Theorem psubclsetN 35222

Description: The set of closed projective subspaces in a Hilbert lattice. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
psubclset.a	|- A = ( Atoms ` K )
psubclset.p	|- ._|_ = ( _|_P ` K )
psubclset.c	|- C = ( PSubCl ` K )

Assertion

Ref	Expression
psubclsetN	|- ( K e. B -> C = { s | ( s C_ A /\ ( ._|_ ` ( ._|_ ` s ) ) = s ) } )

Distinct variable groups:   A, s   K, s

Allowed substitution hints:   B( s)   C( s)   ._|_ ( s)

Proof of Theorem psubclsetN

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( K e. B -> K e. _V )
2		psubclset.c	. . 3 |- C = ( PSubCl ` K )
3		fveq2 6191	. . . . . . . 8 |- ( k = K -> ( Atoms ` k ) = ( Atoms ` K ) )
4		psubclset.a	. . . . . . . 8 |- A = ( Atoms ` K )
5	3, 4	syl6eqr 2674	. . . . . . 7 |- ( k = K -> ( Atoms ` k ) = A )
6	5	sseq2d 3633	. . . . . 6 |- ( k = K -> ( s C_ ( Atoms ` k ) <-> s C_ A ) )
7		fveq2 6191	. . . . . . . . 9 |- ( k = K -> ( _|_P ` k ) = ( _|_P ` K ) )
8		psubclset.p	. . . . . . . . 9 |- ._|_ = ( _|_P ` K )
9	7, 8	syl6eqr 2674	. . . . . . . 8 |- ( k = K -> ( _|_P ` k ) = ._|_ )
10	9	fveq1d 6193	. . . . . . . 8 |- ( k = K -> ( ( _|_P ` k ) ` s ) = ( ._|_ ` s ) )
11	9, 10	fveq12d 6197	. . . . . . 7 |- ( k = K -> ( ( _|_P ` k ) ` ( ( _|_P ` k ) ` s ) ) = ( ._|_ ` ( ._|_ ` s ) ) )
12	11	eqeq1d 2624	. . . . . 6 |- ( k = K -> ( ( ( _|_P ` k ) ` ( ( _|_P ` k ) ` s ) ) = s <-> ( ._|_ ` ( ._|_ ` s ) ) = s ) )
13	6, 12	anbi12d 747	. . . . 5 |- ( k = K -> ( ( s C_ ( Atoms ` k ) /\ ( ( _|_P ` k ) ` ( ( _|_P ` k ) ` s ) ) = s ) <-> ( s C_ A /\ ( ._|_ ` ( ._|_ ` s ) ) = s ) ) )
14	13	abbidv 2741	. . . 4 |- ( k = K -> { s | ( s C_ ( Atoms ` k ) /\ ( ( _|_P ` k ) ` ( ( _|_P ` k ) ` s ) ) = s ) } = { s | ( s C_ A /\ ( ._|_ ` ( ._|_ ` s ) ) = s ) } )
15		df-psubclN 35221	. . . 4 |- PSubCl = ( k e. _V |-> { s | ( s C_ ( Atoms ` k ) /\ ( ( _|_P ` k ) ` ( ( _|_P ` k ) ` s ) ) = s ) } )
16		fvex 6201	. . . . . . 7 |- ( Atoms ` K ) e. _V
17	4, 16	eqeltri 2697	. . . . . 6 |- A e. _V
18	17	pwex 4848	. . . . 5 |- ~P A e. _V
19		selpw 4165	. . . . . . . 8 |- ( s e. ~P A <-> s C_ A )
20	19	anbi1i 731	. . . . . . 7 |- ( ( s e. ~P A /\ ( ._|_ ` ( ._|_ ` s ) ) = s ) <-> ( s C_ A /\ ( ._|_ ` ( ._|_ ` s ) ) = s ) )
21	20	abbii 2739	. . . . . 6 |- { s | ( s e. ~P A /\ ( ._|_ ` ( ._|_ ` s ) ) = s ) } = { s | ( s C_ A /\ ( ._|_ ` ( ._|_ ` s ) ) = s ) }
22		ssab2 3686	. . . . . 6 |- { s | ( s e. ~P A /\ ( ._|_ ` ( ._|_ ` s ) ) = s ) } C_ ~P A
23	21, 22	eqsstr3i 3636	. . . . 5 |- { s | ( s C_ A /\ ( ._|_ ` ( ._|_ ` s ) ) = s ) } C_ ~P A
24	18, 23	ssexi 4803	. . . 4 |- { s | ( s C_ A /\ ( ._|_ ` ( ._|_ ` s ) ) = s ) } e. _V
25	14, 15, 24	fvmpt 6282	. . 3 |- ( K e. _V -> ( PSubCl ` K ) = { s | ( s C_ A /\ ( ._|_ ` ( ._|_ ` s ) ) = s ) } )
26	2, 25	syl5eq 2668	. 2 |- ( K e. _V -> C = { s | ( s C_ A /\ ( ._|_ ` ( ._|_ ` s ) ) = s ) } )
27	1, 26	syl 17	1 |- ( K e. B -> C = { s | ( s C_ A /\ ( ._|_ ` ( ._|_ ` s ) ) = s ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   ` cfv 5888   Atomscatm 34550   _|_PcpolN 35188   PSubClcpscN 35220

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-pow 4843  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-pw 4160  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-psubclN 35221

This theorem is referenced by:  ispsubclN  35223