Theorem thlval 20039

Description: Value of the Hilbert lattice. (Contributed by Mario Carneiro, 25-Oct-2015.)

Hypotheses

Ref	Expression
thlval.k	|- K = (toHL ` W )
thlval.c	|- C = ( CSubSp ` W )
thlval.i	|- I = (toInc ` C )
thlval.o	|- ._|_ = ( ocv ` W )

Assertion

Ref	Expression
thlval	|- ( W e. V -> K = ( I sSet <. ( oc ` ndx ) , ._|_ >. ) )

Proof of Theorem thlval

Dummy variable h is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( W e. V -> W e. _V )
2		thlval.k	. . 3 |- K = (toHL ` W )
3		fveq2 6191	. . . . . . . 8 |- ( h = W -> ( CSubSp ` h ) = ( CSubSp ` W ) )
4		thlval.c	. . . . . . . 8 |- C = ( CSubSp ` W )
5	3, 4	syl6eqr 2674	. . . . . . 7 |- ( h = W -> ( CSubSp ` h ) = C )
6	5	fveq2d 6195	. . . . . 6 |- ( h = W -> (toInc ` ( CSubSp ` h ) ) = (toInc ` C ) )
7		thlval.i	. . . . . 6 |- I = (toInc ` C )
8	6, 7	syl6eqr 2674	. . . . 5 |- ( h = W -> (toInc ` ( CSubSp ` h ) ) = I )
9		fveq2 6191	. . . . . . 7 |- ( h = W -> ( ocv ` h ) = ( ocv ` W ) )
10		thlval.o	. . . . . . 7 |- ._|_ = ( ocv ` W )
11	9, 10	syl6eqr 2674	. . . . . 6 |- ( h = W -> ( ocv ` h ) = ._|_ )
12	11	opeq2d 4409	. . . . 5 |- ( h = W -> <. ( oc ` ndx ) , ( ocv ` h ) >. = <. ( oc ` ndx ) , ._|_ >. )
13	8, 12	oveq12d 6668	. . . 4 |- ( h = W -> ( (toInc ` ( CSubSp ` h ) ) sSet <. ( oc ` ndx ) , ( ocv ` h ) >. ) = ( I sSet <. ( oc ` ndx ) , ._|_ >. ) )
14		df-thl 20009	. . . 4 |- toHL = ( h e. _V |-> ( (toInc ` ( CSubSp ` h ) ) sSet <. ( oc ` ndx ) , ( ocv ` h ) >. ) )
15		ovex 6678	. . . 4 |- ( I sSet <. ( oc ` ndx ) , ._|_ >. ) e. _V
16	13, 14, 15	fvmpt 6282	. . 3 |- ( W e. _V -> (toHL ` W ) = ( I sSet <. ( oc ` ndx ) , ._|_ >. ) )
17	2, 16	syl5eq 2668	. 2 |- ( W e. _V -> K = ( I sSet <. ( oc ` ndx ) , ._|_ >. ) )
18	1, 17	syl 17	1 |- ( W e. V -> K = ( I sSet <. ( oc ` ndx ) , ._|_ >. ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   ` cfv 5888  (class class class)co 6650   ndxcnx 15854   sSet csts 15855   occoc 15949  toInccipo 17151   ocvcocv 20004   CSubSpccss 20005  toHLcthl 20006

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-ov 6653  df-thl 20009

This theorem is referenced by:  thlbas  20040  thlle  20041  thloc  20043