Theorem lshpset 34265

Description: The set of all hyperplanes of a left module or left vector space. The vector v is called a generating vector for the hyperplane. (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
lshpset	|- ( W e. X -> H = { s e. S | ( s =/= V /\ E. v e. V ( N ` ( s u. { v } ) ) = V ) } )

Distinct variable groups:   S, s   v, V   v, s, W

Allowed substitution hints:   S( v)   H( v, s)   N( v, s)   V( s)   X( v, s)

Proof of Theorem lshpset

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lshpset.h	. 2 |- H = (LSHyp ` W )
2		elex 3212	. . 3 |- ( W e. X -> W e. _V )
3		fveq2 6191	. . . . . 6 |- ( w = W -> ( LSubSp ` w ) = ( LSubSp ` W ) )
4		lshpset.s	. . . . . 6 |- S = ( LSubSp ` W )
5	3, 4	syl6eqr 2674	. . . . 5 |- ( w = W -> ( LSubSp ` w ) = S )
6		fveq2 6191	. . . . . . . 8 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
7		lshpset.v	. . . . . . . 8 |- V = ( Base ` W )
8	6, 7	syl6eqr 2674	. . . . . . 7 |- ( w = W -> ( Base ` w ) = V )
9	8	neeq2d 2854	. . . . . 6 |- ( w = W -> ( s =/= ( Base ` w ) <-> s =/= V ) )
10		fveq2 6191	. . . . . . . . . 10 |- ( w = W -> ( LSpan ` w ) = ( LSpan ` W ) )
11		lshpset.n	. . . . . . . . . 10 |- N = ( LSpan ` W )
12	10, 11	syl6eqr 2674	. . . . . . . . 9 |- ( w = W -> ( LSpan ` w ) = N )
13	12	fveq1d 6193	. . . . . . . 8 |- ( w = W -> ( ( LSpan ` w ) ` ( s u. { v } ) ) = ( N ` ( s u. { v } ) ) )
14	13, 8	eqeq12d 2637	. . . . . . 7 |- ( w = W -> ( ( ( LSpan ` w ) ` ( s u. { v } ) ) = ( Base ` w ) <-> ( N ` ( s u. { v } ) ) = V ) )
15	8, 14	rexeqbidv 3153	. . . . . 6 |- ( w = W -> ( E. v e. ( Base ` w ) ( ( LSpan ` w ) ` ( s u. { v } ) ) = ( Base ` w ) <-> E. v e. V ( N ` ( s u. { v } ) ) = V ) )
16	9, 15	anbi12d 747	. . . . 5 |- ( w = W -> ( ( s =/= ( Base ` w ) /\ E. v e. ( Base ` w ) ( ( LSpan ` w ) ` ( s u. { v } ) ) = ( Base ` w ) ) <-> ( s =/= V /\ E. v e. V ( N ` ( s u. { v } ) ) = V ) ) )
17	5, 16	rabeqbidv 3195	. . . 4 |- ( w = W -> { s e. ( LSubSp ` w ) | ( s =/= ( Base ` w ) /\ E. v e. ( Base ` w ) ( ( LSpan ` w ) ` ( s u. { v } ) ) = ( Base ` w ) ) } = { s e. S | ( s =/= V /\ E. v e. V ( N ` ( s u. { v } ) ) = V ) } )
18		df-lshyp 34264	. . . 4 |- LSHyp = ( w e. _V |-> { s e. ( LSubSp ` w ) | ( s =/= ( Base ` w ) /\ E. v e. ( Base ` w ) ( ( LSpan ` w ) ` ( s u. { v } ) ) = ( Base ` w ) ) } )
19		fvex 6201	. . . . . 6 |- ( LSubSp ` W ) e. _V
20	4, 19	eqeltri 2697	. . . . 5 |- S e. _V
21	20	rabex 4813	. . . 4 |- { s e. S | ( s =/= V /\ E. v e. V ( N ` ( s u. { v } ) ) = V ) } e. _V
22	17, 18, 21	fvmpt 6282	. . 3 |- ( W e. _V -> (LSHyp ` W ) = { s e. S | ( s =/= V /\ E. v e. V ( N ` ( s u. { v } ) ) = V ) } )
23	2, 22	syl 17	. 2 |- ( W e. X -> (LSHyp ` W ) = { s e. S | ( s =/= V /\ E. v e. V ( N ` ( s u. { v } ) ) = V ) } )
24	1, 23	syl5eq 2668	1 |- ( W e. X -> H = { s e. S | ( s =/= V /\ E. v e. V ( N ` ( s u. { v } ) ) = V ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   _Vcvv 3200   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:  islshp  34266