Theorem islshp 31977

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 31976	. . 3 |- ( W e. X -> H = { s e. S | ( s =/= V /\ E. v e. V ( N ` ( s u. { v } ) ) = V ) } )
6	5	eleq2d 2472	. 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 2684	. . . . 5 |- ( s = U -> ( s =/= V <-> U =/= V ) )
8		uneq1 3589	. . . . . . . 8 |- ( s = U -> ( s u. { v } ) = ( U u. { v } ) )
9	8	fveq2d 5852	. . . . . . 7 |- ( s = U -> ( N ` ( s u. { v } ) ) = ( N ` ( U u. { v } ) ) )
10	9	eqeq1d 2404	. . . . . 6 |- ( s = U -> ( ( N ` ( s u. { v } ) ) = V <-> ( N ` ( U u. { v } ) ) = V ) )
11	10	rexbidv 2917	. . . . 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 709	. . . 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 3206	. . 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 978	. . 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 252	. 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 261	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 184   /\ wa 367   /\ w3a 974   = wceq 1405   e. wcel 1842   =/= wne 2598   E.wrex 2754   {crab 2757   u. cun 3411   {csn 3971   ` cfv 5568   Basecbs 14839   LSubSpclss 17896   LSpanclspn 17935  LSHypclsh 31973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576  df-lshyp 31975

This theorem is referenced by:  islshpsm  31978  lshplss  31979  lshpne  31980  lshpnel2N  31983  lkrshp  32103  lshpset2N  32117  dochsatshp  34451