Theorem lhpset 35281

Description: The set of co-atoms (lattice hyperplanes). (Contributed by NM, 11-May-2012.)

Hypotheses

Ref	Expression
lhpset.b	|- B = ( Base ` K )
lhpset.u	|- .1. = ( 1. ` K )
lhpset.c	|- C = ( <o ` K )
lhpset.h	|- H = ( LHyp ` K )

Assertion

Ref	Expression
lhpset	|- ( K e. A -> H = { w e. B | w C .1. } )

Distinct variable groups:   w, B   w, C   w, K   w, .1.

Allowed substitution hints:   A( w)   H( w)

Proof of Theorem lhpset

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( K e. A -> K e. _V )
2		lhpset.h	. . 3 |- H = ( LHyp ` K )
3		fveq2 6191	. . . . . 6 |- ( k = K -> ( Base ` k ) = ( Base ` K ) )
4		lhpset.b	. . . . . 6 |- B = ( Base ` K )
5	3, 4	syl6eqr 2674	. . . . 5 |- ( k = K -> ( Base ` k ) = B )
6		eqidd 2623	. . . . . 6 |- ( k = K -> w = w )
7		fveq2 6191	. . . . . . 7 |- ( k = K -> ( <o ` k ) = ( <o ` K ) )
8		lhpset.c	. . . . . . 7 |- C = ( <o ` K )
9	7, 8	syl6eqr 2674	. . . . . 6 |- ( k = K -> ( <o ` k ) = C )
10		fveq2 6191	. . . . . . 7 |- ( k = K -> ( 1. ` k ) = ( 1. ` K ) )
11		lhpset.u	. . . . . . 7 |- .1. = ( 1. ` K )
12	10, 11	syl6eqr 2674	. . . . . 6 |- ( k = K -> ( 1. ` k ) = .1. )
13	6, 9, 12	breq123d 4667	. . . . 5 |- ( k = K -> ( w ( <o ` k ) ( 1. ` k ) <-> w C .1. ) )
14	5, 13	rabeqbidv 3195	. . . 4 |- ( k = K -> { w e. ( Base ` k ) | w ( <o ` k ) ( 1. ` k ) } = { w e. B | w C .1. } )
15		df-lhyp 35274	. . . 4 |- LHyp = ( k e. _V |-> { w e. ( Base ` k ) | w ( <o ` k ) ( 1. ` k ) } )
16		fvex 6201	. . . . . 6 |- ( Base ` K ) e. _V
17	4, 16	eqeltri 2697	. . . . 5 |- B e. _V
18	17	rabex 4813	. . . 4 |- { w e. B | w C .1. } e. _V
19	14, 15, 18	fvmpt 6282	. . 3 |- ( K e. _V -> ( LHyp ` K ) = { w e. B | w C .1. } )
20	2, 19	syl5eq 2668	. 2 |- ( K e. _V -> H = { w e. B | w C .1. } )
21	1, 20	syl 17	1 |- ( K e. A -> H = { w e. B | w C .1. } )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   class class class wbr 4653   ` cfv 5888   Basecbs 15857   1.cp1 17038   <o ccvr 34549   LHypclh 35270

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-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-lhyp 35274

This theorem is referenced by:  islhp  35282