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Theorem lspsnel6 19695
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.)
Hypotheses
Ref Expression
lspsnel5.v 𝑉 = (Base‘𝑊)
lspsnel5.s 𝑆 = (LSubSp‘𝑊)
lspsnel5.n 𝑁 = (LSpan‘𝑊)
lspsnel5.w (𝜑𝑊 ∈ LMod)
lspsnel5.a (𝜑𝑈𝑆)
Assertion
Ref Expression
lspsnel6 (𝜑 → (𝑋𝑈 ↔ (𝑋𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈)))

Proof of Theorem lspsnel6
StepHypRef Expression
1 lspsnel5.a . . . 4 (𝜑𝑈𝑆)
2 lspsnel5.v . . . . 5 𝑉 = (Base‘𝑊)
3 lspsnel5.s . . . . 5 𝑆 = (LSubSp‘𝑊)
42, 3lssel 19638 . . . 4 ((𝑈𝑆𝑋𝑈) → 𝑋𝑉)
51, 4sylan 580 . . 3 ((𝜑𝑋𝑈) → 𝑋𝑉)
6 lspsnel5.w . . . . 5 (𝜑𝑊 ∈ LMod)
76adantr 481 . . . 4 ((𝜑𝑋𝑈) → 𝑊 ∈ LMod)
81adantr 481 . . . 4 ((𝜑𝑋𝑈) → 𝑈𝑆)
9 simpr 485 . . . 4 ((𝜑𝑋𝑈) → 𝑋𝑈)
10 lspsnel5.n . . . . 5 𝑁 = (LSpan‘𝑊)
113, 10lspsnss 19691 . . . 4 ((𝑊 ∈ LMod ∧ 𝑈𝑆𝑋𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈)
127, 8, 9, 11syl3anc 1363 . . 3 ((𝜑𝑋𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈)
135, 12jca 512 . 2 ((𝜑𝑋𝑈) → (𝑋𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈))
142, 10lspsnid 19694 . . . . 5 ((𝑊 ∈ LMod ∧ 𝑋𝑉) → 𝑋 ∈ (𝑁‘{𝑋}))
156, 14sylan 580 . . . 4 ((𝜑𝑋𝑉) → 𝑋 ∈ (𝑁‘{𝑋}))
16 ssel 3958 . . . 4 ((𝑁‘{𝑋}) ⊆ 𝑈 → (𝑋 ∈ (𝑁‘{𝑋}) → 𝑋𝑈))
1715, 16syl5com 31 . . 3 ((𝜑𝑋𝑉) → ((𝑁‘{𝑋}) ⊆ 𝑈𝑋𝑈))
1817impr 455 . 2 ((𝜑 ∧ (𝑋𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈)) → 𝑋𝑈)
1913, 18impbida 797 1 (𝜑 → (𝑋𝑈 ↔ (𝑋𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wss 3933  {csn 4557  cfv 6348  Basecbs 16471  LModclmod 19563  LSubSpclss 19632  LSpanclspn 19672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-0g 16703  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044  df-lmod 19565  df-lss 19633  df-lsp 19673
This theorem is referenced by:  lspsnel5  19696  lsmelval2  19786  dihjat1lem  38444
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