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Theorem lshpdisj 37478
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 20583 . . . . . . . . 9 (π‘Š ∈ LVec β†’ π‘Š ∈ LMod)
31, 2syl 17 . . . . . . . 8 (πœ‘ β†’ π‘Š ∈ LMod)
43adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝑣 ∈ π‘ˆ) β†’ π‘Š ∈ LMod)
5 lshpdisj.x . . . . . . . 8 (πœ‘ β†’ 𝑋 ∈ 𝑉)
65adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝑣 ∈ π‘ˆ) β†’ 𝑋 ∈ 𝑉)
7 eqid 2737 . . . . . . . 8 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
8 eqid 2737 . . . . . . . 8 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
9 lshpdisj.v . . . . . . . 8 𝑉 = (Baseβ€˜π‘Š)
10 eqid 2737 . . . . . . . 8 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
11 lshpdisj.n . . . . . . . 8 𝑁 = (LSpanβ€˜π‘Š)
127, 8, 9, 10, 11lspsnel 20480 . . . . . . 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 585 . . . . . 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 37474 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ Β¬ 𝑋 ∈ π‘ˆ)
1918ad2antrr 725 . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . 16 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
221ad2antrr 725 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β‰  0 ) β†’ π‘Š ∈ LVec)
2321, 15, 3, 16lshplss 37472 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ π‘ˆ ∈ (LSubSpβ€˜π‘Š))
2423ad2antrr 725 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β‰  0 ) β†’ π‘ˆ ∈ (LSubSpβ€˜π‘Š))
255ad2antrr 725 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β‰  0 ) β†’ 𝑋 ∈ 𝑉)
263adantr 482 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ π‘Š ∈ LMod)
27 simpr 486 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
285adantr 482 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ 𝑋 ∈ 𝑉)
299, 10, 7, 8, 11, 26, 27, 28lspsneli 20478 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) ∈ (π‘β€˜{𝑋}))
3029adantr 482 . . . . . . . . . . . . . . . 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 486 . . . . . . . . . . . . . . . 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 20601 . . . . . . . . . . . . . . 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 324 . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . 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 2963 . . . . . . . . . . . 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 2826 . . . . . . . . . . . . 13 (𝑣 = (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β†’ (𝑣 ∈ π‘ˆ ↔ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) ∈ π‘ˆ))
37 eqeq1 2741 . . . . . . . . . . . . 13 (𝑣 = (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β†’ (𝑣 = 0 ↔ (π‘˜( ·𝑠 β€˜π‘Š)𝑋) = 0 ))
3836, 37imbi12d 345 . . . . . . . . . . . 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 247 . . . . . . . . . . 11 ((πœ‘ ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ (𝑣 = (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β†’ (𝑣 ∈ π‘ˆ β†’ 𝑣 = 0 )))
4039ex 414 . . . . . . . . . 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 419 . . . . . . 7 (((πœ‘ ∧ 𝑣 ∈ π‘ˆ) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ (𝑣 = (π‘˜( ·𝑠 β€˜π‘Š)𝑋) β†’ 𝑣 = 0 ))
4443rexlimdva 3153 . . . . . 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 455 . . . 4 (πœ‘ β†’ ((𝑣 ∈ π‘ˆ ∧ 𝑣 ∈ (π‘β€˜{𝑋})) β†’ 𝑣 = 0 ))
47 elin 3931 . . . 4 (𝑣 ∈ (π‘ˆ ∩ (π‘β€˜{𝑋})) ↔ (𝑣 ∈ π‘ˆ ∧ 𝑣 ∈ (π‘β€˜{𝑋})))
48 velsn 4607 . . . 4 (𝑣 ∈ { 0 } ↔ 𝑣 = 0 )
4946, 47, 483imtr4g 296 . . 3 (πœ‘ β†’ (𝑣 ∈ (π‘ˆ ∩ (π‘β€˜{𝑋})) β†’ 𝑣 ∈ { 0 }))
5049ssrdv 3955 . 2 (πœ‘ β†’ (π‘ˆ ∩ (π‘β€˜{𝑋})) βŠ† { 0 })
519, 21, 11lspsncl 20454 . . . . 5 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜{𝑋}) ∈ (LSubSpβ€˜π‘Š))
523, 5, 51syl2anc 585 . . . 4 (πœ‘ β†’ (π‘β€˜{𝑋}) ∈ (LSubSpβ€˜π‘Š))
5321lssincl 20442 . . . 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 1372 . . 3 (πœ‘ β†’ (π‘ˆ ∩ (π‘β€˜{𝑋})) ∈ (LSubSpβ€˜π‘Š))
5520, 21lss0ss 20425 . . 3 ((π‘Š ∈ LMod ∧ (π‘ˆ ∩ (π‘β€˜{𝑋})) ∈ (LSubSpβ€˜π‘Š)) β†’ { 0 } βŠ† (π‘ˆ ∩ (π‘β€˜{𝑋})))
563, 54, 55syl2anc 585 . 2 (πœ‘ β†’ { 0 } βŠ† (π‘ˆ ∩ (π‘β€˜{𝑋})))
5750, 56eqssd 3966 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 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆƒwrex 3074   ∩ cin 3914   βŠ† wss 3915  {csn 4591  β€˜cfv 6501  (class class class)co 7362  Basecbs 17090  Scalarcsca 17143   ·𝑠 cvsca 17144  0gc0g 17328  LSSumclsm 19423  LModclmod 20338  LSubSpclss 20408  LSpanclspn 20448  LVecclvec 20579  LSHypclsh 37466
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-tpos 8162  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-3 12224  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-ress 17120  df-plusg 17153  df-mulr 17154  df-0g 17330  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-submnd 18609  df-grp 18758  df-minusg 18759  df-sbg 18760  df-subg 18932  df-lsm 19425  df-mgp 19904  df-ur 19921  df-ring 19973  df-oppr 20056  df-dvdsr 20077  df-unit 20078  df-invr 20108  df-drng 20201  df-lmod 20340  df-lss 20409  df-lsp 20449  df-lvec 20580  df-lshyp 37468
This theorem is referenced by:  lshpsmreu  37600  lshpkrlem5  37605
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