Theorem islshp 32464

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 32463	. . 3 |- ( W e. X -> H = { s e. S | ( s =/= V /\ E. v e. V ( N ` ( s u. { v } ) ) = V ) } )
6	5	eleq2d 2505	. 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 2611	. . . . 5 |- ( s = U -> ( s =/= V <-> U =/= V ) )
8		uneq1 3498	. . . . . . . 8 |- ( s = U -> ( s u. { v } ) = ( U u. { v } ) )
9	8	fveq2d 5690	. . . . . . 7 |- ( s = U -> ( N ` ( s u. { v } ) ) = ( N ` ( U u. { v } ) ) )
10	9	eqeq1d 2446	. . . . . 6 |- ( s = U -> ( ( N ` ( s u. { v } ) ) = V <-> ( N ` ( U u. { v } ) ) = V ) )
11	10	rexbidv 2731	. . . . 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 710	. . . 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 3112	. . 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 969	. . 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 369   /\ w3a 965   = wceq 1369   e. wcel 1756   =/= wne 2601   E.wrex 2711   {crab 2714   u. cun 3321   {csn 3872   ` cfv 5413   Basecbs 14166   LSubSpclss 16990   LSpanclspn 17029  LSHypclsh 32460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-iota 5376  df-fun 5415  df-fv 5421  df-lshyp 32462

This theorem is referenced by:  islshpsm  32465  lshplss  32466  lshpne  32467  lshpnel2N  32470  lkrshp  32590  lshpset2N  32604  dochsatshp  34936