Theorem psubclsetN 33546

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 3066	. 2 |- ( K e. B -> K e. _V )
2		psubclset.c	. . 3 |- C = ( PSubCl ` K )
3		fveq2 5888	. . . . . . . 8 |- ( k = K -> ( Atoms ` k ) = ( Atoms ` K ) )
4		psubclset.a	. . . . . . . 8 |- A = ( Atoms ` K )
5	3, 4	syl6eqr 2514	. . . . . . 7 |- ( k = K -> ( Atoms ` k ) = A )
6	5	sseq2d 3472	. . . . . 6 |- ( k = K -> ( s C_ ( Atoms ` k ) <-> s C_ A ) )
7		fveq2 5888	. . . . . . . . 9 |- ( k = K -> ( _|_P ` k ) = ( _|_P ` K ) )
8		psubclset.p	. . . . . . . . 9 |- ._|_ = ( _|_P ` K )
9	7, 8	syl6eqr 2514	. . . . . . . 8 |- ( k = K -> ( _|_P ` k ) = ._|_ )
10	9	fveq1d 5890	. . . . . . . 8 |- ( k = K -> ( ( _|_P ` k ) ` s ) = ( ._|_ ` s ) )
11	9, 10	fveq12d 5894	. . . . . . 7 |- ( k = K -> ( ( _|_P ` k ) ` ( ( _|_P ` k ) ` s ) ) = ( ._|_ ` ( ._|_ ` s ) ) )
12	11	eqeq1d 2464	. . . . . 6 |- ( k = K -> ( ( ( _|_P ` k ) ` ( ( _|_P ` k ) ` s ) ) = s <-> ( ._|_ ` ( ._|_ ` s ) ) = s ) )
13	6, 12	anbi12d 722	. . . . 5 |- ( k = K -> ( ( s C_ ( Atoms ` k ) /\ ( ( _|_P ` k ) ` ( ( _|_P ` k ) ` s ) ) = s ) <-> ( s C_ A /\ ( ._|_ ` ( ._|_ ` s ) ) = s ) ) )
14	13	abbidv 2580	. . . 4 |- ( k = K -> { s | ( s C_ ( Atoms ` k ) /\ ( ( _|_P ` k ) ` ( ( _|_P ` k ) ` s ) ) = s ) } = { s | ( s C_ A /\ ( ._|_ ` ( ._|_ ` s ) ) = s ) } )
15		df-psubclN 33545	. . . 4 |- PSubCl = ( k e. _V |-> { s | ( s C_ ( Atoms ` k ) /\ ( ( _|_P ` k ) ` ( ( _|_P ` k ) ` s ) ) = s ) } )
16		fvex 5898	. . . . . . 7 |- ( Atoms ` K ) e. _V
17	4, 16	eqeltri 2536	. . . . . 6 |- A e. _V
18	17	pwex 4600	. . . . 5 |- ~P A e. _V
19		selpw 3970	. . . . . . . 8 |- ( s e. ~P A <-> s C_ A )
20	19	anbi1i 706	. . . . . . 7 |- ( ( s e. ~P A /\ ( ._|_ ` ( ._|_ ` s ) ) = s ) <-> ( s C_ A /\ ( ._|_ ` ( ._|_ ` s ) ) = s ) )
21	20	abbii 2578	. . . . . 6 |- { s | ( s e. ~P A /\ ( ._|_ ` ( ._|_ ` s ) ) = s ) } = { s | ( s C_ A /\ ( ._|_ ` ( ._|_ ` s ) ) = s ) }
22		ssab2 3525	. . . . . 6 |- { s | ( s e. ~P A /\ ( ._|_ ` ( ._|_ ` s ) ) = s ) } C_ ~P A
23	21, 22	eqsstr3i 3475	. . . . 5 |- { s | ( s C_ A /\ ( ._|_ ` ( ._|_ ` s ) ) = s ) } C_ ~P A
24	18, 23	ssexi 4562	. . . 4 |- { s | ( s C_ A /\ ( ._|_ ` ( ._|_ ` s ) ) = s ) } e. _V
25	14, 15, 24	fvmpt 5971	. . 3 |- ( K e. _V -> ( PSubCl ` K ) = { s | ( s C_ A /\ ( ._|_ ` ( ._|_ ` s ) ) = s ) } )
26	2, 25	syl5eq 2508	. 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 375   = wceq 1455   e. wcel 1898   {cab 2448   _Vcvv 3057   C_ wss 3416   ~Pcpw 3963   ` cfv 5601   Atomscatm 32874   _|_PcpolN 33512   PSubClcpscN 33544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-psubclN 33545

This theorem is referenced by:  ispsubclN  33547