Theorem islshp 34266

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 34265	. . 3 |- ( W e. X -> H = { s e. S | ( s =/= V /\ E. v e. V ( N ` ( s u. { v } ) ) = V ) } )
6	5	eleq2d 2687	. 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 2856	. . . . 5 |- ( s = U -> ( s =/= V <-> U =/= V ) )
8		uneq1 3760	. . . . . . . 8 |- ( s = U -> ( s u. { v } ) = ( U u. { v } ) )
9	8	fveq2d 6195	. . . . . . 7 |- ( s = U -> ( N ` ( s u. { v } ) ) = ( N ` ( U u. { v } ) ) )
10	9	eqeq1d 2624	. . . . . 6 |- ( s = U -> ( ( N ` ( s u. { v } ) ) = V <-> ( N ` ( U u. { v } ) ) = V ) )
11	10	rexbidv 3052	. . . . 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 747	. . . 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 3363	. . 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 1042	. . 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 267	. 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 276	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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   u. cun 3572   {csn 4177   ` cfv 5888   Basecbs 15857   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:  islshpsm  34267  lshplss  34268  lshpne  34269  lshpnel2N  34272  lkrshp  34392  lshpset2N  34406  dochsatshp  36740