Theorem lshpne 34269

Description: A hyperplane is not equal to the vector space. (Contributed by NM, 4-Jul-2014.)

Hypotheses

Ref	Expression
lshpne.v	|- V = ( Base ` W )
lshpne.h	|- H = (LSHyp ` W )
lshpne.w	|- ( ph -> W e. LMod )
lshpne.u	|- ( ph -> U e. H )

Assertion

Ref	Expression
lshpne	|- ( ph -> U =/= V )

Proof of Theorem lshpne

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lshpne.u	. . 3 |- ( ph -> U e. H )
2		lshpne.w	. . . 4 |- ( ph -> W e. LMod )
3		lshpne.v	. . . . 5 |- V = ( Base ` W )
4		eqid 2622	. . . . 5 |- ( LSpan ` W ) = ( LSpan ` W )
5		eqid 2622	. . . . 5 |- ( LSubSp ` W ) = ( LSubSp ` W )
6		lshpne.h	. . . . 5 |- H = (LSHyp ` W )
7	3, 4, 5, 6	islshp 34266	. . . 4 |- ( W e. LMod -> ( U e. H <-> ( U e. ( LSubSp ` W ) /\ U =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( U u. { v } ) ) = V ) ) )
8	2, 7	syl 17	. . 3 |- ( ph -> ( U e. H <-> ( U e. ( LSubSp ` W ) /\ U =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( U u. { v } ) ) = V ) ) )
9	1, 8	mpbid 222	. 2 |- ( ph -> ( U e. ( LSubSp ` W ) /\ U =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( U u. { v } ) ) = V ) )
10	9	simp2d 1074	1 |- ( ph -> U =/= V )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   u. cun 3572   {csn 4177   ` cfv 5888   Basecbs 15857   LModclmod 18863   LSubSpclss 18932   LSpanclspn 18971  LSHypclsh 34262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-lshyp 34264

This theorem is referenced by:  lshpnel  34270  lshpcmp  34275  lkrshp3  34393  lkrshp4  34395  dochshpncl  36673  dochlkr  36674  dochkrshp  36675  dochsatshpb  36741