Theorem lineset 35024

Description: The set of lines in a Hilbert lattice. (Contributed by NM, 19-Sep-2011.)

Hypotheses

Ref	Expression
lineset.l	|- .<_ = ( le ` K )
lineset.j	|- .\/ = ( join ` K )
lineset.a	|- A = ( Atoms ` K )
lineset.n	|- N = ( Lines ` K )

Assertion

Ref	Expression
lineset	|- ( K e. B -> N = { s | E. q e. A E. r e. A ( q =/= r /\ s = { p e. A | p .<_ ( q .\/ r ) } ) } )

Distinct variable groups:   q, p, r, s, A   K, p, q, r, s   .\/ , s   .<_ , s

Allowed substitution hints:   B( s, r, q, p)   .\/ ( r, q, p)   .<_ ( r, q, p)   N( s, r, q, p)

Proof of Theorem lineset

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( K e. B -> K e. _V )
2		lineset.n	. . 3 |- N = ( Lines ` K )
3		fveq2 6191	. . . . . . 7 |- ( k = K -> ( Atoms ` k ) = ( Atoms ` K ) )
4		lineset.a	. . . . . . 7 |- A = ( Atoms ` K )
5	3, 4	syl6eqr 2674	. . . . . 6 |- ( k = K -> ( Atoms ` k ) = A )
6		fveq2 6191	. . . . . . . . . . . . 13 |- ( k = K -> ( le ` k ) = ( le ` K ) )
7		lineset.l	. . . . . . . . . . . . 13 |- .<_ = ( le ` K )
8	6, 7	syl6eqr 2674	. . . . . . . . . . . 12 |- ( k = K -> ( le ` k ) = .<_ )
9	8	breqd 4664	. . . . . . . . . . 11 |- ( k = K -> ( p ( le ` k ) ( q ( join ` k ) r ) <-> p .<_ ( q ( join ` k ) r ) ) )
10		fveq2 6191	. . . . . . . . . . . . . 14 |- ( k = K -> ( join ` k ) = ( join ` K ) )
11		lineset.j	. . . . . . . . . . . . . 14 |- .\/ = ( join ` K )
12	10, 11	syl6eqr 2674	. . . . . . . . . . . . 13 |- ( k = K -> ( join ` k ) = .\/ )
13	12	oveqd 6667	. . . . . . . . . . . 12 |- ( k = K -> ( q ( join ` k ) r ) = ( q .\/ r ) )
14	13	breq2d 4665	. . . . . . . . . . 11 |- ( k = K -> ( p .<_ ( q ( join ` k ) r ) <-> p .<_ ( q .\/ r ) ) )
15	9, 14	bitrd 268	. . . . . . . . . 10 |- ( k = K -> ( p ( le ` k ) ( q ( join ` k ) r ) <-> p .<_ ( q .\/ r ) ) )
16	5, 15	rabeqbidv 3195	. . . . . . . . 9 |- ( k = K -> { p e. ( Atoms ` k ) | p ( le ` k ) ( q ( join ` k ) r ) } = { p e. A | p .<_ ( q .\/ r ) } )
17	16	eqeq2d 2632	. . . . . . . 8 |- ( k = K -> ( s = { p e. ( Atoms ` k ) | p ( le ` k ) ( q ( join ` k ) r ) } <-> s = { p e. A | p .<_ ( q .\/ r ) } ) )
18	17	anbi2d 740	. . . . . . 7 |- ( k = K -> ( ( q =/= r /\ s = { p e. ( Atoms ` k ) | p ( le ` k ) ( q ( join ` k ) r ) } ) <-> ( q =/= r /\ s = { p e. A | p .<_ ( q .\/ r ) } ) ) )
19	5, 18	rexeqbidv 3153	. . . . . 6 |- ( k = K -> ( E. r e. ( Atoms ` k ) ( q =/= r /\ s = { p e. ( Atoms ` k ) | p ( le ` k ) ( q ( join ` k ) r ) } ) <-> E. r e. A ( q =/= r /\ s = { p e. A | p .<_ ( q .\/ r ) } ) ) )
20	5, 19	rexeqbidv 3153	. . . . 5 |- ( k = K -> ( E. q e. ( Atoms ` k ) E. r e. ( Atoms ` k ) ( q =/= r /\ s = { p e. ( Atoms ` k ) | p ( le ` k ) ( q ( join ` k ) r ) } ) <-> E. q e. A E. r e. A ( q =/= r /\ s = { p e. A | p .<_ ( q .\/ r ) } ) ) )
21	20	abbidv 2741	. . . 4 |- ( k = K -> { s | E. q e. ( Atoms ` k ) E. r e. ( Atoms ` k ) ( q =/= r /\ s = { p e. ( Atoms ` k ) | p ( le ` k ) ( q ( join ` k ) r ) } ) } = { s | E. q e. A E. r e. A ( q =/= r /\ s = { p e. A | p .<_ ( q .\/ r ) } ) } )
22		df-lines 34787	. . . 4 |- Lines = ( k e. _V |-> { s | E. q e. ( Atoms ` k ) E. r e. ( Atoms ` k ) ( q =/= r /\ s = { p e. ( Atoms ` k ) | p ( le ` k ) ( q ( join ` k ) r ) } ) } )
23		fvex 6201	. . . . . 6 |- ( Atoms ` K ) e. _V
24	4, 23	eqeltri 2697	. . . . 5 |- A e. _V
25		df-sn 4178	. . . . . . 7 |- { { p e. A | p .<_ ( q .\/ r ) } } = { s | s = { p e. A | p .<_ ( q .\/ r ) } }
26		snex 4908	. . . . . . 7 |- { { p e. A | p .<_ ( q .\/ r ) } } e. _V
27	25, 26	eqeltrri 2698	. . . . . 6 |- { s | s = { p e. A | p .<_ ( q .\/ r ) } } e. _V
28		simpr 477	. . . . . . 7 |- ( ( q =/= r /\ s = { p e. A | p .<_ ( q .\/ r ) } ) -> s = { p e. A | p .<_ ( q .\/ r ) } )
29	28	ss2abi 3674	. . . . . 6 |- { s | ( q =/= r /\ s = { p e. A | p .<_ ( q .\/ r ) } ) } C_ { s | s = { p e. A | p .<_ ( q .\/ r ) } }
30	27, 29	ssexi 4803	. . . . 5 |- { s | ( q =/= r /\ s = { p e. A | p .<_ ( q .\/ r ) } ) } e. _V
31	24, 24, 30	ab2rexex2 7160	. . . 4 |- { s | E. q e. A E. r e. A ( q =/= r /\ s = { p e. A | p .<_ ( q .\/ r ) } ) } e. _V
32	21, 22, 31	fvmpt 6282	. . 3 |- ( K e. _V -> ( Lines ` K ) = { s | E. q e. A E. r e. A ( q =/= r /\ s = { p e. A | p .<_ ( q .\/ r ) } ) } )
33	2, 32	syl5eq 2668	. 2 |- ( K e. _V -> N = { s | E. q e. A E. r e. A ( q =/= r /\ s = { p e. A | p .<_ ( q .\/ r ) } ) } )
34	1, 33	syl 17	1 |- ( K e. B -> N = { s | E. q e. A E. r e. A ( q =/= r /\ s = { p e. A | p .<_ ( q .\/ r ) } ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   E.wrex 2913   {crab 2916   _Vcvv 3200   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   lecple 15948   joincjn 16944   Atomscatm 34550   Linesclines 34780

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-lines 34787

This theorem is referenced by:  isline  35025