949 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-0g 17382 df-mgm 18556 df-sgrp 18605 df-mnd 18621 df-grp 18817 df-lmod 20460 df-lss 20530 df-lsp 20570 |
This theorem is referenced by: lspsnel5 20593 lsmelval2 20683 dihjat1lem 40236 |
Copyright terms: Public domain | W3C validator |