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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 19712 | . . . 4 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) |
5 | 1, 4 | sylan 582 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) |
6 | lspsnel5.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
7 | 6 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑊 ∈ LMod) |
8 | 1 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑈 ∈ 𝑆) |
9 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑈) | |
10 | lspsnel5.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑊) | |
11 | 3, 10 | lspsnss 19765 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈) |
12 | 7, 8, 9, 11 | syl3anc 1367 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈) |
13 | 5, 12 | jca 514 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑋 ∈ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈)) |
14 | 2, 10 | lspsnid 19768 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) |
15 | 6, 14 | sylan 582 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) |
16 | ssel 3964 | . . . 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 457 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈)) → 𝑋 ∈ 𝑈) |
19 | 13, 18 | impbida 799 | 1 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ⊆ wss 3939 {csn 4570 ‘cfv 6358 Basecbs 16486 LModclmod 19637 LSubSpclss 19706 LSpanclspn 19746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-lmod 19639 df-lss 19707 df-lsp 19747 |
This theorem is referenced by: lspsnel5 19770 lsmelval2 19860 dihjat1lem 38568 |
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