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Theorem lshpdisj 37845
Description: A hyperplane and the span in the hyperplane definition are disjoint. (Contributed by NM, 3-Jul-2014.)
Hypotheses
Ref Expression
lshpdisj.v 𝑉 = (Baseβ€˜π‘Š)
lshpdisj.o 0 = (0gβ€˜π‘Š)
lshpdisj.n 𝑁 = (LSpanβ€˜π‘Š)
lshpdisj.p βŠ• = (LSSumβ€˜π‘Š)
lshpdisj.h 𝐻 = (LSHypβ€˜π‘Š)
lshpdisj.w (πœ‘ β†’ π‘Š ∈ LVec)
lshpdisj.u (πœ‘ β†’ π‘ˆ ∈ 𝐻)
lshpdisj.x (πœ‘ β†’ 𝑋 ∈ 𝑉)
lshpdisj.e (πœ‘ β†’ (π‘ˆ βŠ• (π‘β€˜{𝑋})) = 𝑉)
Assertion
Ref Expression
lshpdisj (πœ‘ β†’ (π‘ˆ ∩ (π‘β€˜{𝑋})) = { 0 })
Proof of Theorem lshpdisj
Dummy variables 𝑣 π‘˜ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lshpdisj.w . . . . . . . . 9 (πœ‘ β†’ π‘Š ∈ LVec)
2 lveclmod 20709 . . . . . . . . 9 (π‘Š ∈ LVec β†’ π‘Š ∈ LMod)
31, 2syl 17 . . . . . . . 8 (πœ‘ β†’ π‘Š ∈ LMod)
43adantr 481 . . . . . . 7 ((πœ‘ ∧ 𝑣 ∈ π‘ˆ) β†’ π‘Š ∈ LMod)
5 lshpdisj.x . . . . . . . 8 (πœ‘ β†’ 𝑋 ∈ 𝑉)
65adantr 481 . . . . . . 7 ((πœ‘ ∧ 𝑣 ∈ π‘ˆ) β†’ 𝑋 ∈ 𝑉)
7 eqid 2732 . . . . . . . 8 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
8 eqid 2732 . . . . . . . 8 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
9 lshpdisj.v . . . . . . . 8 𝑉 = (Baseβ€˜π‘Š)
10 eqid 2732 . . . . . . . 8 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
11 lshpdisj.n . . . . . . . 8 𝑁 = (LSpanβ€˜π‘Š)
127, 8, 9, 10, 11lspsnel 20606 . . . . . . 7 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (𝑣 ∈ (π‘β€˜{𝑋}) ↔ βˆƒπ‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))𝑣 = (π‘˜( ·𝑠 β€˜π‘Š)𝑋)))
134, 6, 12syl2anc 584 . . . . . 6 ((πœ‘ ∧ 𝑣 ∈ π‘ˆ) β†’ (𝑣 ∈ (π‘β€˜{𝑋}) ↔ βˆƒπ‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))𝑣 = (π‘˜( ·𝑠 β€˜π‘Š)𝑋)))
14 lshpdisj.p . . . . . . . . . . . . . . . . 17 βŠ• = (LSSumβ€˜π‘Š)
15 lshpdisj.h . . . . . . . . . . . . . . . . 17 𝐻 = (LSHypβ€˜π‘Š)
16 lshpdisj.u . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ π‘ˆ ∈ 𝐻)
17 lshpdisj.e . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ (π‘ˆ βŠ• (π‘β€˜{𝑋})) = 𝑉)
189, 11, 14, 15, 3, 16, 5, 17lshpnel 37841 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ Β¬ 𝑋 ∈ π‘ˆ)
1918ad2antrr 724 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β‰  0 ) β†’ Β¬ 𝑋 ∈ π‘ˆ)
20 lshpdisj.o . . . . . . . . . . . . . . . 16 0 = (0gβ€˜π‘Š)
21 eqid 2732 . . . . . . . . . . . . . . . 16 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
221ad2antrr 724 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β‰  0 ) β†’ π‘Š ∈ LVec)
2321, 15, 3, 16lshplss 37839 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ π‘ˆ ∈ (LSubSpβ€˜π‘Š))
2423ad2antrr 724 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β‰  0 ) β†’ π‘ˆ ∈ (LSubSpβ€˜π‘Š))
255ad2antrr 724 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β‰  0 ) β†’ 𝑋 ∈ 𝑉)
263adantr 481 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ π‘Š ∈ LMod)
27 simpr 485 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
285adantr 481 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ 𝑋 ∈ 𝑉)
299, 10, 7, 8, 11, 26, 27, 28lspsneli 20604 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) ∈ (π‘β€˜{𝑋}))
3029adantr 481 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β‰  0 ) β†’ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) ∈ (π‘β€˜{𝑋}))
31 simpr 485 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β‰  0 ) β†’ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β‰  0 )
329, 20, 21, 11, 22, 24, 25, 30, 31lspsnel4 20729 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β‰  0 ) β†’ (𝑋 ∈ π‘ˆ ↔ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) ∈ π‘ˆ))
3319, 32mtbid 323 . . . . . . . . . . . . . 14 (((πœ‘ ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β‰  0 ) β†’ Β¬ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) ∈ π‘ˆ)
3433ex 413 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ ((π‘˜( ·𝑠 β€˜π‘Š)𝑋) β‰  0 β†’ Β¬ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) ∈ π‘ˆ))
3534necon4ad 2959 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ ((π‘˜( ·𝑠 β€˜π‘Š)𝑋) ∈ π‘ˆ β†’ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) = 0 ))
36 eleq1 2821 . . . . . . . . . . . . 13 (𝑣 = (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β†’ (𝑣 ∈ π‘ˆ ↔ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) ∈ π‘ˆ))
37 eqeq1 2736 . . . . . . . . . . . . 13 (𝑣 = (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β†’ (𝑣 = 0 ↔ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) = 0 ))
3836, 37imbi12d 344 . . . . . . . . . . . 12 (𝑣 = (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β†’ ((𝑣 ∈ π‘ˆ β†’ 𝑣 = 0 ) ↔ ((π‘˜( ·𝑠 β€˜π‘Š)𝑋) ∈ π‘ˆ β†’ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) = 0 )))
3935, 38syl5ibrcom 246 . . . . . . . . . . 11 ((πœ‘ ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ (𝑣 = (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β†’ (𝑣 ∈ π‘ˆ β†’ 𝑣 = 0 )))
4039ex 413 . . . . . . . . . 10 (πœ‘ β†’ (π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) β†’ (𝑣 = (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β†’ (𝑣 ∈ π‘ˆ β†’ 𝑣 = 0 ))))
4140com23 86 . . . . . . . . 9 (πœ‘ β†’ (𝑣 = (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β†’ (π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) β†’ (𝑣 ∈ π‘ˆ β†’ 𝑣 = 0 ))))
4241com24 95 . . . . . . . 8 (πœ‘ β†’ (𝑣 ∈ π‘ˆ β†’ (π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) β†’ (𝑣 = (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β†’ 𝑣 = 0 ))))
4342imp31 418 . . . . . . 7 (((πœ‘ ∧ 𝑣 ∈ π‘ˆ) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ (𝑣 = (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β†’ 𝑣 = 0 ))
4443rexlimdva 3155 . . . . . 6 ((πœ‘ ∧ 𝑣 ∈ π‘ˆ) β†’ (βˆƒπ‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))𝑣 = (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β†’ 𝑣 = 0 ))
4513, 44sylbid 239 . . . . 5 ((πœ‘ ∧ 𝑣 ∈ π‘ˆ) β†’ (𝑣 ∈ (π‘β€˜{𝑋}) β†’ 𝑣 = 0 ))
4645expimpd 454 . . . 4 (πœ‘ β†’ ((𝑣 ∈ π‘ˆ ∧ 𝑣 ∈ (π‘β€˜{𝑋})) β†’ 𝑣 = 0 ))
47 elin 3963 . . . 4 (𝑣 ∈ (π‘ˆ ∩ (π‘β€˜{𝑋})) ↔ (𝑣 ∈ π‘ˆ ∧ 𝑣 ∈ (π‘β€˜{𝑋})))
48 velsn 4643 . . . 4 (𝑣 ∈ { 0 } ↔ 𝑣 = 0 )
4946, 47, 483imtr4g 295 . . 3 (πœ‘ β†’ (𝑣 ∈ (π‘ˆ ∩ (π‘β€˜{𝑋})) β†’ 𝑣 ∈ { 0 }))
5049ssrdv 3987 . 2 (πœ‘ β†’ (π‘ˆ ∩ (π‘β€˜{𝑋})) βŠ† { 0 })
519, 21, 11lspsncl 20580 . . . . 5 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜{𝑋}) ∈ (LSubSpβ€˜π‘Š))
523, 5, 51syl2anc 584 . . . 4 (πœ‘ β†’ (π‘β€˜{𝑋}) ∈ (LSubSpβ€˜π‘Š))
5321lssincl 20568 . . . 4 ((π‘Š ∈ LMod ∧ π‘ˆ ∈ (LSubSpβ€˜π‘Š) ∧ (π‘β€˜{𝑋}) ∈ (LSubSpβ€˜π‘Š)) β†’ (π‘ˆ ∩ (π‘β€˜{𝑋})) ∈ (LSubSpβ€˜π‘Š))
543, 23, 52, 53syl3anc 1371 . . 3 (πœ‘ β†’ (π‘ˆ ∩ (π‘β€˜{𝑋})) ∈ (LSubSpβ€˜π‘Š))
5520, 21lss0ss 20551 . . 3 ((π‘Š ∈ LMod ∧ (π‘ˆ ∩ (π‘β€˜{𝑋})) ∈ (LSubSpβ€˜π‘Š)) β†’ { 0 } βŠ† (π‘ˆ ∩ (π‘β€˜{𝑋})))
563, 54, 55syl2anc 584 . 2 (πœ‘ β†’ { 0 } βŠ† (π‘ˆ ∩ (π‘β€˜{𝑋})))
5750, 56eqssd 3998 1 (πœ‘ β†’ (π‘ˆ ∩ (π‘β€˜{𝑋})) = { 0 })
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆƒwrex 3070   ∩ cin 3946   βŠ† wss 3947  {csn 4627  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  Scalarcsca 17196   ·𝑠 cvsca 17197  0gc0g 17381  LSSumclsm 19496  LModclmod 20463  LSubSpclss 20534  LSpanclspn 20574  LVecclvec 20705  LSHypclsh 37833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-tpos 8207  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-submnd 18668  df-grp 18818  df-minusg 18819  df-sbg 18820  df-subg 18997  df-lsm 19498  df-mgp 19982  df-ur 19999  df-ring 20051  df-oppr 20142  df-dvdsr 20163  df-unit 20164  df-invr 20194  df-drng 20309  df-lmod 20465  df-lss 20535  df-lsp 20575  df-lvec 20706  df-lshyp 37835
This theorem is referenced by:  lshpsmreu  37967  lshpkrlem5  37972
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