Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lshplss Structured version   Unicode version

Theorem lshplss 29717
 Description: A hyperplane is a subspace. (Contributed by NM, 3-Jul-2014.)
Hypotheses
Ref Expression
lshplss.s
lshplss.h LSHyp
lshplss.w
lshplss.u
Assertion
Ref Expression
lshplss

Proof of Theorem lshplss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 lshplss.u . . 3
2 lshplss.w . . . 4
3 eqid 2436 . . . . 5
4 eqid 2436 . . . . 5
5 lshplss.s . . . . 5
6 lshplss.h . . . . 5 LSHyp
73, 4, 5, 6islshp 29715 . . . 4
82, 7syl 16 . . 3
91, 8mpbid 202 . 2
109simp1d 969 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   w3a 936   wceq 1652   wcel 1725   wne 2599  wrex 2699   cun 3311  csn 3807  cfv 5447  cbs 13462  clmod 15943  clss 16001  clspn 16040  LSHypclsh 29711 This theorem is referenced by:  lshpnel  29719  lshpnelb  29720  lshpne0  29722  lshpdisj  29723  lshpcmp  29724  lshpsmreu  29845  lshpkrlem1  29846  lshpkrlem5  29850  lshpkr  29853  dochshpncl  32120  dochshpsat  32190  lclkrlem2f  32248 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-iota 5411  df-fun 5449  df-fv 5455  df-lshyp 29713
 Copyright terms: Public domain W3C validator