Theorem islshp 32545

Description: The predicate "is a hyperplane" (of a left module or left vector space). (Contributed by NM, 29-Jun-2014.)

Hypotheses

Ref	Expression
lshpset.v	|- V = ( Base ` W )
lshpset.n	|- N = ( LSpan ` W )
lshpset.s	|- S = ( LSubSp ` W )
lshpset.h	|- H = (LSHyp ` W )

Assertion

Ref	Expression
islshp	|- ( W e. X -> ( U e. H <-> ( U e. S /\ U =/= V /\ E. v e. V ( N ` ( U u. { v } ) ) = V ) ) )

Distinct variable groups:   v, V   v, W   v, U

Allowed substitution hints:   S( v)   H( v)   N( v)   X( v)

Proof of Theorem islshp

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lshpset.v	. . . 4 |- V = ( Base ` W )
2		lshpset.n	. . . 4 |- N = ( LSpan ` W )
3		lshpset.s	. . . 4 |- S = ( LSubSp ` W )
4		lshpset.h	. . . 4 |- H = (LSHyp ` W )
5	1, 2, 3, 4	lshpset 32544	. . 3 |- ( W e. X -> H = { s e. S | ( s =/= V /\ E. v e. V ( N ` ( s u. { v } ) ) = V ) } )
6	5	eleq2d 2514	. 2 |- ( W e. X -> ( U e. H <-> U e. { s e. S | ( s =/= V /\ E. v e. V ( N ` ( s u. { v } ) ) = V ) } ) )
7		neeq1 2686	. . . . 5 |- ( s = U -> ( s =/= V <-> U =/= V ) )
8		uneq1 3581	. . . . . . . 8 |- ( s = U -> ( s u. { v } ) = ( U u. { v } ) )
9	8	fveq2d 5869	. . . . . . 7 |- ( s = U -> ( N ` ( s u. { v } ) ) = ( N ` ( U u. { v } ) ) )
10	9	eqeq1d 2453	. . . . . 6 |- ( s = U -> ( ( N ` ( s u. { v } ) ) = V <-> ( N ` ( U u. { v } ) ) = V ) )
11	10	rexbidv 2901	. . . . 5 |- ( s = U -> ( E. v e. V ( N ` ( s u. { v } ) ) = V <-> E. v e. V ( N ` ( U u. { v } ) ) = V ) )
12	7, 11	anbi12d 717	. . . 4 |- ( s = U -> ( ( s =/= V /\ E. v e. V ( N ` ( s u. { v } ) ) = V ) <-> ( U =/= V /\ E. v e. V ( N ` ( U u. { v } ) ) = V ) ) )
13	12	elrab 3196	. . 3 |- ( U e. { s e. S | ( s =/= V /\ E. v e. V ( N ` ( s u. { v } ) ) = V ) } <-> ( U e. S /\ ( U =/= V /\ E. v e. V ( N ` ( U u. { v } ) ) = V ) ) )
14		3anass 989	. . 3 |- ( ( U e. S /\ U =/= V /\ E. v e. V ( N ` ( U u. { v } ) ) = V ) <-> ( U e. S /\ ( U =/= V /\ E. v e. V ( N ` ( U u. { v } ) ) = V ) ) )
15	13, 14	bitr4i 256	. 2 |- ( U e. { s e. S | ( s =/= V /\ E. v e. V ( N ` ( s u. { v } ) ) = V ) } <-> ( U e. S /\ U =/= V /\ E. v e. V ( N ` ( U u. { v } ) ) = V ) )
16	6, 15	syl6bb 265	1 |- ( W e. X -> ( U e. H <-> ( U e. S /\ U =/= V /\ E. v e. V ( N ` ( U u. { v } ) ) = V ) ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   E.wrex 2738   {crab 2741   u. cun 3402   {csn 3968   ` cfv 5582   Basecbs 15121   LSubSpclss 18155   LSpanclspn 18194  LSHypclsh 32541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-iota 5546  df-fun 5584  df-fv 5590  df-lshyp 32543

This theorem is referenced by:  islshpsm  32546  lshplss  32547  lshpne  32548  lshpnel2N  32551  lkrshp  32671  lshpset2N  32685  dochsatshp  35019