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Theorem lspsnel6 19206
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 19147 . . . 4 ((𝑈𝑆𝑋𝑈) → 𝑋𝑉)
51, 4sylan 561 . . 3 ((𝜑𝑋𝑈) → 𝑋𝑉)
6 lspsnel5.w . . . . 5 (𝜑𝑊 ∈ LMod)
76adantr 466 . . . 4 ((𝜑𝑋𝑈) → 𝑊 ∈ LMod)
81adantr 466 . . . 4 ((𝜑𝑋𝑈) → 𝑈𝑆)
9 simpr 471 . . . 4 ((𝜑𝑋𝑈) → 𝑋𝑈)
10 lspsnel5.n . . . . 5 𝑁 = (LSpan‘𝑊)
113, 10lspsnss 19202 . . . 4 ((𝑊 ∈ LMod ∧ 𝑈𝑆𝑋𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈)
127, 8, 9, 11syl3anc 1475 . . 3 ((𝜑𝑋𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈)
135, 12jca 495 . 2 ((𝜑𝑋𝑈) → (𝑋𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈))
142, 10lspsnid 19205 . . . . 5 ((𝑊 ∈ LMod ∧ 𝑋𝑉) → 𝑋 ∈ (𝑁‘{𝑋}))
156, 14sylan 561 . . . 4 ((𝜑𝑋𝑉) → 𝑋 ∈ (𝑁‘{𝑋}))
16 ssel 3744 . . . 4 ((𝑁‘{𝑋}) ⊆ 𝑈 → (𝑋 ∈ (𝑁‘{𝑋}) → 𝑋𝑈))
1715, 16syl5com 31 . . 3 ((𝜑𝑋𝑉) → ((𝑁‘{𝑋}) ⊆ 𝑈𝑋𝑈))
1817impr 442 . 2 ((𝜑 ∧ (𝑋𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈)) → 𝑋𝑈)
1913, 18impbida 794 1 (𝜑 → (𝑋𝑈 ↔ (𝑋𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  wss 3721  {csn 4314  cfv 6031  Basecbs 16063  LModclmod 19072  LSubSpclss 19141  LSpanclspn 19183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-0g 16309  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-grp 17632  df-lmod 19074  df-lss 19142  df-lsp 19184
This theorem is referenced by:  lspsnel5  19207  lsmelval2  19297  dihjat1lem  37231
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