Theorem islshp 32982

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 32981	. . 3 |- ( W e. X -> H = { s e. S | ( s =/= V /\ E. v e. V ( N ` ( s u. { v } ) ) = V ) } )
6	5	eleq2d 2524	. 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 2733	. . . . 5 |- ( s = U -> ( s =/= V <-> U =/= V ) )
8		uneq1 3614	. . . . . . . 8 |- ( s = U -> ( s u. { v } ) = ( U u. { v } ) )
9	8	fveq2d 5806	. . . . . . 7 |- ( s = U -> ( N ` ( s u. { v } ) ) = ( N ` ( U u. { v } ) ) )
10	9	eqeq1d 2456	. . . . . 6 |- ( s = U -> ( ( N ` ( s u. { v } ) ) = V <-> ( N ` ( U u. { v } ) ) = V ) )
11	10	rexbidv 2868	. . . . 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 3224	. . 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 1370   e. wcel 1758   =/= wne 2648   E.wrex 2800   {crab 2803   u. cun 3437   {csn 3988   ` cfv 5529   Basecbs 14295   LSubSpclss 17139   LSpanclspn 17178  LSHypclsh 32978

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-lshyp 32980

This theorem is referenced by:  islshpsm  32983  lshplss  32984  lshpne  32985  lshpnel2N  32988  lkrshp  33108  lshpset2N  33122  dochsatshp  35454